time problem solving with solution

Problems on Calculating Time

Here we will learn to solve different types of problems on calculating time.

We know, the formula to find out time = distance/speed

Word problems on calculating time:

1. A car travel 60 km in 30 minutes. In how much time will it cover 100 km?

Solution:

Using the unitary method;

Time taken to cover 60 km = 90 minutes

Time taken to cover 1 km = 90/60 minutes

Time taken to cover 100 km = 90/60 × 100 = 150 minutes

Formula of speed = distance/time

= 60 km/(3/2) hr [given 1 hour 30 min = 1 30/60 = 1 ½ hours

= 3/2 hours]

= 60/1 × 2/3 km/hr = 40 km/hr

Now, using the formula of time = distance/speed = 100 km/40 km/hr = 5/2 hours

= 5/2 × 60 minutes, (Since 1 hour = 60 minutes)

= 150 minutes

2. Victor covers 210 km by car at a speed of 70 km/hr. find the time taken to cover this distance.

Solution:

70 km is covered in 1 hour.

1 km is covered in 1/70 hours.

210 km is covered in 1/70 × 210 hours = 3 hours

Given: speed = 70 km/hr, distance covered = 210 km

Time taken = Distance/ Speed = 210/70 hours = 3 hours

3. A train covers a distance of 36 km in 15 minutes. Find the time taken by it to cover the same distance if its speed is decreased by 9 km/hr.

Distance covered by train = 36 km

Time taken = 15 minutes = 15/60 hr = ¼ hr.

Therefore speed of train = Distance covered/time taken

= 36/(1/4) km/hr

= 36/1 × 4/1

= 144 km/hr

Reduced speed = 144 – 9 = 135

Therefore, required time = distance covered/speed

= 36/135 × 60 minutes

= 16 minutes

4. A man is walking at a speed of 6 km per hour. After every km, he takes rest for 2 minutes. How much time will it take to cover a distance of 4 km?

Rest time = Number of rests × time of each rest

= 3 × 2 minutes

= 6 minutes

Total time to cover 4 km = 4/6 × 60 + 6 minutes

= (40 + 6) minutes

Speed of Train

Relationship between Speed, Distance and Time

Conversion of Units of Speed

Problems on Calculating Speed

Problems on Calculating Distance

Two Objects Move in Same Direction

Two Objects Move in Opposite Direction

Train Passes a Moving Object in the Same Direction

Train Passes a Moving Object in the Opposite Direction

Train Passes through a Pole

Train Passes through a Bridge

Two Trains Passes in the Same Direction

Two Trains Passes in the Opposite Direction

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20 Elapsed Time Word Problems

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Elapsed time is the amount of time that passes between the beginning and the end of an event. The concept of elapsed time fits nicely in the elementary school curriculum. Beginning in third grade, students should be able to tell and write time to the nearest minute and solve word problems involving addition and subtraction of time. Reinforce these essential skills with the following elapsed time word problems and games.

Elapsed Time Word Problems

These quick and easy elapsed time word problems are perfect for parents and teachers who want to help students practice elapsed time to the nearest minute with simple mental math problems. Answers are listed below.

Sam and his mom arrive at the doctor’s office at 2:30 p.m. They see the doctor at 3:10 p.m. How long was their wait?
Dad says dinner will be ready in 35 minutes. It’s 5:30 p.m. now. What time will dinner be ready?
Becky is meeting her friend at the library at 12:45 p.m. It takes her 25 minutes to get to the library. What time will she need to leave her house to arrive on time?
Ethan’s birthday party started at 4:30 p.m. The last guest left at 6:32 p.m. How long did Ethan’s party last?
Kayla put cupcakes in the oven at 3:41 p.m. The directions say that the cupcakes need to bake for 38 minutes. What time will Kayla need to take them out of the oven?
Dakota arrived at school at 7:59 a.m. He left at 2:33 p.m. How long was Dakota at school?
Dylan started working on homework at 5:45 p.m. It took him 1 hour and 57 minutes to complete it. What time did Dylan complete his homework?
Dad arrives home at 4:50 p.m. He left work 40 minutes ago. What time did Dad get off work?
Jessica’s family is traveling from Atlanta, Georgia to New York by plane. Their flight leaves at 11:15 a.m. and should take 2 hours and 15 minutes. What time will their plane arrive in New York?
Jordan got to football practice at 7:05 p.m. Steve showed up 11 minutes later. What time did Steve get to practice?
Jack ran a marathon in 2 hours and 17 minutes. He crossed the finish line at 10:33 a.m. What time did the race start?
Marci was babysitting for her cousin. Her cousin was gone for 3 hours and 40 minutes. Marci left at 9:57 p.m. What time did she start babysitting?
Caleb and his friends went to see a movie at 7:35 p.m. They left at 10:05 p.m. How long was the movie?
Francine got to work at 8:10 a.m. She left at 3:45 p.m. How long did Francine work?
Brandon went to bed at 9:15 p.m. It took him 23 minutes to fall asleep. What time did Brandon fall asleep?
Kelli had to wait in a long, slow-moving line to purchase a popular new video game that was just released. She got in line at 9:15 a.m. She left with the game at 11:07 a.m. How long did Kelli wait in line?
Jaydon went to batting practice Saturday morning at 8:30 a.m. He left at 11:42 a.m. How long was he at batting practice?
Ashton got behind on her reading assignment, so she had to read four chapters last night. She started at 8:05 p.m. and finished at 9:15 p.m. How long did it take Ashton to catch up on her assignment?
Natasha has a dentist appointment at 10:40 a.m. It should last 35 minutes. What time will she finish?
Mrs. Kennedy’s 3rd-grade class is going to the aquarium on a field trip. They are scheduled to arrive at 9:10 a.m. and leave at 1:40 p.m. How long will they spend at the aquarium?

Elapsed Time Games

Try these games and activities at home to help your children practice elapsed time.

Daily Schedule

Let your children keep track of their schedule and ask them to figure the elapsed time for each activity. For example, how long did your child spend eating breakfast, reading, taking a bath, or playing video games?

How Long Will It Take?

Give your kids practice with elapsed time by encouraging them to figure out how long daily activities take. For example, the next time you order a pizza online or by phone, you'll probably be given an estimated delivery time. Use that information to create a word problem that's relevant to your child's life, such as, "It's 5:40 p.m. now and the pizza shop says the pizza will be here at 6:20 p.m. How long will it take for the pizza to arrive?"

Order a set of time dice from online retailers or teacher supply stores. The set contains two twelve-sided dice, one with numbers representing the hours and the other with numbers representing minutes. Take turns rolling the time dice with your child. Each player should roll twice, then calculate the elapsed time between the two resulting dice times. (A pencil and paper will come in handy, as you'll want to jot down the time of the first roll.)

Elapsed Time Word Problem Answers

2 hours and 2 minutes
6 hours and 34 minutes
2 hours and 30 minutes
7 hours and 35 minutes
1 hour and 52 minutes
3 hours and 12 minutes
1 hour and 10 minutes
4 hours and 30 minutes
10 Math Tricks That Will Blow Your Mind
How To Tell Time in Spanish
Sinking of the RMS Titanic
A Timeline of the Sinking of the Titanic
How to Tell Time in Italian
When Is the Spring Equinox?
Free Math Word Problem Worksheets for Fifth-Graders
What Time Is the SAT Test?
Proportions Word Problems Worksheet: Answers and Explanations
World War II: Operation Ten-Go
Telling Time in French
Building an Effective Plan of Improvement for Teachers
Learn About the Munich Olympic Massacre
Fundamental Lessons for Telling the Time
Practice Multiplication Skills With Times Tables Worksheets
What Is Uptime in Web Hosting?

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Time Word Problems Worksheets

Time riddles (harder).

Welcome to our Time Word Problems Worksheets page. Here you find our selection of more challenging Time Riddles to help your child learn to read, record and solve problems involving time and clocks.

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Looking for some 24 hour clock problem sheets? Do you need some challenging time problems for older children? Then hopefully you have found the right place!

The printable time sheets in this section involve being able to tell the time to the nearest minute, as well as converting times between the 12 and 24 clock.

These worksheets are great to use when your child is confident telling the time and needs to extend their knowledge by solving time problems.

The sheets are also very good at developing an understanding of mathematical language associated with time.

Each sheet has 2 different Time Riddles with 8 possible solutions.

The aim of each puzzle is to use the clues to work out the correct solution.

Using these sheets will help your child to:

read times to the nearest minute;
convert analogue to digital times;
use 'past' and 'to' language correctly to tell the time;
convert times between the 12 and 24 hour clock;
solve problems involving time;

Our time word problems worksheets will help you practice applying your time skills and knowledge to solve problems.

Time Word Problems Worksheets : to the nearest minute

Time Riddles 4a
PDF version
Time Riddles 4b
Time Riddles 4c

Time Word Problems Worksheets: 24 hour clock

The printable time worksheets in this section involve converting times between the 12 and 24 hour clock.

24 Hour Time Riddles Sheet 1
24 Hour Time Riddles Sheet 2
24 Hour Time Riddles Sheet 3

Extension Activity Ideas

If you are looking for a way to extend learning with these Time Riddles, why not...

Get children to work in partners with one child choosing one of the 8 possible times and the other child asking 'yes/no' questions. E.g. Is it earlier than...? Is the minute hand on number 6...? Is it later than 4pm? ... etc
Children could write their own set of clues down to identify one of the clocks.

Looking for something easier?

Here you will find our selection of easier Time Riddles.

The sheets in this section are similar to those on this page, but involve telling the times: o'clock and half-past, quarter past and quarter to, and to 5 minute intervals.

Time Riddles (easier)

Need help telling the time?

Here you will find our selection of telling time clock worksheets to help your child to learn their o'clock, half-past, quarter past and to, and 5 minute intervals.

read o'clock, half-past times;
read quarter past and quarter to;
read time going up in 5 minute intervals;
convert analogue times to digital;

The sheets in this section are a great way of starting your child off with learning to tell the time.

Telling Time Worksheets o'clock and half past
Clock Worksheets - Quarter Past and Quarter To
Telling Time 5 minute intervals
Telling Time Worksheets Grade 4 (1 minute intervals)

Time Interval Worksheets

These sheets will help you learn to add and subtract hours and minutes from times as well as working out a range of time intervals.

There are also sheets to help you practice adding and subtracting time intervals.

Add and Subtract Time Worksheets
Elapsed Time Worksheets
Online Age Calculator

Do you want to know exactly how old you are to the nearest minute?

Have you been alive for more than a billion seconds?

Do you know how many days old you are?

Our online age calculator will tell you all you need to know...

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Time Interval Word Problems

Concept Development Look back at your work on Application Problem. We know that Lilly finished after Patrick. Let’s use a number line to figure out how many more minutes than Patrick Lilly took to finish.

Label the first tick mark 0 and the last tick mark 60. Label the hours and 5-minute intervals. T: Plot the times 5:31 p.m. and 5:43 p.m. Find the difference between Patrick and Lilly’s times.

How many more minutes than Patrick did it take Lilly to finish her chores? 12 minutes more.

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Course: 4th grade > Unit 14

Converting units of time
Convert to smaller units (sec, min, & hr)
Time word problem: travel time

Time word problem: Susan's break

Time conversion word problems
Converting units of time review (seconds, minutes, & hours)

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Video transcript

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Chapter 9: Radicals

9.10 Rate Word Problems: Work and Time

If it takes Felicia 4 hours to paint a room and her daughter Katy 12 hours to paint the same room, then working together, they could paint the room in 3 hours. The equation used to solve problems of this type is one of reciprocals. It is derived as follows:

[latex]\text{rate}\times \text{time}=\text{work done}[/latex]

For this problem:

[latex]\begin{array}{rrrl} \text{Felicia's rate: }&F_{\text{rate}}\times 4 \text{ h}&=&1\text{ room} \\ \\ \text{Katy's rate: }&K_{\text{rate}}\times 12 \text{ h}&=&1\text{ room} \\ \\ \text{Isolating for their rates: }&F&=&\dfrac{1}{4}\text{ h and }K = \dfrac{1}{12}\text{ h} \end{array}[/latex]

To make this into a solvable equation, find the total time [latex](T)[/latex] needed for Felicia and Katy to paint the room. This time is the sum of the rates of Felicia and Katy, or:

[latex]\begin{array}{rcrl} \text{Total time: } &T \left(\dfrac{1}{4}\text{ h}+\dfrac{1}{12}\text{ h}\right)&=&1\text{ room} \\ \\ \text{This can also be written as: }&\dfrac{1}{4}\text{ h}+\dfrac{1}{12}\text{ h}&=&\dfrac{1 \text{ room}}{T} \\ \\ \text{Solving this yields:}&0.25+0.083&=&\dfrac{1 \text{ room}}{T} \\ \\ &0.333&=&\dfrac{1 \text{ room}}{T} \\ \\ &t&=&\dfrac{1}{0.333}\text{ or }\dfrac{3\text{ h}}{\text{room}} \end{array}[/latex]

Example 9.10.1

Karl can clean a room in 3 hours. If his little sister Kyra helps, they can clean it in 2.4 hours. How long would it take Kyra to do the job alone?

The equation to solve is:

[latex]\begin{array}{rrrrl} \dfrac{1}{3}\text{ h}&+&\dfrac{1}{K}&=&\dfrac{1}{2.4}\text{ h} \\ \\ &&\dfrac{1}{K}&=&\dfrac{1}{2.4}\text{ h}-\dfrac{1}{3}\text{ h}\\ \\ &&\dfrac{1}{K}&=&0.0833\text{ or }K=12\text{ h} \end{array}[/latex]

Example 9.10.2

Doug takes twice as long as Becky to complete a project. Together they can complete the project in 10 hours. How long will it take each of them to complete the project alone?

[latex]\begin{array}{rrl} \dfrac{1}{R}+\dfrac{1}{2R}&=&\dfrac{1}{10}\text{ h,} \\ \text{where Doug's rate (} \dfrac{1}{D}\text{)}& =& \dfrac{1}{2}\times \text{ Becky's (}\dfrac{1}{R}\text{) rate.} \\ \\ \text{Sum the rates: }\dfrac{1}{R}+\dfrac{1}{2R}&=&\dfrac{2}{2R} + \dfrac{1}{2R} = \dfrac{3}{2R} \\ \\ \text{Solve for R: }\dfrac{3}{2R}&=&\dfrac{1}{10}\text{ h} \\ \text{which means }\dfrac{1}{R}&=&\dfrac{1}{10}\times\dfrac{2}{3}\text{ h} \\ \text{so }\dfrac{1}{R}& =& \dfrac{2}{30} \\ \text{ or }R &= &\dfrac{30}{2} \end{array}[/latex]

This means that the time it takes Becky to complete the project alone is [latex]15\text{ h}[/latex].

Since it takes Doug twice as long as Becky, the time for Doug is [latex]30\text{ h}[/latex].

Example 9.10.3

Joey can build a large shed in 10 days less than Cosmo can. If they built it together, it would take them 12 days. How long would it take each of them working alone?

[latex]\begin{array}{rl} \text{The equation to solve:}& \dfrac{1}{(C-10)}+\dfrac{1}{C}=\dfrac{1}{12}, \text{ where }J=C-10 \\ \\ \text{Multiply each term by the LCD:}&(C-10)(C)(12) \\ \\ \text{This leaves}&12C+12(C-10)=C(C-10) \\ \\ \text{Multiplying this out:}&12C+12C-120=C^2-10C \\ \\ \text{Which simplifies to}&C^2-34C+120=0 \\ \\ \text{Which will factor to}& (C-30)(C-4) = 0 \end{array}[/latex]

Cosmo can build the large shed in either 30 days or 4 days. Joey, therefore, can build the shed in 20 days or −6 days (rejected).

The solution is Cosmo takes 30 days to build and Joey takes 20 days.

Example 9.10.4

Clark can complete a job in one hour less than his apprentice. Together, they do the job in 1 hour and 12 minutes. How long would it take each of them working alone?

[latex]\begin{array}{rl} \text{Convert everything to hours:} & 1\text{ h }12\text{ min}=\dfrac{72}{60} \text{ h}=\dfrac{6}{5}\text{ h}\\ \\ \text{The equation to solve is} & \dfrac{1}{A}+\dfrac{1}{A-1}=\dfrac{1}{\dfrac{6}{5}}=\dfrac{5}{6}\\ \\ \text{Therefore the equation is} & \dfrac{1}{A}+\dfrac{1}{A-1}=\dfrac{5}{6} \\ \\ \begin{array}{r} \text{To remove the fractions, } \\ \text{multiply each term by the LCD} \end{array} & (A)(A-1)(6)\\ \\ \text{This leaves} & 6(A)+6(A-1)=5(A)(A-1) \\ \\ \text{Multiplying this out gives} & 6A-6+6A=5A^2-5A \\ \\ \text{Which simplifies to} & 5A^2-17A +6=0 \\ \\ \text{This will factor to} & (5A-2)(A-3)=0 \end{array}[/latex]

The apprentice can do the job in either [latex]\dfrac{2}{5}[/latex] h (reject) or 3 h. Clark takes 2 h.

Example 9.10.5

A sink can be filled by a pipe in 5 minutes, but it takes 7 minutes to drain a full sink. If both the pipe and the drain are open, how long will it take to fill the sink?

The 7 minutes to drain will be subtracted.

[latex]\begin{array}{rl} \text{The equation to solve is} & \dfrac{1}{5}-\dfrac{1}{7}=\dfrac{1}{X} \\ \\ \begin{array}{r} \text{To remove the fractions,} \\ \text{multiply each term by the LCD}\end{array} & (5)(7)(X)\\ \\ \text{This leaves } & (7)(X)-(5)(X)=(5)(7)\\ \\ \text{Multiplying this out gives} & 7X-5X=35\\ \\ \text{Which simplifies to} & 2X=35\text{ or }X=\dfrac{35}{2}\text{ or }17.5 \end{array}[/latex]

17.5 min or 17 min 30 sec is the solution

For Questions 1 to 8, write the formula defining the relation. Do Not Solve!!

Bill’s father can paint a room in 2 hours less than it would take Bill to paint it. Working together, they can complete the job in 2 hours and 24 minutes. How much time would each require working alone?
Of two inlet pipes, the smaller pipe takes four hours longer than the larger pipe to fill a pool. When both pipes are open, the pool is filled in three hours and forty-five minutes. If only the larger pipe is open, how many hours are required to fill the pool?
Jack can wash and wax the family car in one hour less than it would take Bob. The two working together can complete the job in 1.2 hours. How much time would each require if they worked alone?
If Yousef can do a piece of work alone in 6 days, and Bridgit can do it alone in 4 days, how long will it take the two to complete the job working together?
Working alone, it takes John 8 hours longer than Carlos to do a job. Working together, they can do the job in 3 hours. How long would it take each to do the job working alone?
Working alone, Maryam can do a piece of work in 3 days that Noor can do in 4 days and Elana can do in 5 days. How long will it take them to do it working together?
Raj can do a piece of work in 4 days and Rubi can do it in half the time. How long would it take them to do the work together?
A cistern can be filled by one pipe in 20 minutes and by another in 30 minutes. How long would it take both pipes together to fill the tank?

For Questions 9 to 20, find and solve the equation describing the relationship.

If an apprentice can do a piece of work in 24 days, and apprentice and instructor together can do it in 6 days, how long would it take the instructor to do the work alone?
A carpenter and his assistant can do a piece of work in 3.75 days. If the carpenter himself could do the work alone in 5 days, how long would the assistant take to do the work alone?
If Sam can do a certain job in 3 days, while it would take Fred 6 days to do the same job, how long would it take them, working together, to complete the job?
Tim can finish a certain job in 10 hours. It takes his wife JoAnn only 8 hours to do the same job. If they work together, how long will it take them to complete the job?
Two people working together can complete a job in 6 hours. If one of them works twice as fast as the other, how long would it take the slower person, working alone, to do the job?
If two people working together can do a job in 3 hours, how long would it take the faster person to do the same job if one of them is 3 times as fast as the other?
A water tank can be filled by an inlet pipe in 8 hours. It takes twice that long for the outlet pipe to empty the tank. How long would it take to fill the tank if both pipes were open?
A sink can be filled from the faucet in 5 minutes. It takes only 3 minutes to empty the sink when the drain is open. If the sink is full and both the faucet and the drain are open, how long will it take to empty the sink?
It takes 10 hours to fill a pool with the inlet pipe. It can be emptied in 15 hours with the outlet pipe. If the pool is half full to begin with, how long will it take to fill it from there if both pipes are open?
A sink is ¼ full when both the faucet and the drain are opened. The faucet alone can fill the sink in 6 minutes, while it takes 8 minutes to empty it with the drain. How long will it take to fill the remaining ¾ of the sink?
A sink has two faucets: one for hot water and one for cold water. The sink can be filled by a cold-water faucet in 3.5 minutes. If both faucets are open, the sink is filled in 2.1 minutes. How long does it take to fill the sink with just the hot-water faucet open?
A water tank is being filled by two inlet pipes. Pipe A can fill the tank in 4.5 hours, while both pipes together can fill the tank in 2 hours. How long does it take to fill the tank using only pipe B?

Answer Key 9.10

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Distance, Time and Speed Word Problems | GMAT GRE Maths

Before you get into distance, time and speed word problems, check out how you can add a little power to your resume by getting our mini-MBA certificate.

Problems involving Time, Distance and Speed are solved based on one simple formula.

Distance = Speed * Time

Which implies →

Speed = Distance / Time and

Time = Distance / Speed

Let us take a look at some simple examples of distance, time and speed problems. Example 1. A boy walks at a speed of 4 kmph. How much time does he take to walk a distance of 20 km?

Time = Distance / speed = 20/4 = 5 hours. Example 2. A cyclist covers a distance of 15 miles in 2 hours. Calculate his speed.

Speed = Distance/time = 15/2 = 7.5 miles per hour. Example 3. A car takes 4 hours to cover a distance, if it travels at a speed of 40 mph. What should be its speed to cover the same distance in 1.5 hours?

Distance covered = 4*40 = 160 miles

Speed required to cover the same distance in 1.5 hours = 160/1.5 = 106.66 mph Now, take a look at the following example:

Example 4. If a person walks at 4 mph, he covers a certain distance. If he walks at 9 mph, he covers 7.5 miles more. How much distance did he actually cover?

Now we can see that the direct application of our usual formula Distance = Speed * Time or its variations cannot be done in this case and we need to put in extra effort to calculate the given parameters.

Let us see how this question can be solved.

For these kinds of questions, a table like this might make it easier to solve.

Distance	Speed	Time
d	4	t
d+7.5	9	t

Let the distance covered by that person be ‘d’.

Walking at 4 mph and covering a distance ‘d’ is done in a time of ‘d/4’

IF he walks at 9 mph, he covers 7.5 miles more than the actual distance d, which is ‘d+7.5’.

He does this in a time of (d+7.5)/9.

Since the time is same in both the cases →

d/4 = (d+7.5)/9 → 9d = 4(d+7.5) → 9d=4d+30 → d = 6.

So, he covered a distance of 6 miles in 1.5 hours. Example 5. A train is going at 1/3 of its usual speed and it takes an extra 30 minutes to reach its destination. Find its usual time to cover the same distance.

Here, we see that the distance is same.

Let us assume that its usual speed is ‘s’ and time is ‘t’, then

Distance	Speed	Time
d	s	t min
d	S+1/3	t+30 min

s*t = (1/3)s*(t+30) → t = t/3 + 10 → t = 15.

So the actual time taken to cover the distance is 15 minutes.

Note: Note the time is expressed in terms of ‘minutes’. When we express distance in terms of miles or kilometers, time is expressed in terms of hours and has to be converted into appropriate units of measurement.

Solved Questions on Trains

Example 1. X and Y are two stations which are 320 miles apart. A train starts at a certain time from X and travels towards Y at 70 mph. After 2 hours, another train starts from Y and travels towards X at 20 mph. At what time do they meet?

Let the time after which they meet be ‘t’ hours.

Then the time travelled by second train becomes ‘t-2’.

Distance covered by first train+Distance covered by second train = 320 miles

70t+20(t-2) = 320

Solving this gives t = 4.

So the two trains meet after 4 hours. Example 2. A train leaves from a station and moves at a certain speed. After 2 hours, another train leaves from the same station and moves in the same direction at a speed of 60 mph. If it catches up with the first train in 4 hours, what is the speed of the first train?

Let the speed of the first train be ‘s’.

Distance covered by the first train in (2+4) hours = Distance covered by second train in 4 hours

Therefore, 6s = 60*4

Solving which gives s=40.

So the slower train is moving at the rate of 40 mph.

Questions on Boats/Airplanes

For problems with boats and streams,

Speed of the boat upstream (against the current) = Speed of the boat in still water – speed of the stream

[As the stream obstructs the speed of the boat in still water, its speed has to be subtracted from the usual speed of the boat]

Speed of the boat downstream (along with the current) = Speed of the boat in still water + speed of the stream

[As the stream pushes the boat and makes it easier for the boat to reach the destination faster, speed of the stream has to be added]

Similarly, for airplanes travelling with/against the wind,

Speed of the plane with the wind = speed of the plane + speed of the wind

Speed of the plane against the wind = speed of the plane – speed of the wind

Let us look at some examples.

Example 1. A man travels at 3 mph in still water. If the current’s velocity is 1 mph, it takes 3 hours to row to a place and come back. How far is the place?

Let the distance be ‘d’ miles.

Time taken to cover the distance upstream + Time taken to cover the distance downstream = 3

Speed upstream = 3-1 = 2 mph

Speed downstream = 3+1 = 4 mph

So, our equation would be d/2 + d/4 = 3 → solving which, we get d = 4 miles. Example 2. With the wind, an airplane covers a distance of 2400 kms in 4 hours and against the wind in 6 hours. What is the speed of the plane and that of the wind?

Let the speed of the plane be ‘a’ and that of the wind be ‘w’.

Our table looks like this:

	Distance	Speed	Time
With the wind	2400	a+w	4
Against the wind	2400	a-w	6

4(a+w) = 2400 and 6(a-w) = 2400

Expressing one unknown variable in terms of the other makes it easier to solve, which means

a+w = 600 → w=600-a

Substituting the value of w in the second equation,

a-(600-a) = 400 → a = 500

The speed of the plane is 500 kmph and that of the wind is 100 kmph.

More solved examples on Speed, Distance and Time

Example 1. A boy travelled by train which moved at the speed of 30 mph. He then boarded a bus which moved at the speed of 40 mph and reached his destination. The entire distance covered was 100 miles and the entire duration of the journey was 3 hours. Find the distance he travelled by bus.

	Distance	Speed	Time
Train	d	30	t
Bus	100-d	40	3-t

Let the time taken by the train be ‘t’. Then that of bus is ‘3-t’.

The entire distance covered was 100 miles

So, 30t + 40(3-t) = 100

Solving which gives t=2.

Substituting the value of t in 40(3-t), we get the distance travelled by bus is 40 miles.

Alternatively, we can add the time and equate it to 3 hours, which directly gives the distance.

d/30 + (100-d)/40 = 3

Solving which gives d = 60, which is the distance travelled by train. 100-60 = 40 miles is the distance travelled by bus. Example 2. A plane covered a distance of 630 miles in 6 hours. For the first part of the trip, the average speed was 100 mph and for the second part of the trip, the average speed was 110 mph. what is the time it flew at each speed?

Our table looks like this.

	Distance	Speed	Time
1 part of journey	d	100	t
2 part of journey	630-d	110	6-t

Assuming the distance covered in the 1 st part of journey to be ‘d’, the distance covered in the second half becomes ‘630-d’.

Assuming the time taken for the first part of the journey to be ‘t’, the time taken for the second half becomes ‘6-t’.

From the first equation, d=100t

The second equation is 630-d = 110(6-t).

Substituting the value of d from the first equation, we get

630-100t = 110(6-t)

Solving this gives t=3.

So the plane flew the first part of the journey in 3 hours and the second part in 3 hours. Example 2. Two persons are walking towards each other on a walking path that is 20 miles long. One is walking at the rate of 3 mph and the other at 4 mph. After how much time will they meet each other?

	Distance	Speed	Time
First person	d	3	t
Second person	20-d	4	t

Assuming the distance travelled by the first person to be ‘d’, the distance travelled by the second person is ’20-d’.

The time is ‘t’ for both of them because when they meet, they would have walked for the same time.

Since time is same, we can equate as

d/3 = (20-d)/4

Solving this gives d=60/7 miles (8.5 miles approximately)

Then t = 20/7 hours

So the two persons meet after 2 6/7 hours.

Practice Questions for you to solve

Problem 1: Click here

A boat covers a certain distance in 2 hours, while it comes back in 3 hours. If the speed of the stream is 4 kmph, what is the boat’s speed in still water?

A) 30 kmph B) 20 kmph C) 15 kmph D) 40 kmph

Answer 1: Click here

Explanation

Let the speed of the boat be ‘s’ kmph.

Then, 2(s+4) = 3(s-4) → s = 20

Problem 2: Click here

A cyclist travels for 3 hours, travelling for the first half of the journey at 12 mph and the second half at 15 mph. Find the total distance he covered.

A) 30 miles B) 35 miles C) 40 miles D) 180 miles

Answer 2: Click here

Since it is mentioned, that the first ‘half’ of the journey is covered in 12 mph and the second in 15, the equation looks like

(d/2)/12 + (d/2)/15 = 3

Solving this gives d = 40 miles

17 thoughts on “Distance, Time and Speed Word Problems | GMAT GRE Maths”

Meera walked to school at a speed of 3 miles per hour. Once she reached the school, she realized that she forgot to bring her books, so rushed back home at a speed of 6 miles per hour. She then walked back to school at a speed of 4 miles per hour. All the times, she walked in the same route. please explain above problem

When she walks faster the time she takes to reach her home and school is lower. There is nothing wrong with the statement. They never mentioned how long she took every time.

a man covers a distance on a toy train.if the train moved 4km/hr faster,it would take 30 min. less. if it moved 2km/hr slower, it would have taken 20 min. more .find the distance.

Let the speed be x. and time be y. A.T.Q, (x+4)(y-1/2)=d and (x-2)(y+1/3)=d. Equate these two and get the answer

Could you explain how ? you have two equations and there are 3 variables.

The 3rd equation is d=xy. Now, you have 3 equations with 3 unknowns. The variables x and y represent the usual speed and usual time to travel distance d.

Speed comes out to be 20 km/hr and the time taken is 3 hrs. The distance traveled is 60 km.

(s + 4) (t – 1/2)= st 1…new equotion = -1/2s + 4t = 2

(s – 2) (t + 1/3)= st 2…new equotion = 1/3s – 2t = 2/3

Multiply all by 6 1… -3s + 24t = 12 2… 2s – 12t = 4 Next, use elimination t= 3 Find s: -3s + 24t = 12 -3s + 24(3) = 12 -3s = -60 s= 20

st or distance = 3 x 20 = 60 km/h

It’s probably the average speed that we are looking for here. Ave. Speed= total distance/ total time. Since it’s harder to look for one variable since both are absent, you can use, 3d/ d( V2V3 + V1V3 + V1V2/ V1V2V3)

2 girls meenu and priya start at the same time to ride from madurai to manamadurai, 60 km away.meenu travels 4kmph slower than priya. priya reaches manamadurai and at turns back meeting meenu 12km from manamaduai. find meenu’s speed?

Hi, when the two girls meet, they have taken equal time to travel their respective distance. So, we just need to equate their time equations

Distance travelled by Meenu = 60 -12 = 48 Distance travelled by Priya = 60 + 12 = 72 Let ‘s’ be the speed of Meenu

Time taken by Meenu => t1 = 48/s Time taken by Priya => t2 = 72/(s+4)

t1=t2 Thus, 48/s = 72/(s+4) => 24s = 192 => s = 8Km/hr

A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 KMS away from A at the same time. On the way, however the train lost about 12.5 minutes while stopping at the stations. The speed of the car is:

Let speed of the CAR BE x kmph.

Then, speed of the train = 3/2(x) .’. 75/x – 75/(3/2)x= 125/(10*60) — subtracting the times travelled by two them hence trains wastage time

therefore x= 120 kmph

A cyclist completes a distance of 60 km at the same speed throughout. She travels 10 km in one hour. She stops every 20 km for one hour to have a break. What are the two variables involved in this situation?

For the answer, not variables: 60km divided by 10km/h=6 hours 60 divided by 20= 3 hours 3 hours+6 hours= 9 hours Answer: 9 hours

Let the length of the train to prod past a point be the intrinsic distance (D) of the train and its speed be S. Its speed, S in passing the electric pole of negligible length is = D/12. The length of the platform added to the intrinsic length of the train. So, the total distance = D + 200. The time = 20 secs. The Speed, S = (D + 200)/20 At constant speed, D/12 = (D + 200)/20 Cross-multiplying, 20D = 12D + 200*12 20D – 12D = 200*12; 8D = 200*12 D = 200*12/8 = 300m. 4th Aug, 2018

Can anyone solve this? Nathan and Philip agree to meet up at the park at 5:00 pm. Nathan lives 300 m due north of the park, and Philip lives 500 m due west of the park. Philip leaves his house at 4:54 pm and walks towards the park at a pace of 1.5 m/s, but Nathan loses track of time and doesn’t leave until 4:59 pm. Trying to avoid being too late, he jogs towards the park at 2.5 m/s. At what rate is the distance between the two friends changing 30 seconds after Nathan has departed?

Time Measurement Problems: Simple and Complex Forms

In order to solve time measurement problems, we first need to know what the units of time are and what they are equivalent to.

If you already know them, keep on reading! If not, you can click here on this link for a review.

As you can see in the table above, there are bigger and smaller units than a day. The smaller ones are hours, minutes and seconds; these are part of a sexagesimal system. This means that they go from 60 to 60, for example, an hour is 60 minutes and 1 minute is 60 seconds.

1 h = 60 min 1 min = 60 s 1 hr = 3600 s

It’s also important to remember that these time measurements (sexagesimal system) can be expressed in two ways:

Simple: using only one unit. For example: 8715 s
Complex: using several units simultaneously. For example: 2 h 25 min 15 s. In the complex form, minutes and seconds can never surpass 60.

Is it possible to go from one type of unit to another?

The answer is yes. A measurement that’s written in the complex form can be written in the simplified form and vice versa. Let’s take a look at how to do that.

Time Measurement Problems: From Complex to Simple

We need to convert every part of the expression to the same unit (the one the problem asks for) and later, add them all together.

Express 2h 25 min 15 s in seconds:

2 h 25 min 15 s = 8715 s

Time measurement problems: from simple to complex.

We need to divide the measurement and the successive quotient (if necessary) by 60.

Express 8715 s in the complex form:

8715 s = 2h 25 min 15 s

In the example, we can see that the remainder of the first division is the seconds of the new expression, and the quotient (the minutes) is divided again by 60. The quotient of the second division is the hours and the minutes are the remainder.

With this information, we can now solve the following problem:

The winner of the school’s go-kart race finished with a time of 1h 53 min 58 s. The racer in last place finished 365 s after the winner. What was the final time of the racer in last place? Express the answer in complex form.

We can see that the winner’s time is expressed in complex form and the difference of his time and the last qualifier’s time is expressed in simple form. The problem asks us to express the answer in the complex form, so the first thing we need to do is convert 365 seconds to the complex form.

In order to convert 365 seconds to the complex form, we have to start by dividing it by 60. The remainder that we get from dividing will be the seconds and the quotient, the minutes and because the number is smaller than 60, we don’t have to keep dividing.

365 s = 6 min 5 s

The next step is to add everything in order to find out the total race time of the person that came in last place:

Now, we add the seconds to the seconds, the minutes to the minutes, and the hours to the hours. The result is: 1h 59 min 63 s

So, are we done?

Not yet! Remember that in the complex form of time measurement, the seconds, and minutes must be less than 60. So, we still have to do one more step:

63 s = 60 s + 3 s and 60 s = 1 min, therefore, we write 63 s as 1 min and 3 s.

We need to add that extra minute to the 59 minutes that we already had and we’re left with 1h 60 min 3 s.

Now we have 60 min and 60 min = 1 h

So then, 1 h 60 min 3 s = 2 h 0 min 3 s

And we finally have the answer expressed in the complex form: The total time of the last qualifier was 2 h 0 min 3 s

One last question before we finish…

Is 1 min 30.132 s a correct expression?

Although it’s not very common, yes, it is a correct expression. In fact, it was the classification time of Fernando Alonso (Formula 1 racer) in Australia’s Grand Prix in 2014. He completed the track in 1 minute, 30 seconds, 1 hundredth, 3 tenths, and 2 milliseconds.

The tenths, hundredths, and millionths of a second (submultiples of a second) are used when time needs to be measured extremely precisely, like in a Formula 1 race!

If you know of more situations that call for submultiples of a second, don’t think twice about letting us know and remember that you can log onto Smartick if you want to know more about time measurement problems.

By the way, one week is 7 days, or 168 hours, or 10080 minutes, or 604,800 seconds, or even 604, 800, 000 milliseconds!

Learn More:

Learn How to Measure Time and the Units Associated
Learn How to Solve a Time Word Problem
Conversion Capacity Problems in the Metric System
Practice Length Measurement Problems
Divisibility by 4: How Do We Know If a Number Is Divisible by 4?
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Time Word Problems Worksheets

This is one of those math topics that we all will master at one point; I can assure you. Do you know how many jobs that I had as young person that I despised? I would spend all day, when I was not working, calculating how long until I had to go to work. When I was at work, I would spend every free second, I had calculating how long until I could go home. The word problems that we present you with will be fun and engaging, in most cases. You will learn to stretch the concept of time in many different directions. This selection of lessons and worksheets helps students learn how to breakdown and make sense of story-based math problems that include an aspect of time within them.

Aligned Standard: 3.MD.A.1

Mark's Exercises - Mark starts his exercise at 6:45 p.m. He finished it at 8:25 p.m. How long does he take to exercise?
Step-by-Step Lesson - How long does Mark do his exercises for?
Guided Lesson - How long is the wrestling match, cooking and television watching taking?
Guided Lesson Explanation - See if my explanation of how we circumvent the AM to PM switch works for you.
Practice Worksheet - These questions took me a whole day. I hope it was well spent.
Matching Worksheet - Match the event duration and the times in the different columns.
Time Word Problems 5 Pack - Plenty of worksheets for you to work on this skill with.
Back In Time Word Problems Five Pack of Worksheets - How long ago was that invented or did that event happen? These are all relative to the time given.
Answer Keys - These are for all the unlocked materials above.

Homework Sheets

I had to rewrite number 3 several times. The wording was just not there.

Homework 1 - Jerry has to get the cake from the bakery to the wedding hall. The drive will take him 1 hour and 20 minutes. If the cake needs to be there by 4:15, what time must he leave by?
Homework 2 - Kyle is able to build a new patio at the rate of 1 square foot per 10 minutes. How long will it take Kyle to finish a 15 x 20 foot patio?
Homework 3 - Dave is one of the best mechanics in town. Below you will see a time table of how long it takes Dave to make certain repairs. Answer the questions below based on the time table.

Practice Worksheets

The practice problems should be fun for kids, assuming they know their math.

Practice 1 - Tara goes to the pool at 2 p.m. She swims for an hour and 25 minutes then she walks home. It takes her 10 minutes. What time did she return home?
Practice 2 - Jerry goes to the beach at 3 o'clock, and he makes 5 sand castles. It took him 25 minutes to create each sand castle. What time was he done?
Practice 3 - Ned runs the Nerd Hut. They fix all kinds of computer problems. They charge $40 an hour to make repairs or software adjustments. You can see a list of the most common repairs and tweaks they make. It also shows you how long each task takes.

Math Skill Quizzes

See if the timeline in number 1 is helpful.

Quiz 1 - A workable parachute was invented in 1783 by Jean Pierre Blanchard. If the year is 2015, how old is the invention of the parachute?
Quiz 2 - Jeff can install 50 shingles every hour. He got to the job site at 7 a.m. this morning and installed 400 shingles. What time was he done?

Things to Consider When Solving Time Word Problems

When you are tackling word problems that have involve measures of time there are some things you should get in the habit of doing to have a greater chance of success. It begins with noting what each time denotes. I often see students write down times and perform operations between those times, but they neglect to indicate what events occurred at those times. This is often why they choose the wrong operation to perform between them. Another major problem that I see when students are performing these types of problems is that they do not clearly separate units of time. For instance, if a time value consist of 4 minutes and 22 seconds, they do not clearly indicate the separation between minutes and seconds. Get in the habit of doing that and it will make these problems much easier to take head on.

What is the Meaning of AM and PM in Time?

Aren't clocks useful? They help you tell time! Whether you need to understand how much time is needed for you to complete your homework, or how much time is needed for you to finish lunch, being able to tell time is very important. When we are talking about what time of the day it is, we usually say its 3 pm or 8 am. But do you ever wonder what this am and pm means? Let's find out!

Today, the clocks that we use divide the day into 12-hour time increments. Why is that? Because the number of hours in a day is 24 and thus, can be divided further into 12 hours. Isn't that odd? Because our clocks show time enough only for one of the 12-hour period. Hence, to solve this problem, two portions of 12 hours are shown on the analog by using the AM and PM suffixes.

The word AM. is Latin for Ante Meridiem, which means before noon. The term PM is Latin for Post Meridiam, which means after noon. This is how the 12-hour increments are displayed in the same analog.

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Time and Work Formula and Solved Problems

The basic formula for solving is: 1/r + 1/s = 1/h
Let us take a case, say a person Hrithik
Let us say that in 1 day Hrithik will do 1/20 th of the work and 1 day Dhoni will do 1/30 th of the work. Now if they are working together they will be doing 1/20 + 1/30 = 5/60 = 1/12 th of the work in 1 day. Now try to analyze, if two persons are doing 1/12 th of the work on first day, they will do 1/12 th of the work on second day, 1/12 th of the work on third day and so on. Now adding all that when they would have worked for 12 days 12/12 = 1 i.e. the whole work would have been over. Thus the concept works in direct as well as in reverse condition.
The conclusion of the concept is if a person does a work in ‘r’ days, then in 1 day- 1/r th of the work is done and if 1/s th of the work is done in 1 day, then the work will be finished in ‘s’ days. Thus working together both can finish 1/h (1/r + 1/s = 1/h) work in 1 day & this complete the task in ’h’ hours.
The same can also be interpreted in another manner i.e. If one person does a piece of work in x days and another person does it in y days. Then together they can finish that work in xy/(x+y) days
In case of three persons taking x, y and z days respectively, They can finish the work together in xyz/(xy + yz + xz) days

Time and work problems

Time and work concepts, time and work problems (easy), time and work problems (difficult), most popular articles - ps.

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TIME AND WORK PROBLEMS

1. If a person can do a piece of work in ‘m’ days, he can do ¹⁄ m part of the work in 1 day.

2. If the number of persons engaged to do a piece of work be increased (or decreased) in a certain ratio the time required to do the same work will be decreased (or increased) in the same ratio.

3. If A is twice as good a workman as B, then A will take half the time taken by B to do a certain piece of work.

4. Time and work are always in direct proportion.

more work ----> more time

less work ----> less time

5. A takes m days and B takes n days to complete a work. If they work together, then the formula to find the number of days taken by them to complete the work is

Problem 1 :

A can do a piece of work in 15 days while B can do it in 10 days. How long will they take together to do it?

Using the above formula, if they work together, number of days taken to complete the work is

Problem 2 :

A and B can complete a work in 6 days . B and C can complete the same work in 8 days. C and A can complete in 12 days. How many days will take for A, B and C combined together to complete the same amount of work ?

From the given information, we can have

(A + B)'s 1 day work = ⅙

(B + C)'s 1 day work = ⅛

(A + C)'s 1 day work = ¹⁄₁₂

(A + B + B + C + A + C)'s 1 day work = ⅙ + ⅛ + ¹⁄₁₂

(2A + 2B + 2C)'s 1 day work = ⅙ + ⅛ + ¹⁄₁₂

2 ⋅ (A + B + C)'s 1 day work = ⅙ + ⅛ + ¹⁄₁₂

L.C.M of (6, 8, 12) = 24.

2 (A + B + C)'s 1 day work = ⁴⁄₂₄ + ³⁄₂₄ + ²⁄₂₄

2 (A + B + C)'s 1 day work = ⁹⁄₂₄

2 (A + B + C)'s 1 day work = ⅜

(A + B + C)'s 1 day work = ³⁄₁₆

Time taken by A, B and C together to complete the work is

= 5 ⅓ days

Problem 3 :

A and B can do a work in 15 days, B and C in 30 days and A and C in 18 days. They work together for 9 days and then A left. In how many more days, can B and C finish the remaining work ?

(A + B)'s 1 day work = ¹⁄₁₅

(B + C)'s 1 day work = ¹⁄₃₀

(A + C)'s 1 day work = ¹⁄₁₈

(A + B + B + C + A + C)'s 1 day work = ¹⁄₁₅ + ¹⁄₃₀ + ¹⁄₁₈

(2A + 2B + 2C)'s 1 day work = ¹⁄₁₅ + ¹⁄₃₀ + ¹⁄₁₈

2 (A + B + C)'s 1 day work = ¹⁄₁₅ + ¹⁄₃₀ + ¹⁄₁₈

L.C.M of (15, 30, 18) = 90.

2 (A + B + C)'s 1 day work = ⁶⁄₉₀ + ³⁄₉₀ + ⁵⁄₉₀

2 (A + B + C)'s 1 day work = ¹⁴⁄₉₀

2 (A + B + C)'s 1 day work = ⁷⁄₄₅

(A + B + C)'s 1 day work = ⁷⁄₉₀

Then, the amount of work completed by A, B and C together in 9 days is

= 9 ⋅ ⁷⁄₉₀

= ⁷⁄₁₀

Amount of work left for B and C to complete is

= ³⁄₁₀

Number of days that B will take to finish the work is

= amount of work/part of the work done in 1 day

= ³⁄₁₀ ÷ ¹⁄₃₀

= ³⁄₁₀ ⋅ ³⁰⁄₁

Problem 4 :

A contractor decided to complete the work in 90 days and employed 50 men at the beginning and 20 men additionally after 20 days and got the work completed as per schedule. If he had not employed the additional men, how many extra days would he have needed to complete the work?

The work has to completed in 90 days (as per schedule).

Total no. of men appointed initially = 50.

Given : 50 men have already worked for 20 days and completed a part of the work.

If the remaining work is done by 70 men (50 + 20 = 70), the work can be completed in 70 days and the total work can be completed in 90 days as per the schedule.

Let 'x' be the no. of days required when the remaining work is done by 50 men.

For the remaining work,

70 men ----> 70 days

50 men -----> x days

The above one is a inverse variation.

Because, when no. of men is decreased, no. of days will be increased.

By inverse variation, we have

70 ⋅ 70 = 50 ⋅ x

So, if the remaining work is done by 50 men, it can be completed in 98 days.

So, extra days needed = 98 - 70 = 28 days.

Problem 5 :

Three taps A, B and C can fill a tank in 10, 15 and 20 hours respectively. If A is open all the time and B and C are open for one hour each alternately, find the time taken to fill the tank.

A's 1 hour work = ⅒

B's 1 hour work = ¹⁄₁₅

C's 1 hour work = ¹⁄₂₀

In the first hour, we have

(A + B)'s work = ⅒ + ¹⁄₁₅

(A + B)'s work = ³⁄₃₀ + ²⁄₃₀

(A + B)'s work = ⁵⁄₃₀

(A + B)'s work = ⅙

In the second hour, we have

(A + C)'s work = ⅒ + ¹⁄₂₀

(A + C)'s work = ²⁄₂₀ + ¹⁄₂₀

(A + C)'s work = ³⁄₂₀

Amount of work done in each two hours is

= ¹⁰⁄₆₀ + ⁹⁄₆₀

Amount of work done :

In the first 2 hours : ¹⁹⁄₆₀

In the first 4 hours : ¹⁹⁄₆₀ + ¹⁹⁄₆₀ = ³⁸⁄₆₀

In the first 6 hours : ¹⁹⁄₆₀ + ¹⁹⁄₆₀ + ¹⁹⁄₆₀ = ⁵⁷⁄₆₀

After 6 hours, the remaining work will be

= ¹⁄₂₀

¹⁄₂₀ is the small amount of work left and A alone can complete this.

Time taken by A to complete this 1/20 part of the work is

= amount of work/part of work done in 1 hour

= ¹⁄₂₀ ÷ ⅒

= ¹⁄₂₀ ⋅ ¹⁰⁄₁

= ½ hours

So, A will will take half an hour (or 30 minutes) to complete the remaining work ¹⁄₂₀ .

So, total time taken to complete the work is

= 6 hours + 30 minutes

= 6 ½ hours

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Time problems

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This problem solving activity has a number focus.

These students are to sit an exam that begins at 9:30 pm sharp. Which of them will get there i n time for the start of the exam?

Derek’s watch is 15 minutes fast but he thinks that it is 10 minutes fast.

Marilyn’s watch is 15 minutes fast but she thinks it is 20 minutes fast.

Sara’s watch is 15 minutes slow but she thinks it is 10 minutes fast.

Tim’s watch is 15 minutes slow but he thinks that it is 10 minutes slow.

Understand how positive and negative numbers can be used in an unusual practical problem.
Devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, look for patterns, draw a diagram, make a table, use algebra).

This logic problem, involving positive and negative numbers in a practical situation, may challenge students and generate interesting discussion. Students need to possess knowledge of time to work successfully with this problem.

Copymaster of the problem (English)
Copymaster of the problem (Māori)
A clock (a digital version could be used)

The Problem

These students are to sit an exam that begins at 9:30 pm sharp. Which of them will get there in time for the start of the exam?

Teaching Sequence

Pose the problem for students to work on in their groups. Allow discussion.
When groups believe they know about at least one of the examinees, pause for discussion.
Take a vote on which of the four people get to the exam late and have students give reasons for their choices.
Reach a consensus.
Give time for the students finish the problem and begin on the extension as appropriate.
Have students share their results with reference to their written solutions. Encourage them to use the clock to model their solutions.

There are other possibilities for people’s watches. They could be fast but they think they’re slow; they could be slow but they think they’re fast; they could be slow and their owners think that they are slower than that, and so on. Investigate the other possibilities.

There are many ways of solving this problem. Here is one approach, using a table for each examinee. Derek.

If it was 9:30, Derek’s watch would show 9:45 because it is 15 minutes fast. But Derek thinks it is 10 minutes fast. So he thinks that the actual time is 9:35. That gives the first row of the table.

But Derek wants to be at the exam at 9:30. So he will go there when he thinks that it is 9:30. But when he thinks that it is 9:30, his watch will be showing 9:40. However, the actual time then is 9:25. So Derek gets there in time.

Marilyn.

Marilyn will be 5 minutes late.

Sara, who will very very late.

Tim. There are a lot of people coming late to this exam!

Solution to the Extension

The possibilities are:

watch fast by x; person thinks it is fast by y with y > x;
watch fast by x; person thinks it is fast by y with y < x;
watch fast by x; person thinks it is fast by y with y = x;
watch fast by x; person thinks it is slow by y with y > x;
watch fast by x; person thinks it is slow by y with y < x;
watch fast by x; person thinks it is slow by y with y = x;
watch slow by x; person thinks it is fast by y with y > x;
watch slow by x; person thinks it is fast by y with y < x;
watch slow by x; person thinks it is fast by y with y = x;
watch slow by x; person thinks it is slow by y with y > x;
watch slow by x; person thinks it is slow by y with y < x;
watch fast by x; person thinks it is slow by y with y = x.

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Tell me about a time you solved a problem – 7 sample answers & more

What else is a typical human life, if not a succession of problems we have to solve ? Of course, this depends on a few things. One’s perspective for a start, because for some people everything is a problem, while others take life more easily, and do not worry much about tomorrow. Other thing is your present situation , economically speaking. Sad but true, hundreds of millions fight to barely survive every day –that’s their main and only problem. No time or capacity to focus on anything else. But an average Joe or Jane in the US (or any other “first world” country) has their share of problems too. Job problems, relationship issues, debts, and so on. At the end of the day, life’s not a walk in the park for almost anyone, and we have to deal with stuff, and solve problems. But what problem should you talk about in an interview?

It depends. In an ideal case, you should describe a problem you solved in your past job , one that demonstrates your skills and experience in the field , and the fact that you solved it should help convince the hiring managers that you are the right person for the job , and have the necessary qualification and skills. This is easy to say, but often hard to do. Maybe this is your first job application ever , or you worked in a different field before. Or you actually failed to solve problems in your previous job, and it is the main reason why you find yourself without any job at the moment. In any of these cases, you can talk about problems you solved in your personal life, at school, or even problems related to your mentality, self-confidence, and other.

Let’s have a look at 7 sample answers to this tricky question right now. I tried to come up with a nice mix of answers for problems from different areas of life, including some out-of-the-box answers. Hopefully at least one of them will resonate with you, and give you an inspiration for your own unique interview answer. Just do not forget that your attitude matters more than anything else for the hiring managers. Regardless of the problem you narrate, they should get an impression that you faced it with your head high, and tried your very best –regardless of the eventual outcome.

7 example answers to “Tell me about a time you solved a problem” interview question

The last one I remember was in my last job in a restaurant . I found it really hard to get along with one of the colleagues. For some reason they didn’t like me, the communication didn’t really work, and on the top of that they did not work as hard as other people on the shift did. I could ask the manager to address the issue, but I decided to solve the problem on my own . Asked the colleague out for a coffee during the break, and openly, without emotions, explained them my point of view and why we should start to cooperate and communicate normally, in order to make things better for everyone. I also showed interest in their feelings and opinions, trying to understand them better. It turned out to be just the first of several talks we had together, and at the end, we became friends and everything improved.
My last job of a process engineer was all about solving problems . Because I had to find ways of making the production process more efficient. Hence I not only solved problems–I identified them , identified areas for improvement, and then tried to implement some solutions. A really difficult one was to solve a bottleneck we faced in the manufacturing process of the printers. Two steps in the production process were slowing the entire process down a lot, and I had to come up with new processes , trying to increase our daily production capacity. This required a lot of studying, testing, experimenting, and also failing. But eventually I came up with some improvements which resulted in 10 percent increase of the daily production capacity of the plant , which was something huge for my employer. On the top of that, I learned a lot in the process, and hope to benefit from it in my new job with you .
This is my first job application , but I recall many problems I had to solve at school . For example coming up with difficult presentations, or even passing the exam from a subject I struggled with . But I always remained calm, and simply did my best to prepare for the exam, or to get the presentation done. This often meant sacrifices, such as spending long hours studying , instead of enjoying some leisure time activities with my friends. But it eventually paid off , I graduated, and now I have a chance to apply for this great job with you. I honestly believe that problems belong to life , and as long as we don’t get scared and face them with courage and enthusiasm, we will solve most of them.
To be honest with you, my entire life is a problem to solve right now . I’ve lost my partner a year ago, which was a hard blow. I haven’t been able to start a new relationship ever since. What’s more, I’ve been struggling with debts, have some health issues –which can be related to the other issues I face, and so on. But I am still here, I haven’t resigned on life , and I try my best to get back on track and sort my problems out. Getting this great job with you will help me a lot , since it will help me out with the debt I have, which will take some burden out of my shoulders, and allow me to move forward also with other things I have to address.
In my opinion, the primary cause of unhappiness is never really the situation, but our thoughts about it . Look, I have my issues and problems, just like everyone else. But I focus on the good things I have i life–a family, decent health, luck to be living in a rich country with opportunities always present. On the top of that, I try my best to live in the present moment , always giving 100% of my focus and attention to the task I am working on . And the magic does happen, because not only I am rarely stressed about things, I also mange to solve all the problems I face, be it at work or in my personal life.
My biggest problem was lack of self-confidence . It impacted me in all areas of life, including my job search. To be honest , I battled with this problem for years, never quite knowing how to address it. Eventually I decided to do two things. First, to visit a psychologist and start working with her. And secondly, to work on my physical condition and appearance, because I knew it had an impact on my self-confidence. It has been about a year since I made this move, and it really did wonders for me. As you can see now, I can speak here with you without nerves, and I believe I can achieve things in life . With some problems it may take you longer to solve them, but if one doesn’t give up and persist in their efforts, they will eventually overcome them. I am a living proof of it.
I am solving one right now . Have been without a job for more than 12 months, and I find it hard to encounter anything new. In my late fifties, many companies do not even give me a chance to interview with them and prove that I feel much younger than that, and can still work hard every day. I am extremely grateful that you gave me an opportunity to interview here, because I do not only want a job, I need one . And I sincerely hope that I will solve this problem soon. Because I am giving it my best shot, sending out dozens of applications every week, and trying my best to be prepared for the eventual interviews…

Ready to answer this one? I hope so! Do not forget to check also 7 sample answers to other tricky questions you may face in your interview:

Tell me about a time when you missed an obvious solution to a problem.
Are you a problem solver?
Tell me about a time when you had a simple solution to a complex problem.
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40 problem-solving techniques and processes

All teams and organizations encounter challenges. Approaching those challenges without a structured problem solving process can end up making things worse.

Proven problem solving techniques such as those outlined below can guide your group through a process of identifying problems and challenges , ideating on possible solutions , and then evaluating and implementing the most suitable .

In this post, you'll find problem-solving tools you can use to develop effective solutions. You'll also find some tips for facilitating the problem solving process and solving complex problems.

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What is problem solving?

Problem solving is a process of finding and implementing a solution to a challenge or obstacle. In most contexts, this means going through a problem solving process that begins with identifying the issue, exploring its root causes, ideating and refining possible solutions before implementing and measuring the impact of that solution.

For simple or small problems, it can be tempting to skip straight to implementing what you believe is the right solution. The danger with this approach is that without exploring the true causes of the issue, it might just occur again or your chosen solution may cause other issues.

Particularly in the world of work, good problem solving means using data to back up each step of the process, bringing in new perspectives and effectively measuring the impact of your solution.

Effective problem solving can help ensure that your team or organization is well positioned to overcome challenges, be resilient to change and create innovation. In my experience, problem solving is a combination of skillset, mindset and process, and it’s especially vital for leaders to cultivate this skill.

A group of people looking at a poster with notes on it

What is the seven step problem solving process?

A problem solving process is a step-by-step framework from going from discovering a problem all the way through to implementing a solution.

With practice, this framework can become intuitive, and innovative companies tend to have a consistent and ongoing ability to discover and tackle challenges when they come up.

You might see everything from a four step problem solving process through to seven steps. While all these processes cover roughly the same ground, I’ve found a seven step problem solving process is helpful for making all key steps legible.

We’ll outline that process here and then follow with techniques you can use to explore and work on that step of the problem solving process with a group.

The seven-step problem solving process is:

1. Problem identification

The first stage of any problem solving process is to identify the problem(s) you need to solve. This often looks like using group discussions and activities to help a group surface and effectively articulate the challenges they’re facing and wish to resolve.

Be sure to align with your team on the exact definition and nature of the problem you’re solving. An effective process is one where everyone is pulling in the same direction – ensure clarity and alignment now to help avoid misunderstandings later.

2. Problem analysis and refinement

The process of problem analysis means ensuring that the problem you are seeking to solve is the right problem . Choosing the right problem to solve means you are on the right path to creating the right solution.

At this stage, you may look deeper at the problem you identified to try and discover the root cause at the level of people or process. You may also spend some time sourcing data, consulting relevant parties and creating and refining a problem statement.

Problem refinement means adjusting scope or focus of the problem you will be aiming to solve based on what comes up during your analysis. As you analyze data sources, you might discover that the root cause means you need to adjust your problem statement. Alternatively, you might find that your original problem statement is too big to be meaningful approached within your current project.

Remember that the goal of any problem refinement is to help set the stage for effective solution development and deployment. Set the right focus and get buy-in from your team here and you’ll be well positioned to move forward with confidence.

3. Solution generation

Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or techniquess designed to produce working prototypes of possible solutions.

The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can often come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold.

4. Solution development

No solution is perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically.

During this stage, you will often ask your team to iterate and improve upon your front-running solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.

Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose.

5. Decision making and planning

Nearly there! Once you’ve got a set of possible, you’ll need to make a decision on which to implement. This can be a consensus-based group decision or it might be for a leader or major stakeholder to decide. You’ll find a set of effective decision making methods below.

Once your group has reached consensus and selected a solution, there are some additional actions that also need to be decided upon. You’ll want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.

Set clear accountabilities, actions, timeframes, and follow-ups for your chosen solution. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group.

Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved.

6. Solution implementation

This is what we were waiting for! All problem solving processes have the end goal of implementing an effective and impactful solution that your group has confidence in.

Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way. For some solutions, you might also implement a test with a small group and monitor results before rolling it out to an entire company.

You should have a clear owner for your solution who will oversee the plans you made together and help ensure they’re put into place. This person will often coordinate the implementation team and set-up processes to measure the efficacy of your solution too.

7. Solution evaluation

So you and your team developed a great solution to a problem and have a gut feeling it’s been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback.

You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives.

None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.

Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization.

It’s also worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time.

What does an effective problem solving process look like?

Every effective problem solving process begins with an agenda . In our experience, a well-structured problem solving workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

The format of a workshop ensures that you can get buy-in from your group, encourage free-thinking and solution exploration before making a decision on what to implement following the session.

This Design Sprint 2.0 template is an effective problem solving process from top agency AJ&Smart. It’s a great format for the entire problem solving process, with four-days of workshops designed to surface issues, explore solutions and even test a solution.

Check it for an example of how you might structure and run a problem solving process and feel free to copy and adjust it your needs!

For a shorter process you can run in a single afternoon, this remote problem solving agenda will guide you effectively in just a couple of hours.

Whatever the length of your workshop, by using SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Complete problem-solving methods

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

The Six Thinking Hats #creative thinking #meeting facilitation #problem solving #issue resolution #idea generation #conflict resolution The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on.

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages.

Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly It doesn’t matter where you work and what your job role is, if you work with other people together as a team, you will always encounter the same challenges: Unclear goals and miscommunication that cause busy work and overtime Unstructured meetings that leave attendants tired, confused and without clear outcomes. Frustration builds up because internal challenges to productivity are not addressed Sudden changes in priorities lead to a loss of focus and momentum Muddled compromise takes the place of clear decision- making, leaving everybody to come up with their own interpretation. In short, a lack of structure leads to a waste of time and effort, projects that drag on for too long and frustrated, burnt out teams. AJ&Smart has worked with some of the most innovative, productive companies in the world. What sets their teams apart from others is not better tools, bigger talent or more beautiful offices. The secret sauce to becoming a more productive, more creative and happier team is simple: Replace all open discussion or brainstorming with a structured process that leads to more ideas, clearer decisions and better outcomes. When a good process provides guardrails and a clear path to follow, it becomes easier to come up with ideas, make decisions and solve problems. This is why AJ&Smart created Lightning Decision Jam (LDJ). It’s a simple and short, but powerful group exercise that can be run either in-person, in the same room, or remotely with distributed teams.

Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design.

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition #problem solving #idea generation #creativity #online #remote-friendly A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

The 5 Whys

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges.

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results.

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys #hyperisland #innovation This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold.

World Cafe #hyperisland #innovation #issue analysis World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD) #idea generation #liberating structures #action #issue analysis #remote-friendly DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology #action plan #idea generation #problem solving #issue analysis #large group #online #remote-friendly Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around.

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish.

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis #problem solving ##root cause analysis #decision making #online facilitation A process to help identify and understand the origins of problems, issues or observations.

Problem Tree

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them.

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree #define intentions #create #design #issue analysis A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward.

SWOT analysis #gamestorming #problem solving #action #meeting facilitation The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results.

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause.

Agreement-Certainty Matrix #issue analysis #liberating structures #problem solving You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic . A problem is simple when it can be solved reliably with practices that are easy to duplicate. It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably. A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail. Chaotic is when the context is too turbulent to identify a path forward. A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.” The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process.

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID #gamestorming #project planning #issue analysis #problem solving When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!

Speed Boat #gamestorming #problem solving #action Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How #idea generation #issue analysis #problem solving #online #creative thinking #remote-friendly A questioning method for generating, explaining, investigating ideas.

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed.

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It! #gamestorming #problem solving #action Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills.

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO!

LEGO Challenge #hyperisland #team A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems.

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken.

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What? #issue analysis #innovation #liberating structures You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

Journalists

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists #vision #big picture #issue analysis #remote-friendly This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for brainstorming solutions

Now you have the context and background of the problem you are trying to solving, now comes the time to start ideating and thinking about how you’ll solve the issue.

Here, you’ll want to encourage creative, free thinking and speed. Get as many ideas out as possible and explore different perspectives so you have the raw material for the next step.

Looking at a problem from a new angle can be one of the most effective ways of creating an effective solution. TRIZ is a problem-solving tool that asks the group to consider what they must not do in order to solve a challenge.

By reversing the discussion, new topics and taboo subjects often emerge, allowing the group to think more deeply and create ideas that confront the status quo in a safe and meaningful way. If you’re working on a problem that you’ve tried to solve before, TRIZ is a great problem-solving method to help your team get unblocked.

Making Space with TRIZ #issue analysis #liberating structures #issue resolution You can clear space for innovation by helping a group let go of what it knows (but rarely admits) limits its success and by inviting creative destruction. TRIZ makes it possible to challenge sacred cows safely and encourages heretical thinking. The question “What must we stop doing to make progress on our deepest purpose?” induces seriously fun yet very courageous conversations. Since laughter often erupts, issues that are otherwise taboo get a chance to be aired and confronted. With creative destruction come opportunities for renewal as local action and innovation rush in to fill the vacuum. Whoosh!

Mindspin

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly.

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation.

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex.

MindSpin #teampedia #idea generation #problem solving #action A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed.

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable.

The Creativity Dice #creativity #problem solving #thiagi #issue analysis Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Idea and Concept Development

Brainstorming without structure can quickly become chaotic or frustrating. In a problem-solving context, having an ideation framework to follow can help ensure your team is both creative and disciplined.

In this method, you’ll find an idea generation process that encourages your group to brainstorm effectively before developing their ideas and begin clustering them together. By using concepts such as Yes and…, more is more and postponing judgement, you can create the ideal conditions for brainstorming with ease.

Idea & Concept Development #hyperisland #innovation #idea generation Ideation and Concept Development is a process for groups to work creatively and collaboratively to generate creative ideas. It’s a general approach that can be adapted and customized to suit many different scenarios. It includes basic principles for idea generation and several steps for groups to work with. It also includes steps for idea selection and development.

Problem-solving techniques for developing and refining solutions

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to develop and refine your ideas in order to bring them closer to a solution that actually solves the problem.

Use these problem-solving techniques when you want to help your team think through their ideas and refine them as part of your problem solving process.

Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result.

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution.

Improved Solutions #creativity #thiagi #problem solving #action #team You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged.

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch #design sprint #innovation #idea generation #remote-friendly The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper, Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

Ensuring that everyone in a group is able to contribute to a discussion is vital during any problem solving process. Not only does this ensure all bases are covered, but its then easier to get buy-in and accountability when people have been able to contribute to the process.

1-2-4-All is a tried and tested facilitation technique where participants are asked to first brainstorm on a topic on their own. Next, they discuss and share ideas in a pair before moving into a small group. Those groups are then asked to present the best idea from their discussion to the rest of the team.

This method can be used in many different contexts effectively, though I find it particularly shines in the idea development stage of the process. Giving each participant time to concretize their ideas and develop them in progressively larger groups can create a great space for both innovation and psychological safety.

1-2-4-All #idea generation #liberating structures #issue analysis With this facilitation technique you can immediately include everyone regardless of how large the group is. You can generate better ideas and more of them faster than ever before. You can tap the know-how and imagination that is distributed widely in places not known in advance. Open, generative conversation unfolds. Ideas and solutions are sifted in rapid fashion. Most importantly, participants own the ideas, so follow-up and implementation is simplified. No buy-in strategies needed! Simple and elegant!

15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change.

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change.

15% Solutions #action #liberating structures #remote-friendly You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference. 15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change. With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Problem-solving techniques for making decisions and planning

After your group is happy with the possible solutions you’ve developed, now comes the time to choose which to implement. There’s more than one way to make a decision and the best option is often dependant on the needs and set-up of your group.

Sometimes, it’s the case that you’ll want to vote as a group on what is likely to be the most impactful solution. Other times, it might be down to a decision maker or major stakeholder to make the final decision. Whatever your process, here’s some techniques you can use to help you make a decision during your problem solving process.

How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process.

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud.

How-Now-Wow Matrix #gamestorming #idea generation #remote-friendly When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice.

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them.

Impact and Effort Matrix #gamestorming #decision making #action #remote-friendly In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action?

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus.

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively.

Dotmocracy #action #decision making #group prioritization #hyperisland #remote-friendly Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

Straddling the gap between decision making and planning, MoSCoW is a simple and effective method that allows a group team to easily prioritize a set of possible options.

Use this method in a problem solving process by collecting and summarizing all your possible solutions and then categorize them into 4 sections: “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”.

This method is particularly useful when its less about choosing one possible solution and more about prioritorizing which to do first and which may not fit in the scope of your project. In my experience, complex challenges often require multiple small fixes, and this method can be a great way to move from a pile of things you’d all like to do to a structured plan.

MoSCoW #define intentions #create #design #action #remote-friendly MoSCoW is a method that allows the team to prioritize the different features that they will work on. Features are then categorized into “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”. To be used at the beginning of a timeslot (for example during Sprint planning) and when planning is needed.

When it comes to managing the rollout of a solution, clarity and accountability are key factors in ensuring the success of the project. The RAACI chart is a simple but effective model for setting roles and responsibilities as part of a planning session.

Start by listing each person involved in the project and put them into the following groups in order to make it clear who is responsible for what during the rollout of your solution.

Responsibility (Which person and/or team will be taking action?)
Authority (At what “point” must the responsible person check in before going further?)
Accountability (Who must the responsible person check in with?)
Consultation (Who must be consulted by the responsible person before decisions are made?)
Information (Who must be informed of decisions, once made?)

Ensure this information is easily accessible and use it to inform who does what and who is looped into discussions and kept up to date.

RAACI #roles and responsibility #teamwork #project management Clarifying roles and responsibilities, levels of autonomy/latitude in decision making, and levels of engagement among diverse stakeholders.

Problem-solving warm-up activities

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process. Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute.

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out #team #opening #closing #hyperisland #remote-friendly Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

Doodling Together

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start.

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems.

Doodling Together #collaboration #creativity #teamwork #fun #team #visual methods #energiser #icebreaker #remote-friendly Create wild, weird and often funny postcards together & establish a group’s creative confidence.

Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team!

Show and Tell #gamestorming #action #opening #meeting facilitation Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible.

Constellations #trust #connection #opening #coaching #patterns #system Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic.

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree #thiagi #opening #perspectives #remote-friendly With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Closing activities for a problem-solving process

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round.

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them.

One breath feedback #closing #feedback #action This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

Who What When Matrix

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward.

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved.

Who/What/When Matrix #gamestorming #action #project planning With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out!

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised.

Response Cards #debriefing #closing #structured sharing #questions and answers #thiagi #action It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Tips for effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.

Try different approaches

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Create psychologically safe spaces for discussion

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner.

It can be tough for people to stand up and contribute if the problems or challenges are emotive or personal in nature. Try and create a psychologically safe space for these kinds of discussions and where possible, create regular opportunities for challenges to be brought up organically.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

Save time and effort creating an effective problem solving process

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks to build your agenda. When you make changes or update your agenda, your session timing adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore how to use SessionLab to design effective problem solving workshops or watch this five minute video to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you!

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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The Art of Effective Problem Solving: A Step-by-Step Guide

Author: Daniel Croft

Daniel Croft is an experienced continuous improvement manager with a Lean Six Sigma Black Belt and a Bachelor's degree in Business Management. With more than ten years of experience applying his skills across various industries, Daniel specializes in optimizing processes and improving efficiency. His approach combines practical experience with a deep understanding of business fundamentals to drive meaningful change.

Whether we realise it or not, problem solving skills are an important part of our daily lives. From resolving a minor annoyance at home to tackling complex business challenges at work, our ability to solve problems has a significant impact on our success and happiness. However, not everyone is naturally gifted at problem-solving, and even those who are can always improve their skills. In this blog post, we will go over the art of effective problem-solving step by step.

You will learn how to define a problem, gather information, assess alternatives, and implement a solution, all while honing your critical thinking and creative problem-solving skills. Whether you’re a seasoned problem solver or just getting started, this guide will arm you with the knowledge and tools you need to face any challenge with confidence. So let’s get started!

Problem Solving Methodologies

Individuals and organisations can use a variety of problem-solving methodologies to address complex challenges. 8D and A3 problem solving techniques are two popular methodologies in the Lean Six Sigma framework.

Methodology of 8D (Eight Discipline) Problem Solving:

The 8D problem solving methodology is a systematic, team-based approach to problem solving. It is a method that guides a team through eight distinct steps to solve a problem in a systematic and comprehensive manner.

The 8D process consists of the following steps:

Form a team: Assemble a group of people who have the necessary expertise to work on the problem.
Define the issue: Clearly identify and define the problem, including the root cause and the customer impact.
Create a temporary containment plan: Put in place a plan to lessen the impact of the problem until a permanent solution can be found.
Identify the root cause: To identify the underlying causes of the problem, use root cause analysis techniques such as Fishbone diagrams and Pareto charts.
Create and test long-term corrective actions: Create and test a long-term solution to eliminate the root cause of the problem.
Implement and validate the permanent solution: Implement and validate the permanent solution’s effectiveness.
Prevent recurrence: Put in place measures to keep the problem from recurring.
Recognize and reward the team: Recognize and reward the team for its efforts.

Download the 8D Problem Solving Template

A3 Problem Solving Method:

The A3 problem solving technique is a visual, team-based problem-solving approach that is frequently used in Lean Six Sigma projects. The A3 report is a one-page document that clearly and concisely outlines the problem, root cause analysis, and proposed solution.

The A3 problem-solving procedure consists of the following steps:

Determine the issue: Define the issue clearly, including its impact on the customer.
Perform root cause analysis: Identify the underlying causes of the problem using root cause analysis techniques.
Create and implement a solution: Create and implement a solution that addresses the problem’s root cause.
Monitor and improve the solution: Keep an eye on the solution’s effectiveness and make any necessary changes.

Subsequently, in the Lean Six Sigma framework, the 8D and A3 problem solving methodologies are two popular approaches to problem solving. Both methodologies provide a structured, team-based problem-solving approach that guides individuals through a comprehensive and systematic process of identifying, analysing, and resolving problems in an effective and efficient manner.

Step 1 – Define the Problem

The definition of the problem is the first step in effective problem solving. This may appear to be a simple task, but it is actually quite difficult. This is because problems are frequently complex and multi-layered, making it easy to confuse symptoms with the underlying cause. To avoid this pitfall, it is critical to thoroughly understand the problem.

To begin, ask yourself some clarifying questions:

What exactly is the issue?
What are the problem’s symptoms or consequences?
Who or what is impacted by the issue?
When and where does the issue arise?

Answering these questions will assist you in determining the scope of the problem. However, simply describing the problem is not always sufficient; you must also identify the root cause. The root cause is the underlying cause of the problem and is usually the key to resolving it permanently.

Try asking “why” questions to find the root cause:

What causes the problem?
Why does it continue?
Why does it have the effects that it does?

By repeatedly asking “ why ,” you’ll eventually get to the bottom of the problem. This is an important step in the problem-solving process because it ensures that you’re dealing with the root cause rather than just the symptoms.

Once you have a firm grasp on the issue, it is time to divide it into smaller, more manageable chunks. This makes tackling the problem easier and reduces the risk of becoming overwhelmed. For example, if you’re attempting to solve a complex business problem, you might divide it into smaller components like market research, product development, and sales strategies.

To summarise step 1, defining the problem is an important first step in effective problem-solving. You will be able to identify the root cause and break it down into manageable parts if you take the time to thoroughly understand the problem. This will prepare you for the next step in the problem-solving process, which is gathering information and brainstorming ideas.

Step 2 – Gather Information and Brainstorm Ideas

Gathering information and brainstorming ideas is the next step in effective problem solving. This entails researching the problem and relevant information, collaborating with others, and coming up with a variety of potential solutions. This increases your chances of finding the best solution to the problem.

Begin by researching the problem and relevant information. This could include reading articles, conducting surveys, or consulting with experts. The goal is to collect as much information as possible in order to better understand the problem and possible solutions.

Next, work with others to gather a variety of perspectives. Brainstorming with others can be an excellent way to come up with new and creative ideas. Encourage everyone to share their thoughts and ideas when working in a group, and make an effort to actively listen to what others have to say. Be open to new and unconventional ideas and resist the urge to dismiss them too quickly.

Finally, use brainstorming to generate a wide range of potential solutions. This is the place where you can let your imagination run wild. At this stage, don’t worry about the feasibility or practicality of the solutions; instead, focus on generating as many ideas as possible. Write down everything that comes to mind, no matter how ridiculous or unusual it may appear. This can be done individually or in groups.

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the next step in the problem-solving process, which we’ll go over in greater detail in the following section.

Step 3 – Evaluate Options and Choose the Best Solution

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the third step in effective problem solving, and it entails weighing the advantages and disadvantages of each solution, considering their feasibility and practicability, and selecting the solution that is most likely to solve the problem effectively.

To begin, weigh the advantages and disadvantages of each solution. This will assist you in determining the potential outcomes of each solution and deciding which is the best option. For example, a quick and easy solution may not be the most effective in the long run, whereas a more complex and time-consuming solution may be more effective in solving the problem in the long run.

Consider each solution’s feasibility and practicability. Consider the following:

Can the solution be implemented within the available resources, time, and budget?
What are the possible barriers to implementing the solution?
Is the solution feasible in today’s political, economic, and social environment?

You’ll be able to tell which solutions are likely to succeed and which aren’t by assessing their feasibility and practicability.

Finally, choose the solution that is most likely to effectively solve the problem. This solution should be based on the criteria you’ve established, such as the advantages and disadvantages of each solution, their feasibility and practicability, and your overall goals.

It is critical to remember that there is no one-size-fits-all solution to problems. What is effective for one person or situation may not be effective for another. This is why it is critical to consider a wide range of solutions and evaluate each one based on its ability to effectively solve the problem.

Step 4 – Implement and Monitor the Solution

Communication the missing peice from Lean Six Sigma - Learnleansigma

When you’ve decided on the best solution, it’s time to put it into action. The fourth and final step in effective problem solving is to put the solution into action, monitor its progress, and make any necessary adjustments.

To begin, implement the solution. This may entail delegating tasks, developing a strategy, and allocating resources. Ascertain that everyone involved understands their role and responsibilities in the solution’s implementation.

Next, keep an eye on the solution’s progress. This may entail scheduling regular check-ins, tracking metrics, and soliciting feedback from others. You will be able to identify any potential roadblocks and make any necessary adjustments in a timely manner if you monitor the progress of the solution.

Finally, make any necessary modifications to the solution. This could entail changing the solution, altering the plan of action, or delegating different tasks. Be willing to make changes if they will improve the solution or help it solve the problem more effectively.

It’s important to remember that problem solving is an iterative process, and there may be times when you need to start from scratch. This is especially true if the initial solution does not effectively solve the problem. In these situations, it’s critical to be adaptable and flexible and to keep trying new solutions until you find the one that works best.

To summarise, effective problem solving is a critical skill that can assist individuals and organisations in overcoming challenges and achieving their objectives. Effective problem solving consists of four key steps: defining the problem, generating potential solutions, evaluating alternatives and selecting the best solution, and implementing the solution.

You can increase your chances of success in problem solving by following these steps and considering factors such as the pros and cons of each solution, their feasibility and practicability, and making any necessary adjustments. Furthermore, keep in mind that problem solving is an iterative process, and there may be times when you need to go back to the beginning and restart. Maintain your adaptability and try new solutions until you find the one that works best for you.

Novick, L.R. and Bassok, M., 2005. Problem Solving . Cambridge University Press.

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Daniel Croft

Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website www.learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

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Newsom, lawmakers use cuts, reserves and ‘fiscal emergency’ declaration to solve budget deficit

Gavin Newsom speaks in front of a sign that says "The California Blueprint"

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Gov. Gavin Newsom and Democratic lawmakers struck a deal Saturday to make $16 billion in cuts, declare a statewide fiscal emergency and pull money from the state’s rainy-day reserves to balance a $46.8-billion budget deficit.

The agreement for a $297.7-billion spending plan is the result of weeks of contentious negotiations with labor unions and business interests after weaker-than-anticipated revenue forced Newsom and lawmakers to scale back California’s progressive policy agenda. The shortfall inspired a tug-of-war over coveted state dollars that has caused rifts between the governor and some of his closest allies at the Capitol.

Among the more high-profile changes, the 2024-25 budget plan delays a minimum wage increase for healthcare workers until at least October, cuts $1.1 billion for affordable housing and slashes $750 million in funding for the state prison system.

California’s business community also took a hit, with the three-year suspension of nearly $15 billion in tax breaks a year earlier than Newsom initially proposed.

“This agreement sets the state on a path for long-term fiscal stability — addressing the current shortfall and strengthening budget resilience down the road,” Newsom said in statement. “We’re making sure to preserve programs that serve millions of Californians, including key funding for education, health care, expanded behavioral health services and combating homelessness.”

The deficit marks a dramatic reversal of California’s financial standing from a projected $100-billion surplus two years ago and creates a challenging political narrative for Newsom, who often boasts of the state being an essential economic engine for the nation.

The governor is required by law to declare a statewide budget emergency before he can take money from the reserves to address the deficit. But an emergency declaration gives fodder to critics who have accused Democrats of mismanaging the state’s finances and overspending.

Despite the shortfall, the California economy remains strong, and the state has more revenue to spend than it did when Newsom took office.

“This is not a revenue problem,” said David Crane, president of Govern for California, a nonprofit that seeks to oppose the influence of labor unions on state government. “The deficit is a result of expenditures.”

In April, Newsom touted the fact that the California economy held its position as the fifth largest in the world , saying the state “continues to punch above its weight.”

The state government’s financial problem can be blamed in part on poor revenue projections that led Newsom and lawmakers to allocate more money for programs than they had available to spend.

The state’s progressive tax structure leaves government dependent on revenue from income taxes paid by chief executives and other top earners, which are subject to stock-market fluctuations and difficult to predict. Additionally, the delay of the 2022 tax filing deadline from April to November forced California leaders to craft the current budget without a full understanding of how much state tax revenue had declined.

Newsom anticipated California’s deficit to grow when he signed the budget last year and said he dedicated much of the new money in his spending plan to one-time funding increases that he could easily halt if revenue were to fall. The cuts include $500 million for a loan program to fund affordable student housing at colleges and a reduction of $485 million for work-study programs.

Yet the governor and lawmakers have been criticized for choosing to pull money from the state’s rainy-day fund — $5.1 billion in 2024-25 and $7.1 billion planned the following year — to avoid deeper cuts. Democrats also plan to take $900 million from a safety-net reserve account next year.

Tapping into the state’s piggy bank now has raised concerns about what could happen to programs serving California’s neediest if the economy falls into recession and revenue drops even lower.

Democrats at the Capitol released a broad overview of some of the cuts the Legislature will vote on next week, before the budget takes effect July 1.

Newsom and lawmakers said the agreement includes proposed legislation requiring the state, in the future, to set aside surplus funds for subsequent budget years as a means to protect against revenue swings and a constitutional amendment in 2026 to increase the rainy-day fund. Details were not shared with the announcement.

Here’s what we know so far about the agreement.

Pushing off a healthcare minimum wage hike

Newsom signed a bill into law last year to give healthcare workers a minimum wage increase to $25 per hour. He waited a few weeks to explain that he wouldn’t allow the law to take effect if the state budget crisis worsened .

At the time, the Department of Finance estimated that the law could cost the state $2 billion. Labor unions said the cost was closer to $300 million, if the state required hospitals to cover much of the expenditure.

Newsom’s concerns, which he said he shared with unions before he signed the law, set off months of private negotiations over when to raise wages and how to pay for the increase.

Those talks ended with the budget agreement, which delays the pay hike from taking effect until Oct. 15 at the earliest, instead of this month, as originally planned.

The start date for the pay hike hinges on one of two scenarios: state revenue in the first quarter of the fiscal year coming in 3% above projections, or more federal funding for hospitals through a quality-assurance fee. If neither happens, the increase could be delayed beyond October.

Lawmakers and the governor are essentially using the quality-assurance fee as a mechanism to ensure that hospitals can pay for the increase. Hospitals pay quality-assurance fees; the federal government matches the money, then remits the funding back to hospitals.

The federal increase requested by the state is expected to cover 30% of the cost of higher wages for hospitals.

The budget pegs the state’s cost for the program at $600 million in 2024-25.

Dave Regan, president of SEIU United Healthcare Workers West, said “workers are disappointed” that the wage hike won’t kick in this summer but recognized that legislative leaders and Newsom listened to the union’s concerns despite the shortfall.

California Gov. Gavin Newsom leaves the stage after delivering his budget proposal in Sacramento, Calif., Tuesday, Jan. 10, 2023. California faces a projected budget deficit of $22.5 billion for the coming fiscal year, Newsom announced Tuesday, just days into his second term. It’s a sharp turnaround from last year’s $98 billion surplus. (AP Photo/José Luis Villegas)

As deficit estimate hits $68 billion, Newsom seeks ‘major changes’ to healthcare wage law

Gov. Gavin Newsom said his staff has been working with Democrats in the Legislature on the state’s healthcare minimum wage law in light of budget concerns.

Dec. 7, 2023

No solution on battle over MCO tax

The question of how to use the proceeds of a tax on managed care organizations, known as the MCO tax, turned out to be the most difficult to answer in budget negotiations — so challenging, in fact, that talks fizzled out, and Newsom threatened to oppose a ballot measure backed by some of his closest allies.

The tax applies to health insurance providers that charge fixed monthly payments for services and acts as a mechanism to allow California to collect billions in additional federal funds for Medi-Cal, the healthcare system for low-income residents.

Newsom and lawmakers renewed the tax last June and agreed to use some of the proceeds to raise reimbursement rates to providers who serve Medi-Cal patients. For years, doctors have waged an unsuccessful campaign to raise rates, arguing that the reimbursements are too low, result in a shortage of doctors willing to accept patients and restrict access to care.

But Newsom reversed course and proposed taking more than $6 billion from the Medi-Cal rate increases over multiple years and using the funding instead to avoid cuts to the program.

The change pitted Newsom against a coalition led by the California Medical Assn. and Planned Parenthood, two groups that have supported the governor’s causes and backed his campaigns.

The coalition called for the governor to stick to the agreement he made in 2023 to raise rates for providers. It also is leading a charge to pass a measure on the 2024 ballot that would permanently establish an MCO tax to fund higher reimbursement rates and other Medi-Cal services.

The governor wants the coalition to take the measure off the ballot. He wants the funds to be flexible so the state can use the money if necessary to support the Medi-Cal system in the future.

The coalition has so far declined to take the measure off the ballot, fearing that Democrats would divert the funding again. The talks ended in a stalemate.

The final state budget includes $6.9 billion next year to support the Medi-Cal system.

Newsom and lawmakers agreed to offer a smaller pot of money for “provider rate increases and investments” from the MCO tax, but far less money than was previously set aside. The budget includes $133 million in 2024-25 and a plan to raise that to $728 million in 2025-26 and $1.2 billion the following year.

Democrats said the MCO funding would become “inoperable,” essentially eliminated, if the measure is approved on the 2024 ballot.

Jodi Hicks, chief executive of Planned Parenthood Affiliates of California, said the tenets of the ballot initiative were negotiated last year with safety-net providers and policymakers. The coalition’s measure, if approved in November, would provide the state $2 billion for general Medi-Cal costs, outside of rate increases, annually in 2025 and 2026.

“The intent of the initiative was to make historic investments in the Medi-Cal program and provide for a generational change for patients and how they experience access to care,” Hicks said.

The governor threatened to campaign against the measure as the budget talks soured this year, setting up the possibility that he could challenge his supporters in the November election.

California Gov. Gavin Newsom discusses some of the ways that have been used to reduce the state's budget deficit in his revised 2024-25 state budget unveiled during a news conference in Sacramento, Calif., Friday, May 10, 2024. (AP Photo/Rich Pedroncelli)

Secret negotiations heat up in Sacramento as budget, ballot deadlines loom

California officials must finalize the state budget and the list of measures that will go on the ballot by the end of June. Secret budget conversations are intertwined with deal-making around the 2024 ballot.

June 17, 2024

A pause on business tax breaks

The budget deal limits total tax credits for businesses in the state to $5 million per filer and pauses a net operating loss tax deduction for businesses with income of more than $1 million in 2024, 2025 and 2026.

In a concession to the business community, Newsom and lawmakers are allowing companies to receive refunds for the tax credits after the limits end.

Newsom originally proposed halting and capping the tax breaks beginning in 2025. But Democrats in the Legislature pushed to apply the changes a year earlier, allowing them to avoid cuts to other programs.

The administration said the changes to the tax breaks will increase revenue by nearly $15 billion through 2026.

The early start could hurt businesses who were planning to deduct losses from their 2024 taxes and now have to scramble to scale back on employees or inventory to cover the cost of an unexpectedly higher bill. The limit also marks the second time in five years that the state has capped tax credits, which could turn away companies that operate in California.

Big cut to prisons

Lawmakers previously proposed an additional $1 billion in cuts to the Department of Corrections and Rehabilitation, including at least $12 million in reductions to the governor’s project to transform San Quentin . Newsom’s proposed cuts had included $80.6 million in savings from the newly announced deactivation of 46 housing units at 13 state prisons.

The final agreement drops funding for corrections by $750 million through next budget year, which includes $400 million from Newsom’s plan to cut state operations and eliminate vacant jobs. Overall, the budget reductions to operations and personnel across all of state government saves nearly $2.2 billion.

From left, Gov. Gavin Newsom; Sen. Mike McGuire; and Assembly Speaker Robert Rivas

With deadline nearing, Newsom and lawmakers disagree over solutions to California budget crisis

Gov. Gavin Newsom and Democratic lawmakers work to settle their differences over a few key budget cuts to reduce California’s massive deficit.

June 5, 2024

Newsom supports another round of homelessness grants

In late May, Democrats in the Legislature proposed spending $1 billion more than the governor had budgeted on a sixth round of Homeless Housing, Assistance and Prevention grants to local governments. At the same time, lawmakers proposed cutting $100 million in funding to clean up homeless encampments in the current budget year.

The final budget deal appears to show a compromise.

The deal includes $1 billion in additional homelessness grants, which the governor and lawmakers said would be tied to new accountability measures to make sure local governments use the funding appropriately. The agreement also provides $150 million next year for encampment grants.

Broadband internet access for all — a little later

The COVID-19 pandemic exposed the need to improve access to broadband internet in homes across California when K-12 education shifted from the classroom to remote learning. Low-income families and those who live in rural areas often lack the same level of connectivity as more wealthy communities.

Newsom has sought to make internet access more equitable under a “broadband for all” initiative.

The spending plan delays $550 million in funding for “last mile” work, which connects the network to homes, until the 2027 budget year. The budget agreement still offers $250 million next year for a program to expand and improve the fiber-optic network under “middle-mile” projects, and Democrats intend to provide a total of $2 billion for last-mile work over multiple years.

A funding delay for public schools

Under Proposition 98, approved by voters in 1988, California has a minimum funding guarantee for schools and community colleges.

This year, Newsom proposed an unusual maneuver to re-characterize funding in 2022-23 to reflect the lower-than-expected state revenue.

The California Teachers Assn. said the change would have reduced funding for schools by about $12 billion over two years. The union ran a television ad criticizing Newsom’s proposal to pressure him to reverse course.

Newsom and teachers agreed late last month to a complicated solution that suspends the minimum funding guarantee and delays $5.5 billion in funding until future years.

Newsom’s budget plan saves vital programs for immigrants, but kids and hungry seniors may suffer

June 26, 2024

Gov. Gavin Newsom with three budget chart lines going up and down.

Can California control its boom-and-bust budget?

June 11, 2024

FILE - California Gov. Gavin Newsom answers a reporters question about his revised 2024-25 state budget during a news conference in Sacramento, Calif., Friday, May 10, 2024. The California Teachers Association has criticized Newsom's budget proposal, saying it would wreak havoc on school funding. (AP Photo/Rich Pedroncelli, File)

Teachers criticize Newsom’s budget proposal, saying it would ‘wreak havoc on funding for our schools’

May 17, 2024

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Taryn Luna covers Gov. Gavin Newsom and California politics in Sacramento for the Los Angeles Times.

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In this article, we attempt to provide the numerical solution for a non-linear reaction--advection diffusion equation with fractional-order space-time derivatives in a finite domain. In the proposed scheme, time fractional derivative in Caputo sense is approximated by using the non-standard finite difference method and the fractional space derivative is specifically approximated by using Vieta–Fibonacci polynomials. These approximations generate a system of ordinary differential equations which is converted into an equivalent system of algebraic equations by using collocation method. Finally, the obtained system of algebraic equations is solved to find the dependent variables (unknowns) of the considered problem. The stability and convergence related to the time discreatization of this approach are also discussed. In this study, the effectiveness and precision of the proposed scheme are analyzed with the help of examples, and it is observed that the proposed scheme is sufficiently accurate and efficient technique. Also, the effects of fractional-order derivatives on concentration profiles are discussed.

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A numerical method for nonlinear fractional reaction–advection–diffusion equation with piecewise fractional derivative

An Efficient Non-standard Finite Difference Scheme for Solving Distributed Order Time Fractional Reaction–Diffusion Equation

Collocation Method for Time Fractional Diffusion Equation Based on the Chebyshev Polynomials of Second Kind

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Yang, Q., D. Chen, T. Zhao, and Y. Chen. 2016. Fractional calculus in image processing: A review. Fractional Calculus and Applied Analysis 19(5): 1222–1249.

Article MathSciNet Google Scholar

Odzijewicz, T., A.B. Malinowska, and D.F. Torres. 2012. Fractional calculus of variations in terms of a generalized fractional integral with applications to physics. In Abstract and Applied Analysis , vol. 2012 Hindawi.

Tarasov, V.E. 2019. On history of mathematical economics: Application of fractional calculus. Mathematics 7(6): 509.

Article Google Scholar

Tejado, I., E. Pérez, and D. Valério. 2019. Fractional calculus in economic growth modelling of the group of seven. Fractional Calculus and Applied Analysis 22(1): 139–157.

Prajapati, J., A. Patel, K. Pathak, and A. Shukla. 2012. Fractional calculus approach in the study of instability phenomenon in fluid dynamics. Palestine Journal of Mathematics 1(2): 95–103.

MathSciNet Google Scholar

Matušu, R. 2011. Application of fractional order calculus to control theory. International Journal of Mathematical Models and Methods in Applied Sciences 5(7): 1162–1169.

Google Scholar

Yang, X.-J., D. Baleanu, and H. Srivastava. 2022. Advanced analysis of local fractional calculus applied to the rice theory in fractal fracture mechanics. In Methods of Mathematical Modelling and Computation for Complex Systems , vol. 373, 105–133. Springer.

Baleanu, D., F.A. Ghassabzade, J.J. Nieto, and A. Jajarmi. 2022. On a new and generalized fractional model for a real cholera outbreak. Alexandria Engineering Journal 61(11): 9175–9186.

Defterli, O., D. Baleanu, A. Jajarmi, S.S. Sajjadi, A. Noorhan, and J.H. Asad. 2022. Fractional treatment: an accelerated mass-spring system. Romanian Reports in Physics 74: 122.

Abbas, S., M. Benchohra, J.E. Lazreg, J.J. Nieto, and Y. Zhou. 2023. Fractional Differential Equations and Inclusions: Classical and Advanced Topics . Singapore: World Scientific.

Jajarmi, A., D. Baleanu, S.S. Sajjadi, and J.J. Nieto. 2022. Analysis and some applications of a regularized $\psi$ -Hilfer fractional derivative. Journal of Computational and Applied Mathematics 415: 114476.

Deng, Z.-Q., V.P. Singh, and L. Bengtsson. 2004. Numerical solution of fractional advection–dispersion equation. Journal of Hydraulic Engineering 130(5): 422–431.

Jiang, W., and Y. Lin. 2010. Approximate solution of the fractional advection–dispersion equation. Computer Physics Communications 181(3): 557–561.

Lin, R., F. Liu, V. Anh, and I. Turner. 2009. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Applied Mathematics and Computation 212(2): 435–445.

Pandey, P., S. Kumar, and S. Das. 2019. Approximate analytical solution of coupled fractional order reaction–advection–diffusion equations. The European Physical Journal Plus 134(7): 364.

Jannelli, A., M. Ruggieri, and M.P. Speciale. 2019. Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation. Communications in Nonlinear Science and Numerical Simulation 70: 89–101.

Ginting, V., and Y. Li. 2019. On the fractional diffusion–advection–reaction equation in R. Fractional Calculus and Applied Analysis 22(4): 1039–1062.

Singh, A., and S. Das. 2019. Study and analysis of spatial-time nonlinear fractional-order reaction–advection–diffusion equation. Journal of Porous Media 22(7): 787–798.

Yu, Q., F. Liu, V. Anh, and I. Turner. 2008. Solving linear and non-linear space-time fractional reaction–diffusion equations by the adomian decomposition method. International Journal for Numerical Methods in Engineering 74(1): 138–158.

Ateş, I., and P. Zegeling. 2017. A homotopy perturbation method for fractional-order advection–diffusion–reaction boundary-value problems. Applied Mathematical Modelling 47: 425–441.

Fazio, R., A. Jannelli, and S. Agreste. 2018. A finite difference method on non-uniform meshes for time-fractional advection–diffusion equations with a source term. Applied Sciences 8(6): 960.

Liu, F., P. Zhuang, V. Anh, I. Turner, and K. Burrage. 2007. Stability and convergence of the difference methods for the space-time fractional advection–diffusion equation. Applied Mathematics and Computation 191(1): 12–20.

Lin, Y., and C. Xu. 2007. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics 225(2): 1533–1552.

Zhuang, P., Y. Gu, F. Liu, I. Turner, and P. Yarlagadda. 2011. Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method. International Journal for Numerical Methods in Engineering 88(13): 1346–1362.

Singh, J., D. Kumar, and S.S. Gupta. 2012. Application of homotopy perturbation transform method to linear and non-linear space-time fractional reaction–diffusion equations. The Journal of Mathematics and Computer Science 1: 40–52.

Momani, S., and A. Yıldırım. 2010. Analytical approximate solutions of the fractional convection–diffusion equation with nonlinear source term by He’s homotopy perturbation method. International Journal of Computer Mathematics 87(5): 1057–1065.

Dwivedi, K.D., and S. Das. 2019. Numerical solution of the nonlinear diffusion equation by using non-standard/standard finite difference and Fibonacci collocation methods. The European Physical Journal Plus 134(12): 608.

Heydari, M., and A. Atangana. 2023. A numerical method for nonlinear fractional reaction–advection–diffusion equation with piecewise fractional derivative. Mathematical Sciences 17: 169–181.

Sweilam, N.H., A.A.E. El-Sayed, and S. Boulaaras. 2021. Fractional-order advection–dispersion problem solution via the spectral collocation method and the non-standard finite difference technique. Chaos, Solitons & Fractals 144: 110736.

Abeye, N., M. Ayalew, D. Suthar, S. Purohit, and K. Jangid. 2022. Numerical solution of unsteady state fractional advection–dispersion equation. Arab Journal of Basic and Applied Sciences 29(1): 77–85.

Sharma, R., and Rajeev, 2023. An operational matrix approach to solve a 2D variable-order reaction advection diffusion equation with Vieta-Fibonacci polynomials. Special Topics & Reviews in Porous Media: An International Journal 14(5): 79–96.

Sharma, R., and Rajeev 2023. A numerical approach to solve 2D fractional RADE of variable-order with Vieta–Lucas polynomials. Chinese Journal of Physics 86: 433–446.

Biswas, C., S. Das, A. Singh, and M. Chopra. 2023. Solution of fractional-order reaction–advection–diffusion equation arising in porous media. Journal of Porous Media 26(1): 15–29.

Mickens, R.E. 2000. Applications of Nonstandard Finite Difference Schemes . Singapore: World Scientific.

Agarwal, P., and A. El-Sayed. 2018. Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Physica A: Statistical Mechanics and Its Applications 500: 40–49.

Uddin, M., and S. Haq. 2011. RBFS approximation method for time fractional partial differential equations. Communications in Nonlinear Science and Numerical Simulation 16(11): 4208–4214.

Tajadodi, H. 2020. A numerical approach of fractional advection–diffusion equation with Atangana–Baleanu derivative. Chaos, Solitons & Fractals 130: 109527.

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Acknowledgements

The authors would like to thank the reviewers for their valuable comments, which have helped to improve the quality of this paper.

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Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

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Rashmi Sharma: Conceptualization, Methodology, Writing-Original draft preparation, Software, Supervision. Rajeev: Conceptualization, Methodology, Supervision. All the authors read and approved the final manuscript.

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Sharma, R., Rajeev A numerical approach based on Vieta–Fibonacci polynomials to solve fractional order advection–reaction diffusion problem. J Anal (2024). https://doi.org/10.1007/s41478-024-00804-6

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Received : 01 February 2024

Accepted : 07 June 2024

Published : 25 June 2024

DOI : https://doi.org/10.1007/s41478-024-00804-6

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Title: asymptotic stability of the equilibrium for the free boundary problem of a compressible atmospheric primitive model with physical vacuum.

Abstract: This paper concerns the large time asymptotic behavior of solutions to the free boundary problem of the compressible primitive equations in atmospheric dynamics with physical vacuum. Up to second order of the perturbations of an equilibrium, we have introduced a model of the compressible primitive equations with a specific viscosity and shown that the physical vacuum free boundary problem for this model system has a global-in-time solution converging to an equilibrium exponentially, provided that the initial data is a small perturbation of the equilibrium. More precisely, we introduce a new coordinate system by choosing the enthalpy (the square of sound speed) as the vertical coordinate, and thanks to the hydrostatic balance, the degenerate density at the free boundary admits a representation with separation of variables in the new coordinates. Such a property allows us to establish horizontal derivative estimates without involving the singular vertical derivative of the density profile, which plays a key role in our analysis.

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I recently developed a guide to Progressive Lightmapper troubleshooting in order to help developers get the most out of Baked Global Illumination (GI) in the Unity Editor. Here, I unpack five of the most common lightmapping problems and their solutions, supported by images and links to pages in the Unity Manual . For the full guide, visit the forums .

Cornell Box scene rendered with no Global Illumination (left) and with Baked Global Illumination (right)

If certain prerequisites are not met, the Progressive Lightmapper might fail to generate lighting in your scene. These conditions include, but are not limited to:

No objects marked as GI Contributors
No baked lights in the scene
Shader issues

To correct this, I recommend trying one of the fixes laid out below.

Contribute GI settings under the Mesh Renderer component.

Mark objects you want to lightmap as GI Contributors by following these steps:

Select your GameObject .

Navigate to the Mesh Renderer component .

Unfold the Lighting header.

Check the Contribute to Global Illumination checkbox.

Doing so will enable the Receive Global Illumination parameter underneath. It contains two options:

Lightmaps : Meant for static lightmapped objects – GameObject will receive and contribute GI to lightmaps.
Light Probes : Meant for small props and objects not fit for lightmapping – GameObject will receive GI from Light Probes and will contribute GI to the surrounding lightmaps.

Only Mixed and Baked lights can contribute to Baked GI. Select lights in your scene, then set the Mode to either Mixed or Baked in the Light component . Other properties that are worth checking include:

Color : Dark colors will have low or no GI contribution. Choose bright colors for lights and use the Intensity property to boost or dim them.
Intensity : The higher the intensity, the brighter the light. Ensure that your lights are bright enough for meaningful GI contribution.
Indirect Multiplier : This property controls the intensity of the indirect bounce. Make sure that it is not set to zero. Otherwise, the light will have no contribution to GI at all. Note that setting this value above one will make the lighting in your scene non-compliant to the physically based rendering (PBR) standard.
Visit No Baked Global Illumination in the scene forum

In the Lighting window (accessible via Window > Rendering > Lighting ), make sure that the Lighting Settings Asset field is not left blank. If there is no asset assigned, click on the New Lighting Settings button. This will create and assign an asset, unlocking the properties in the window for editing.

Once this is complete, verify that:

You have ticked the Baked Global Illumination checkbox, which enables Baked GI computations. This checkbox will also expose the Lighting Mode drop-down list.
The Max Bounces value is not set to zero. The higher this value, the more the light will bounce around the environment.
The Indirect Intensity slider is not set to zero. Setting this slider to zero will diminish all indirect lighting in the scene.

Custom shaders could be the reason why GI computation has failed. For debug purposes, use the built-in shaders that come with the Unity Editor. Those are:

Standard Shader : Available in the Built-in Render Pipeline
Lit Shader : Available in the Universal Render Pipeline (URP)
Lit Shader : Available in the High Definition Render Pipeline (HDRP)

If Unity generates lighting after switching to one of the shaders outlined above, the problem might be with the custom shaders . In this case, make sure that the surface shaders contain the LIGHTMAP_ON shader keyword.

Check out the Meta Pass page for details on how to further customize the Baked GI output using shaders.

Other potential fixes

If the above steps have not solved your problem, consider trying these potential fixes:

Select a different lightmapping backend in the Lighting window . If lighting fails to bake when using the Progressive GPU, but succeeds when baking with the Progressive CPU, this might be the result of a hardware or driver problem.
Update the GPU drivers. Please refer to the GPU manufacturer’s page for the correct drivers for your system. (For Linux machines, check the Linux driver setup section in this forum thread ).
Ensure that your GPU meets the minimum requirements. Please refer to this forum thread .
Clear the GI Cache. To clear it, navigate to Preferences > GI Cache and click on the Clean Cache button. Keep in mind that this will delete all lighting data present in the scene, requiring you to regenerate lighting.

Certain objects that appear unlit or out of place might indicate a problem with the scene setup, which often reproduces when dynamic objects have no Light Probes to sample the lighting from. Furthermore, any glossy metallic material in the scene might appear as black if no local Reflection Probes are present.

Cornell Box scene with a non-lightmapped metallic sphere and a statue: Both objects are black when no Light Probes or Reflection Probes are present (left). Diffuse statue is visible when lit by Light Probes (center). Both the reflective sphere and the statue are visible after placing Light Probes and Reflection Probes (right).

Dynamic objects – or GI Contributors receiving GI from Light Probes – need Light Probes to sample indirect lighting data. If none are present, objects will fall back to sampling the Ambient Probe (i.e., the Light and Reflection Probe that is always present in the scene).

To mitigate this, set up a Light Probe network in the scene, adding more probes in areas of high importance. Make sure that there are enough Light Probes to encompass all affected objects, and generate lighting again to see the effect.

Reflective metallic objects might still render as black, even after placing a dense network of Light Probes. To shade such objects, you need to place a Reflection Probe that encompasses the affected object. Generate the lighting again or re-bake the probe in the Reflection Probe Component by clicking the Bake button.

If you observe black areas in the reflections, try increasing the Bounces count. This will increase the number of bounces, thus creating reflections within reflections. You can access this property in Lighting > Environment > Environment Lighting .

Visit Objects are missing lighting forum

If performing the previous steps still does not solve the issue, inspect the Mesh Renderer component of the affected object. Under the Probes section , make sure that the Light Probes and Reflection Properties are set to anything other than Off .

Light Probes and Reflection Probes properties need to be set to Blend Probes (or anything other than Off).

Pure black materials will absorb all direct and indirect light. This is physically correct behavior. In real life, no naturally occurring material is completely black. For example, one of the darkest natural materials, coal, measures at “50, 50, 50” on an RGB luminosity scale.

Adjust your material color values to follow the physically based shading standards. In the Built-in Render Pipeline, you can use the Validate Albedo Scene View Draw Mode to determine whether Albedo values are PBR-compliant. You can use the Rendering Debugger in URP and HDRP to do the same.

If you are working with multiple scenes , check that the scene containing lighting is set as the Active Scene . By default, Unity sets the first loaded scene as the Active Scene, which might have a detrimental effect in the standalone player builds.

The plane in the ceiling and the sphere both have emissive materials applied to them (left). Bloom is enabled as a post-processing effect to give the impression of a glowing material (center). Emissive plane and sphere are both marked as GI Contributors influencing lighting in the scene (right).

There are two types of issues related to emissive material rendering:

Emissive materials do not appear as “glowing,” which indicates a post-processing issue.
Emissive materials are not contributing to Global Illumination, which indicates an issue with object or material setup.

To correct either of these issues, I recommend trying one of the fixes laid out below.

To create the impression of a glowing material, enable Bloom in your post-processing stack of choice. Refer to Built-in RP , URP , or HDRP documentation for tips on how to do this.

If you intend on using emissive objects for lightmapping, make sure that:

You have marked the GameObject in question as a GI Contributor . Due to the self-illuminating nature of emissive objects, you can set their Receive Global Illumination property to Light Probes . This will save space in the lightmap atlas.
The Global Illumination property is set to Baked in the Material Inspector . This property is available under the Emission input. Refer to Built-in RP , URP , or HDRP documentation for more details.

Material Inspector in the Built-in Render Pipeline: We highlighted Emission properties in blue.

In the Lighting window , make sure that the Indirect Intensity property is not set to zero. Setting it to zero will disable all indirect lighting, including baked contribution from baked emissive objects.

Visit Emissive materials not rendering forum

When baking in Non-directional mode, the Unity Editor will not create a separate texture to hold directionality information. This will result in objects looking flat after baking.

It is worth noting that low frequency normal maps are hard to capture using directionality textures. Such textures will appear flat when generating lighting using fully baked lights.

Non-directional lightmaps with a baked light (left) vs Directional lightmaps with a baked light (right): Non-directional lightmaps lack the directionality of incoming light, which provides a good representation of relief when using normal maps.

In the Lighting window , set the Directional Mode property to Directional . This mode will generate a secondary texture that will store dominant light direction. Normal maps will have a good representation of relief, but will lack specular response.

Visit Flat normal maps Forum

Mixed lights provide real-time specular and normal response. The Progressive Lightmapper bakes indirect lighting into a lightmap. This combination ensures the highest quality material response when using baked lighting.

Directional lightmaps illuminated by a Baked spotlight (left) vs Directional lightmaps illuminated by a Mixed spotlight (right): Notice the improved contrast in the normal maps in the scene lit by the Mixed light.

If your project allows for it, switch the light Mode to Mixed in the Light component . Note that Mixed lights have the same performance cost as real-time lights. Depending on the Lighting Mode used, Mixed lights will cast real-time shadows but not baked soft shadows.

Probe-lit GameObjects will often have a better material response than those lit by Baked lights. If your art direction allows for it, set their Receive Global Illumination property to Light Probes in the Mesh Renderer component . Note that you can also use Light Probe Proxy Volume (LPPV) to add a spatial gradient to probe-lit objects.

A simple scene containing several Baked point lights before a lightmap bake (left) vs same scene after baking lightmaps (right): Notice that specular highlights are missing after baking.

One of the inherent limitations of Baked lights is that they do not provide real-time specular response to materials. This means that glossy materials will lack specular highlights after generating lighting.

Unlike Baked lights, Mixed lights provide real-time direct specular response to materials. If specular highlights are important in your scene, switch the light Mode to Mixed in the Light component .

Visit Missing specular response forum

It is possible to imitate specular response from lights by using emissive objects. To do so, follow these steps:

Place a Reflection Probe in your scene.

Right-click in the Hierarchy panel and select 3D Object > Sphere .

Select the newly created object and set its Static Editor Flag to Reflection Probe Static .

In the Project panel, create a new material by right-clicking and selecting Create > Material .

Select the newly created material and enable the Emission checkbox. Set the Global Illumination property to None .

Drag and drop the material onto the sphere to assign it.

Place the sphere in the same position as your light.

Generate lighting.

After following the above steps, you should be able to see the emissive objects captured in the Reflection Probe cubemap. You can hide those objects after baking or set up a Culling Mask in the Camera component .

Same scene but with emissive proxies captured by the Reflection Probe (left) vs an alternative perspective showcasing the placement of said proxies (right)

For more tips on troubleshooting the Progressive Lightmapper, check out the full guide in the forums . If you’d like to discuss this article or share other solutions, feel free to connect with me there or here . Finally, be sure to watch for new technical blogs from other Unity developers as part of the ongoing Tech from the Trenches series .

If we’re all so busy, why isn’t anything getting done?

Have you ever asked why it’s so difficult to get things done in business today—despite seemingly endless meetings and emails? Why it takes so long to make decisions—and even then not necessarily the right ones? You’re not the first to think there must be a better way. Many organizations address these problems by redesigning boxes and lines: who does what and who reports to whom. This exercise tends to focus almost obsessively on vertical command relationships and rarely solves for what, in our experience, is the underlying disease: the poor design and execution of collaborative interactions.

About the authors

This article is a collaborative effort by Aaron De Smet , Caitlin Hewes, Mengwei Luo, J.R. Maxwell , and Patrick Simon , representing views from McKinsey’s People & Organizational Performance Practice.

In our efforts to connect across our organizations, we’re drowning in real-time virtual interaction technology, from Zoom to Slack to Teams, plus group texting, WeChat, WhatsApp, and everything in between. There’s seemingly no excuse to not collaborate. The problem? Interacting is easier than ever, but true, productive, value-creating collaboration is not. And what’s more, where engagement is occurring, its quality is deteriorating. This wastes valuable resources, because every minute spent on a low-value interaction eats into time that could be used for important, creative, and powerful activities.

It’s no wonder a recent McKinsey survey found 80 percent of executives were considering or already implementing changes in meeting structure and cadence in response to the evolution in how people work due to the COVID-19 pandemic. Indeed, most executives say they frequently find themselves spending way too much time on pointless interactions that drain their energy and produce information overload.

Most executives say they frequently find themselves spending way too much time on pointless interactions.

Three critical collaborative interactions

What can be done? We’ve found it’s possible to quickly improve collaborative interactions by categorizing them by type and making a few shifts accordingly. We’ve observed three broad categories of collaborative interactions (exhibit):

Decision making, including complex or uncertain decisions (for example, investment decisions) and cross-cutting routine decisions (such as quarterly business reviews)
Creative solutions and coordination, including innovation sessions (for example, developing new products) and routine working sessions (such as daily check-ins)
Information sharing, including one-way communication (video, for instance) and two-way communication (such as town halls with Q&As)

Below we describe the key shifts required to improve each category of collaborative interaction, as well as tools you can use to pinpoint problems in the moment and take corrective action.

Decision making: Determining decision rights

When you’re told you’re “responsible” for a decision, does that mean you get to decide? What if you’re told you’re “accountable”? Do you cast the deciding vote, or does the person responsible? What about those who must be “consulted”? Sometimes they are told their input will be reflected in the final answer—can they veto a decision if they feel their input was not fully considered?

It’s no wonder one of the key factors for fast, high-quality decisions is to clarify exactly who makes them. Consider a success story at a renewable-energy company. To foster accountability and transparency, the company developed a 30-minute “role card” conversation for managers to have with their direct reports. As part of this conversation, managers explicitly laid out the decision rights and accountability metrics for each direct report. The result? Role clarity enabled easier navigation for employees, sped up decision making, and resulted in decisions that were much more customer focused.

How to define decision rights

We recommend a simple yet comprehensive approach for defining decision rights. We call it DARE, which stands for deciders, advisers, recommenders, and executors:

Deciders are the only ones with a vote (unlike the RACI model, which helps determine who is responsible, accountable, consulted, and informed). If the deciders get stuck, they should jointly agree on how to escalate the decision or figure out a way to move the process along, even if it means agreeing to “disagree and commit.”

Advisers have input and help shape the decision. They have an outsize voice in setting the context of the decision and have a big stake in its outcome—for example, it may affect their profit-and-loss statements—but they don’t get a vote.

Recommenders conduct the analyses, explore the alternatives, illuminate the pros and cons, and ultimately recommend a course of action to advisers and deciders. They see the day-to-day implications of the decision but also have no vote. Best-in-class recommenders offer multiple options and sometimes invite others to suggest more if doing so may lead to better outcomes. A common mistake of recommenders, though, is coming in with only one recommendation (often the status quo) and trying to convince everyone it’s the best path forward. In general, the more recommenders, the better the process—but not in the decision meeting itself.

Executers don’t give input but are deeply involved in implementing the decision. For speed, clarity, and alignment, executers need to be in the room when the decision is made so they can ask clarifying questions and spot flaws that might hinder implementation. Notably, the number of executers doesn’t necessarily depend on the importance of the decision. An M&A decision, for example, might have just two executors: the CFO and a business-unit head.

To make this shift, ensure everyone is crystal clear about who has a voice but no vote or veto. Our research indicates while it is often helpful to involve more people in decision making, not all of them should be deciders—in many cases, just one individual should be the decider (see sidebar “How to define decision rights”). Don’t underestimate the difficulty of implementing this. It often goes against our risk-averse instinct to ensure everyone is “happy” with a decision, particularly our superiors and major stakeholders. Executing and sustaining this change takes real courage and leadership.

Creative solutions and coordination: Open innovation

Routine working sessions are fairly straightforward. What many organizations struggle with is finding innovative ways to identify and drive toward solutions. How often do you tell your teams what to do versus empowering them to come up with solutions? While they may solve the immediate need to “get stuff done,” bureaucracies and micromanagement are a recipe for disaster. They slow down the organizational response to the market and customers, prevent leaders from focusing on strategic priorities, and harm employee engagement. Our research suggests key success factors in winning organizations are empowering employees and spending more time on high-quality coaching interactions.

How microenterprises empower employees to drive innovative solutions

Haier, a Chinese appliance maker, created more than 4,000 microenterprises (MEs) that share common approaches but operate independently. Haier has three types of microenterprises:

Market-facing MEs have roots in Haier’s legacy appliance business, reinvented for today’s customer-centric, web-enabled world. They are expected to grow revenue and profit ten times faster than the industry average.
Incubating MEs focus on emerging markets such as e-gaming or wrapping new business models around familiar products. They currently account for more than 10 percent of Haier’s market capitalization.
“Node” MEs sell market-facing ME products and services such as design, manufacturing, and human-resources support.

Take Haier. The Chinese appliance maker divided itself into more than 4,000 microenterprises with ten to 15 employees each, organized in an open ecosystem of users, inventors, and partners (see sidebar “How microenterprises empower employees to drive innovative solutions”). This shift turned employees into energetic entrepreneurs who were directly accountable for customers. Haier’s microenterprises are free to form and evolve with little central direction, but they share the same approach to target setting, internal contracting, and cross-unit coordination. Empowering employees to drive innovative solutions has taken the company from innovation-phobic to entrepreneurial at scale. Since 2015, revenue from Haier Smart Home, the company’s listed home-appliance business, has grown by more than 18 percent a year, topping 209 billion renminbi ($32 billion) in 2020. The company has also made a string of acquisitions, including the 2016 purchase of GE Appliances, with new ventures creating more than $2 billion in market value.

Empowering others doesn’t mean leaving them alone. Successful empowerment, counterintuitively, doesn’t mean leaving employees alone. Empowerment requires leaders to give employees both the tools and the right level of guidance and involvement. Leaders should play what we call the coach role: coaches don’t tell people what to do but instead provide guidance and guardrails and ensure accountability, while stepping back and allowing others to come up with solutions.

Haier was able to use a variety of tools—including objectives and key results (OKRs) and common problem statements—to foster an agile way of working across the enterprise that focuses innovative organizational energy on the most important topics. Not all companies can do this, and some will never be ready for enterprise agility. But every organization can take steps to improve the speed and quality of decisions made by empowered individuals.

Managers who are great coaches, for example, have typically benefited from years of investment by mentors, sponsors, and organizations. We think all organizations should do more to improve the coaching skills of managers and help them to create the space and time to coach teams, as opposed to filling out reports, presenting in meetings, and other activities that take time away from driving impact through the work of their teams.

But while great coaches take time to develop, something as simple as a daily stand-up or check-in can drive horizontal connectivity, creating the space for teams to understand what others are doing and where they need help to drive work forward without having to specifically task anyone in a hierarchical way. You may also consider how you are driving a focus on outcomes over activities on a near-term and long-term basis. Whether it’s OKRs or something else, how is your organization proactively communicating a focus on impact and results over tasks and activities? What do you measure? How is it tracked? How is the performance of your people and your teams managed against it? Over what time horizons?

The importance of psychological safety. As you start this journey, be sure to take a close look at psychological safety. If employees don’t feel psychologically safe, it will be nearly impossible for leaders and managers to break through disempowering behaviors like constant escalation, hiding problems or risks, and being afraid to ask questions—no matter how skilled they are as coaches.

Employers should be on the lookout for common problems indicating that significant challenges to psychological safety lurk underneath the surface. Consider asking yourself and your teams questions to test the degree of psychological safety you have cultivated: Do employees have space to bring up concerns or dissent? Do they feel that if they make a mistake it will be held against them? Do they feel they can take risks or ask for help? Do they feel others may undermine them? Do employees feel valued for their unique skills and talents? If the answer to any of these is not a clear-cut “yes,” the organization likely has room for improvement on psychological safety and relatedness as a foundation to high-quality interactions within and between teams.

Information sharing: Fit-for-purpose interactions

Do any of these scenarios sound familiar? You spend a significant amount of time in meetings every day but feel like nothing has been accomplished. You jump from one meeting to another and don’t get to think on your own until 7 p.m. You wonder why you need to attend a series of meetings where the same materials are presented over and over again. You’re exhausted.

An increasing number of organizations have begun to realize the urgency of driving ruthless meeting efficiency and of questioning whether meetings are truly required at all to share information. Live interactions can be useful for information sharing, particularly when there is an interpretive lens required to understand the information, when that information is particularly sensitive, or when leaders want to ensure there’s ample time to process it and ask questions. That said, most of us would say that most meetings are not particularly useful and often don’t accomplish their intended objective.

We have observed that many companies are moving to shorter meetings (15 to 30 minutes) rather than the standard default of one-hour meetings in an effort to drive focus and productivity. For example, Netflix launched a redesign effort to drastically improve meeting efficiency, resulting in a tightly controlled meeting protocol. Meetings cannot go beyond 30 minutes. Meetings for one-way information sharing must be canceled in favor of other mechanisms such as a memo, podcast, or vlog. Two-way information sharing during meetings is limited by having attendees review materials in advance, replacing presentations with Q&As. Early data show Netflix has been able to reduce the number of meetings by more than 65 percent, and more than 85 percent of employees favor the approach.

Making meeting time a scarce resource is another strategy organizations are using to improve the quality of information sharing and other types of interactions occurring in a meeting setting. Some companies have implemented no-meeting days. In Japan, Microsoft’s “Work Life Choice Challenge” adopted a four-day workweek, reduced the time employees spend in meetings—and boosted productivity by 40 percent. 1 Bill Chappell, “4-day workweek boosted workers’ productivity by 40%, Microsoft Japan says,” NPR, November 4, 2019, npr.org. Similarly, Shopify uses “No Meeting Wednesdays” to enable employees to devote time to projects they are passionate about and to promote creative thinking. 2 Amy Elisa Jackson, “Feedback & meeting-free Wednesdays: How Shopify beats the competition,” Glassdoor, December 5, 2018, glassdoor.com. And Moveline’s product team dedicates every Tuesday to “Maker Day,” an opportunity to create and solve complex problems without the distraction of meetings. 3 Rebecca Greenfield, “Why your office needs a maker day,” Fast Company , April 17, 2014, fastcompany.com.

Finally, no meeting could be considered well scoped without considering who should participate, as there are real financial and transaction costs to meeting participation. Leaders should treat time spent in meetings as seriously as companies treat financial capital. Every leader in every organization should ask the following questions before attending any meeting: What’s this meeting for? What’s my role? Can I shorten this meeting by limiting live information sharing and focusing on discussion and decision making? We encourage you to excuse yourself from meetings if you don’t have a role in influencing the outcome and to instead get a quick update over email. If you are not essential, the meeting will still be successful (possibly more so!) without your presence. Try it and see what happens.

High-quality, focused interactions can improve productivity, speed, and innovation within any organization—and drive better business performance. We hope the above insights have inspired you to try some new techniques to improve the effectiveness and efficiency of collaboration within your organization.

Aaron De Smet is a senior partner in McKinsey’s New Jersey office; Caitlin Hewes is a consultant in the Atlanta office; Mengwei Luo is an associate partner in the New York office; J.R. Maxwell is a partner in the Washington, DC, office; and Patrick Simon is a partner in the Munich office.

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20 Elapsed Time Word Problems. Elapsed time is the amount of time that passes between the beginning and the end of an event. The concept of elapsed time fits nicely in the elementary school curriculum. Beginning in third grade, students should be able to tell and write time to the nearest minute and solve word problems involving addition and ...
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Time Measurement Problems: Simple and Complex Forms

1 h = 60 min 1 min = 60 s 1 hr = 3600 s

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