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9.3: Work-rate problems

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• Darlene Diaz
• Santiago Canyon College via ASCCC Open Educational Resources Initiative

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If it takes one person \(4\) hours to paint a room and another person \(12\) hours to paint the same room, working together they could paint the room even quicker. As it turns out, they would paint the room in \(3\) hours together. This is reasoned by the following logic. If the first person paints the room in \(4\) hours, she paints \(\dfrac{1}{4}\) of the room each hour. If the second person takes \(12\) hours to paint the room, he paints \(\dfrac{1}{12}\) of the room each hour. So together, each hour they paint \(\dfrac{1}{4}+\dfrac{1}{12}\) of the room. Let’s simplify this sum:

\[\dfrac{3}{12}+\dfrac{1}{12}=\dfrac{4}{12}=\dfrac{1}{3}\nonumber\]

This means each hour, working together, they complete \(\dfrac{1}{3}\) of the room. If \(\dfrac{1}{3}\) of the room is painted each hour, it follows that it will take \(3\) hours to complete the entire room.

Work-Rate Equation

If the first person does a job in time A, a second person does a job in time B, and together they can do a job in time T (total). We can use the work-rate equation:

\[\underset{\text{job per time A}}{\underbrace{\dfrac{1}{A}}}+\underset{\text{job per time B}}{\underbrace{\dfrac{1}{B}}}=\underset{\text{job per time T}}{\underbrace{\dfrac{1}{T}}}\nonumber\]

The Egyptians were the first to work with fractions. When the Egyptians wrote fractions, they were all unit fractions (a numerator of one). They used these types of fractions for about 2,000 years. Some believe that this cumbersome style of using fractions was used for so long out of tradition. Others believe the Egyptians had a way of thinking about and working with fractions that has been completely lost in history.

One Unknown Time

Example 9.3.1.

Adam can clean a room in 3 hours. If his sister Maria helps, they can clean it in \(2\dfrac{2}{5}\) hours. How long will it take Maria to do the job alone?

We use the work-rate equation to model the problem, but before doing this, we can display the information on a table:

Now, let’s set up the equation and solve. Notice, \(\dfrac{1}{2\dfrac{2}{5}}\) is an improper fraction and we can rewrite this as \(\dfrac{1}{\dfrac{12}{5}}=\dfrac{5}{12}\). We first clear denominators, then solve the linear equation as usual.

\[\begin{aligned}\dfrac{1}{3}+\dfrac{1}{t}&=\dfrac{5}{12} \\ \color{blue}{12t}\color{black}{}\cdot\dfrac{1}{3}+\color{blue}{12t}\color{black}{}\cdot\dfrac{1}{t}&=\color{blue}{12t}\color{black}{}\cdot\dfrac{5}{12}\\ 4t+12&=5t \\ 12&=t \\ t&=12\end{aligned}\]

Thus, it would take Maria \(12\) hours to clean the room by herself.

Example 9.3.2

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?

Now, let’s set up the equation and solve. Notice, were are filling the sink and draining it. Since we are draining the sink, we are losing water as the sink fills. Hence, we will subtract the rate in which the sink drains. We first clear denominators, then solve the linear equation as usual.

\[\begin{aligned}\dfrac{1}{5}-\dfrac{1}{7}&=\dfrac{1}{t} \\ \color{blue}{35t}\color{black}{}\cdot\dfrac{1}{5}-\color{blue}{35t}\color{black}{}\cdot\dfrac{1}{7}&=\color{blue}{35t}\color{black}{}\cdot\dfrac{1}{t} \\ 7t-5t&=35 \\ 2t&=35 \\ t&=\dfrac{35}{2}\end{aligned}\]

Thus, it would take \(\dfrac{35}{2}\) minutes to fill the sink, i.e., \(17\dfrac{1}{2}\) minutes.

Two Unknown Times

Example 9.3.3.

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

Now, let’s set up the equation and solve. We first clear denominators, then solve the linear equation as usual.

\[\begin{aligned}\dfrac{1}{2t}+\dfrac{1}{t}&=\dfrac{1}{10} \\ \color{blue}{10t}\color{black}{}\cdot\dfrac{1}{2t}+\color{blue}{10t}\color{black}{}\cdot\dfrac{1}{t}&=\color{blue}{10t}\color{black}{}\cdot\dfrac{1}{10} \\5+10&=t \\ 15&=t \\ t&=15\end{aligned}\]

Thus, it would take Rachel \(15\) hours to complete a project and Mike twice as long, \(30\) hours.

Example 9.3.4

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

Now, let’s set up the equation and solve. We first clear denominators, then solve the equation as usual.

\[\begin{array}{rl}\dfrac{1}{t}+\dfrac{1}{t-10}=\dfrac{1}{12}&\text{Apply the work-rate equation} \\ \color{blue}{12t(t-10)}\color{black}{}\cdot\dfrac{1}{t}+\color{blue}{12t(t-10)}\color{black}{}\cdot\dfrac{1}{t-10}=\color{blue}{12t(t-10)}\color{black}{}\cdot\dfrac{1}{12}&\text{Clear denominators} \\ 12(t-10)+12t=t(t-10)&\text{Distribute} \\ 12t-120+12t=t^2-10t &\text{Combine like terms} \\ 24t-120=t^2-10t&\text{Notice the }t^2\text{ term; solve by factoring} \\ t^2-34t+120=0&\text{Factor} \\ (t-4)(t-30)=0&\text{Apply zero product rule} \\ t-4=0\text{ or }t-30=0&\text{Isolate variable terms} \\ t=4\text{ or }t=30&\text{Solutions}\end{array}\nonumber\]

We obtained \(t = 4\) and \(t = 30\) for the solutions. However, we need to verify these solutions with Cosmo and Brittney’s times. If \(t = 4\), then Brittney’s time would be \(4 − 10 = −6\) days. This makes no sense since days are always positive. Thus, it would take Cosmo \(30\) days to build a shed and Brittney \(10\) less days, \(20\) days.

Example 9.3.5

An electrician 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?

We use the work-rate equation to model the problem, but before doing this, we can display the information on a table. Notice the time given doing the job together: \(1\) hour and \(12\) minutes. Unfortunately, we cannot use this format in the work-rate equation. Hence, we need to convert this to the same time units: \(1\) hour and \(12\) minutes \(= 1\dfrac{12}{60}\) hours \(= 1.2\) hours \(= \dfrac{6}{5}\) hours.

Note, \(\dfrac{1}{\dfrac{6}{5}} = \dfrac{5}{6}\). Now, let’s set up the equation and solve. We first clear denominators, then solve the equation as usual.

\[\begin{array}{rl}\dfrac{1}{t-1}+\dfrac{1}{t}=\dfrac{5}{6}&\text{Apply the work-rate equation} \\ \color{blue}{6t(t-1)}\color{black}{}\cdot\dfrac{1}{t-1}+\color{blue}{6t(t-1)}\color{black}{}\cdot\dfrac{1}{t}=\color{blue}{6t(t-1)}\color{black}{}\cdot\dfrac{5}{6}&\text{Clear denominators} \\ 6t+6(t-1)=5t(t-1)&\text{Distribute} \\ 6t+6t-6=5t^2-5t&\text{Combine like terms} \\ 12t-6=5t^2-5t&\text{Notice the }5t^2\text{ term; solve by factoring} \\ 5t^2-17t+6=0&\text{Factor} \\ (5t-2)(t-3)=0&\text{Apply zero product rule} \\ 5t-2=0\text{ or }t-3=0&\text{Isolate variable terms} \\ t=\dfrac{2}{5}\text{ or }t=3&\text{Solutions}\end{array}\nonumber\]

We obtained \(t = \dfrac{2}{5}\) and \(t = 3\) for the solutions. However, we need to verify these solutions with the electrician and apprentice’s times. If \(t =\dfrac{2}{5}\), then the electrician’s time would be \(\dfrac{2}{5} −1 = −\dfrac{3}{5}\) hours. This makes no sense since hours are always positive. Thus, it would take the apprentice \(3\) hours to complete a job and the electrician \(1\) less hour, \(2\) hours.

Work-Rate Problems Homework

Exercise 9.3.1.

Bill’s father can paint a room in two hours less than Bill can paint it. Working together they can complete the job in two hours and \(24\) minutes. How much time would each require working alone?

Exercise 9.3.2

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?

Exercise 9.3.3

Jack can wash and wax the family car in one hour less than Bob can. The two working together can complete the job in \(1\dfrac{1}{5}\) hours. How much time would each require if they worked alone?

Exercise 9.3.4

If A can do a piece of work alone in \(6\) days and B can do it alone in \(4\) days, how long will it take the two working together to complete the job?

Exercise 9.3.5

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 will it take each to do the job working alone?

Exercise 9.3.6

A can do a piece of work in \(3\) days, B in \(4\) days, and C in \(5\) days each working alone. How long will it take them to do it working together?

Exercise 9.3.7

A can do a piece of work in \(4\) days and B can do it in half the time. How long will it take them to do the work together?

Exercise 9.3.8

A cistern can be filled by one pipe in \(20\) minutes and by another in \(30\) minutes. How long will it take both pipes together to fill the tank?

Exercise 9.3.9

If A can do a piece of work in \(24\) days and A and B together can do it in \(6\) days, how long would it take B to do the work alone?

Exercise 9.3.10

A carpenter and his assistant can do a piece of work in \(3\dfrac{3}{4}\) days. If the carpenter himself could do the work alone in \(5\) days, how long would the assistant take to do the work alone?

Exercise 9.3.11

If Sam can do a certain job in \(3\) days, while it takes Fred \(6\) days to do the same job, how long will it take them, working together, to complete the job?

Exercise 9.3.12

Tim can finish a certain job in \(10\) hours. It take 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?

Exercise 9.3.13

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 faster person, working alone, to do the job?

Exercise 9.3.14

If two people working together can do a job in \(3\) hours, how long will it take the slower person to do the same job if one of them is \(3\) times as fast as the other?

Exercise 9.3.15

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 will it take to fill the tank if both pipes are open?

Exercise 9.3.16

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?

Exercise 9.3.17

It takes \(10\) hours to fill a pool with the inlet pipe. It can be emptied in \(15\) hrs 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?

Exercise 9.3.18

A sink is \(\dfrac{1}{4}\) 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 \(\dfrac{3}{4}\) of the sink?

Exercise 9.3.19

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?

Exercise 9.3.20

A water tank is being filled by two inlet pipes. Pipe A can fill the tank in \(4\dfrac{1}{2}\) hrs, while both pipes together can fill the tank in \(2\) hours. How long does it take to fill the tank using only pipe B?

Exercise 9.3.21

A tank can be emptied by any one of three caps. The first can empty the tank in \(20\) minutes while the second takes \(32\) minutes. If all three working together could empty the tank in \(8\dfrac{8}{59}\) minutes, how long would the third take to empty the tank?

Exercise 9.3.22

One pipe can fill a cistern in \(1\dfrac{1}{2}\) hours while a second pipe can fill it in \(2\dfrac{1}{3}\) hrs. Three pipes working together fill the cistern in \(42\) minutes. How long would it take the third pipe alone to fill the tank?

Exercise 9.3.23

Sam takes \(6\) hours longer than Susan to wax a floor. Working together they can wax the floor in \(4\) hours. How long will it take each of them working alone to wax the floor?

Exercise 9.3.24

It takes Robert \(9\) hours longer than Paul to rapair a transmission. If it takes them \(2 \dfrac{2}{5}\) hours to do the job if they work together, how long will it take each of them working alone?

Exercise 9.3.25

It takes Sally \(10\dfrac{1}{2}\) minutes longer than Patricia to clean up their dorm room. If they work together they can clean it in \(5\) minutes. How long will it take each of them if they work alone?

Exercise 9.3.26

A takes \(7 \dfrac{1}{2}\) minutes longer than B to do a job. Working together they can do the job in \(9\) minutes. How long does it take each working alone?

Exercise 9.3.27

Secretary A takes \(6\) minutes longer than Secretary B to type \(10\) pages of manuscript. If they divide the job and work together it will take them \(8 \dfrac{3}{4}\) minutes to type \(10\) pages. How long will it take each working alone to type the \(10\) pages?

Exercise 9.3.28

It takes John \(24\) minutes longer than Sally to mow the lawn. If they work together they can mow the lawn in \(9\) minutes. How long will it take each to mow the lawn if they work alone?

Work Rate Problems with Solutions

A set of problems related to work and rate of work is presented with detailed solutions.

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"Work" Word Problems

Painting & Pipes Tubs & Man-Hours Unequal Times Etc.

"Work" problems usually involve situations such as two people working together to paint a house. You are usually told how long each person takes to paint a similarly-sized house, and you are asked how long it will take the two of them to paint the house when they work together.

Many of these problems are not terribly realistic — since when can two laser printers work together on printing one report? — but it's the technique that they want you to learn, not the applicability to "real life".

The method of solution for "work" problems is not obvious, so don't feel bad if you're totally lost at the moment. There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does in a given unit of time . For instance:

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Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours to paint a similarly-sized house. How long would it take the two painters together to paint the house?

To find out how much they can do together per hour , I make the necessary assumption that their labors are additive (in other words, that they never get in each other's way in any manner), and I add together what they can do individually per hour . So, per hour, their labors are:

But the exercise didn't ask me how much they can do per hour; it asked me how long they'll take to finish one whole job, working togets. So now I'll pick the variable " t " to stand for how long they take (that is, the time they take) to do the job together. Then they can do:

This gives me an expression for their combined hourly rate. I already had a numerical expression for their combined hourly rate. So, setting these two expressions equal, I get:

I can solve by flipping the equation; I get:

An hour has sixty minutes, so 0.8 of an hour has forty-eight minutes. Then:

They can complete the job together in 4 hours and 48 minutes.

The important thing to understand about the above example is that the key was in converting how long each person took to complete the task into a rate.

hours to complete job:

first painter: 12

second painter: 8

together: t

Since the unit for completion was "hours", I converted each time to an hourly rate; that is, I restated everything in terms of how much of the entire task could be completed per hour. To do this, I simply inverted each value for "hours to complete job":

completed per hour:

Then, assuming that their per-hour rates were additive, I added the portion that each could do per hour, summed them, and set this equal to the "together" rate:

adding their labor:

As you can see in the above example, "work" problems commonly create rational equations . But the equations themselves are usually pretty simple to solve.

One pipe can fill a pool 1.25 times as fast as a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used?

My first step is to list the times taken by each pipe to fill the pool, and how long the two pipes take together. In this case, I know the "together" time, but not the individual times. One of the pipes' times is expressed in terms of the other pipe's time, so I'll pick a variable to stand for one of these times.

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Since the faster pipe's time to completion is defined in terms of the second pipe's time, I'll pick a variable for the slower pipe's time, and then use this to create an expression for the faster pipe's time:

slow pipe: s

together: 5

Next, I'll convert all of the completion times to per-hour rates:

Then I make the necessary assumption that the pipes' contributions are additive (which is reasonable, in this case), add the two pipes' contributions, and set this equal to the combined per-hour rate:

multiplying through by 20 s (being the lowest common denominator of all the fractional terms):

20 + 25 = 4 s

45/4 = 11.25 = s

They asked me for the time of the slower pipe, so I don't need to find the time for the faster pipe. My answer is:

The slower pipe takes 11.25 hours.

Note: I could have picked a variable for the faster pipe, and then defined the time for the slower pipe in terms of this variable. If you're not sure how you'd do this, then think about it in terms of nicer numbers: If someone goes twice as fast as you, then you take twice as long as he does; if he goes three times as fast as you, then you take three times as long as him. In this case, if he goes 1.25 times as fast, then you take 1.25 times as long. So the variables could have been " f  " for the number of hours the faster pipe takes, and then the number of hours for the slower pipe would have been " 1.25 f  ".

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Chapter 8: Rational Expressions

8.8 Rate Word Problems: Speed, Distance and Time

Distance, rate and time problems are a standard application of linear equations. When solving these problems, use the relationship rate (speed or velocity) times time equals distance .

[latex]r\cdot t=d[/latex]

For example, suppose a person were to travel 30 km/h for 4 h. To find the total distance, multiply rate times time or (30km/h)(4h) = 120 km.

The problems to be solved here will have a few more steps than described above. So to keep the information in the problem organized, use a table. An example of the basic structure of the table is below:

The third column, distance, will always be filled in by multiplying the rate and time columns together. If given a total distance of both persons or trips, put this information in the distance column. Now use this table to set up and solve the following examples.

Example 8.8.1

Joey and Natasha start from the same point and walk in opposite directions. Joey walks 2 km/h faster than Natasha. After 3 hours, they are 30 kilometres apart. How fast did each walk?

The distance travelled by both is 30 km. Therefore, the equation to be solved is:

[latex]\begin{array}{rrrrrrl} 3r&+&3(r&+&2)&=&30 \\ 3r&+&3r&+&6&=&30 \\ &&&-&6&&-6 \\ \hline &&&&\dfrac{6r}{6}&=&\dfrac{24}{6} \\ \\ &&&&r&=&4 \text{ km/h} \end{array}[/latex]

This means that Natasha walks at 4 km/h and Joey walks at 6 km/h.

Example 8.8.2

Nick and Chloe left their campsite by canoe and paddled downstream at an average speed of 12 km/h. They turned around and paddled back upstream at an average rate of 4 km/h. The total trip took 1 hour. After how much time did the campers turn around downstream?

The distance travelled downstream is the same distance that they travelled upstream. Therefore, the equation to be solved is:

[latex]\begin{array}{rrlll} 12(t)&=&4(1&-&t) \\ 12t&=&4&-&4t \\ +4t&&&+&4t \\ \hline \dfrac{16t}{16}&=&\dfrac{4}{16}&& \\ \\ t&=&0.25&& \end{array}[/latex]

This means the campers paddled downstream for 0.25 h and spent 0.75 h paddling back.

Example 8.8.3

Terry leaves his house riding a bike at 20 km/h. Sally leaves 6 h later on a scooter to catch up with him travelling at 80 km/h. How long will it take her to catch up with him?

The distance travelled by both is the same. Therefore, the equation to be solved is:

[latex]\begin{array}{rrrrr} 20(t)&=&80(t&-&6) \\ 20t&=&80t&-&480 \\ -80t&&-80t&& \\ \hline \dfrac{-60t}{-60}&=&\dfrac{-480}{-60}&& \\ \\ t&=&8&& \end{array}[/latex]

This means that Terry travels for 8 h and Sally only needs 2 h to catch up to him.

Example 8.8.4

On a 130-kilometre trip, a car travelled at an average speed of 55 km/h and then reduced its speed to 40 km/h for the remainder of the trip. The trip took 2.5 h. For how long did the car travel 40 km/h?

[latex]\begin{array}{rrrrrrr} 55(t)&+&40(2.5&-&t)&=&130 \\ 55t&+&100&-&40t&=&130 \\ &-&100&&&&-100 \\ \hline &&&&\dfrac{15t}{15}&=&\dfrac{30}{15} \\ \\ &&&&t&=&2 \end{array}[/latex]

This means that the time spent travelling at 40 km/h was 0.5 h.

Distance, time and rate problems have a few variations that mix the unknowns between distance, rate and time. They generally involve solving a problem that uses the combined distance travelled to equal some distance or a problem in which the distances travelled by both parties is the same. These distance, rate and time problems will be revisited later on in this textbook where quadratic solutions are required to solve them.

For Questions 1 to 8, find the equations needed to solve the problems. Do not solve.

• A is 60 kilometres from B. An automobile at A starts for B at the rate of 20 km/h at the same time that an automobile at B starts for A at the rate of 25 km/h. How long will it be before the automobiles meet?
• Two automobiles are 276 kilometres apart and start to travel toward each other at the same time. They travel at rates differing by 5 km/h. If they meet after 6 h, find the rate of each.
• Two trains starting at the same station head in opposite directions. They travel at the rates of 25 and 40 km/h, respectively. If they start at the same time, how soon will they be 195 kilometres apart?
• Two bike messengers, Jerry and Susan, ride in opposite directions. If Jerry rides at the rate of 20 km/h, at what rate must Susan ride if they are 150 kilometres apart in 5 hours?
• A passenger and a freight train start toward each other at the same time from two points 300 kilometres apart. If the rate of the passenger train exceeds the rate of the freight train by 15 km/h, and they meet after 4 hours, what must the rate of each be?
• Two automobiles started travelling in opposite directions at the same time from the same point. Their rates were 25 and 35 km/h, respectively. After how many hours were they 180 kilometres apart?
• A man having ten hours at his disposal made an excursion by bike, riding out at the rate of 10 km/h and returning on foot at the rate of 3 km/h. Find the distance he rode.
• A man walks at the rate of 4 km/h. How far can he walk into the country and ride back on a trolley that travels at the rate of 20 km/h, if he must be back home 3 hours from the time he started?

Solve Questions 9 to 22.

• A boy rides away from home in an automobile at the rate of 28 km/h and walks back at the rate of 4 km/h. The round trip requires 2 hours. How far does he ride?
• A motorboat leaves a harbour and travels at an average speed of 15 km/h toward an island. The average speed on the return trip was 10 km/h. How far was the island from the harbour if the trip took a total of 5 hours?
• A family drove to a resort at an average speed of 30 km/h and later returned over the same road at an average speed of 50 km/h. Find the distance to the resort if the total driving time was 8 hours.
• As part of his flight training, a student pilot was required to fly to an airport and then return. The average speed to the airport was 90 km/h, and the average speed returning was 120 km/h. Find the distance between the two airports if the total flying time was 7 hours.
• Sam starts travelling at 4 km/h from a campsite 2 hours ahead of Sue, who travels 6 km/h in the same direction. How many hours will it take for Sue to catch up to Sam?
• A man travels 5 km/h. After travelling for 6 hours, another man starts at the same place as the first man did, following at the rate of 8 km/h. When will the second man overtake the first?
• A motorboat leaves a harbour and travels at an average speed of 8 km/h toward a small island. Two hours later, a cabin cruiser leaves the same harbour and travels at an average speed of 16 km/h toward the same island. How many hours after the cabin cruiser leaves will it be alongside the motorboat?
• A long distance runner started on a course, running at an average speed of 6 km/h. One hour later, a second runner began the same course at an average speed of 8 km/h. How long after the second runner started will they overtake the first runner?
• Two men are travelling in opposite directions at the rate of 20 and 30 km/h at the same time and from the same place. In how many hours will they be 300 kilometres apart?
• Two trains start at the same time from the same place and travel in opposite directions. If the rate of one is 6 km/h more than the rate of the other and they are 168 kilometres apart at the end of 4 hours, what is the rate of each?
• Two cyclists start from the same point and ride in opposite directions. One cyclist rides twice as fast as the other. In three hours, they are 72 kilometres apart. Find the rate of each cyclist.
• Two small planes start from the same point and fly in opposite directions. The first plane is flying 25 km/h slower than the second plane. In two hours, the planes are 430 kilometres apart. Find the rate of each plane.
• On a 130-kilometre trip, a car travelled at an average speed of 55 km/h and then reduced its speed to 40 km/h for the remainder of the trip. The trip took a total of 2.5 hours. For how long did the car travel at 40 km/h?
• Running at an average rate of 8 m/s, a sprinter ran to the end of a track and then jogged back to the starting point at an average of 3 m/s. The sprinter took 55 s to run to the end of the track and jog back. Find the length of the track.

Answer Key 8.8

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Lesson HOW TO Solve Rate of Work (painting, pool filling, etc) Problems

• "Worker A can finish a job in 3 hours. When working at the same time as Worker B, they can finish the job in 2 hours. How long does it take for Worker B to finish the job if he works alone?"
• "Painters A and B can paint a wall in 10 hours when working at the same time. Painter B works twice as fast as A. How long would it take to each of them to paint it if they worked alone?"
• "A 10,000 litre pool is filled by two pipes: A and B. Pipe A delivers 1,000 litres per hour. When pipe A and B are both on, they can fill an empty pool in 4 hours. How many litres per hour can pipe B deliver?"

Basic Formula

Where does the formula come from, what does it mean that a worker works n times faster than another one, sample problems.

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Work-related Problems

Case 1: Workers have different rates

Work rate = (1 job done) / (Time to finish the job)

Time of doing the job = (1 job done) / (Work rate)

For example Albert can finish a job in A days Bryan can finish the same job in B days Carlo can undo the job in C days  

1/A = rate of Albert 1/B = rate of Bryan -1/C = rate of Carlo  

Albert and Bryan work together until the job is done: (1/A + 1/B)t = 1 Albert is doing the job while Carlo is undoing it until the job is done: (1/A - 1/C)t = 1  

Problem Lejon can finish a job in 6 hours while Romel can do the same job in 3 hours. Working together, how many hours can they finish the job?

$(1/6 + 1/3)t = 1$

$\frac{1}{2}t = 1$

$t = 2 \, \text{ hours}$           answer

Case 2: Workers have equal rates

Work done = no. of workers × time of doing the job

To finish the job

If a job can be done by 10 workers in 5 hours, the work load is 10(5) = 50 man-hours. If 4 workers is doing the job for 6 hours, the work done is 4(6) = 24 man-hours. A remaining of 50 - 24 = 26 man-hours of work still needs to be done.  

Problem Eleven men could finish the job in 15 days. Five men started the job and four men were added at the beginning of the sixth day. How many days will it take them to finish the job?  

Let $x$ = no. of days for them to finish the job $25 + (5 + 4)(x - 5) = 165$

$25 + 9(x - 5) = 165$

$x = 20.56 \, \text{ days}$           answer

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• Number of Hours for Pipe to Fill the Tank if the Drain is Closed
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Rate Work Problems

Related Topics: Lesson Plans and Worksheets for Grade 6 Lesson Plans and Worksheets for all Grades More Lessons for Grade 6 Common Core For Grade 6

Video Solutions to help Grade 6 students learn how to solve constant rate work problems by calculating and comparing unit rates.

New York State Common Core Math Grade 6, Module 1, Lesson 23

Download Grade 6, Module 1, Lesson 23 Worksheets

Lesson 23 Outcome

• Students solve constant rate work problems by calculating and comparing unit rates.

Lesson 23 Summary

• Constant rate problems always count or measure something happening per unit of time. The time is always in the denominator.
• Sometimes the units of time in the denominators of two rates are not the same. One must be converted to the other before calculating the unit rate of each.
• Dividing the numerator by the denominator calculates the unit rate; this number stays in the numerator. The number in the denominator of the equivalent fraction is always 1.

NYS Math Module 1 Grade 6 Lesson 23 Classwork

Example 1: Fresh-Cut Grass Suppose that on a Saturday morning you can cut 3 lawns in 5 hours, and your friend can cut 5 lawns in 8 hours. Your friend claims he is working faster than you. Who is cutting lawns at a faster rate? How do you find out?

Example 2: Restaurant Advertising Next, suppose you own a restaurant. You want to do some advertising, so you hire 2 middle school students to deliver take-out menus around town. One of them, Darla, delivers 350 menus in 2 hours, and another employee, Drew, delivers 510 menus in 3 hours. You promise a $10 bonus to the fastest worker since time is money in the restaurant business. Who gets the bonus?

Example 3: Survival of the Fittest Which runs faster: a cheetah that can run 60 feet in 4 seconds or gazelle that can run 100 feet in 8 seconds?

Example 4: Flying Fingers What if the units of time are not the same in the two rates? The secretary in the main office can type 225 words in 3 minutes, while the computer teacher can type 105 words in 90 seconds. Who types at a faster rate?

Problem Set

• Who walks at a faster rate: someone who walks 60 feet in 10 seconds or someone who walks 42 feet in 6 seconds?
• Who walks at a faster rate: someone who walks 60 feet in 10 seconds or someone who takes 5 seconds to walk 25 feet? Review the lesson summary before answering!
• Which parachute has a slower decent: a red parachute that falls 10 feet in 4 seconds or a blue parachute that falls 12 feet in 6 seconds?
• During the winter of 2012-2013, Buffalo, New York received 22 inches of snow in 12 hours. Oswego, New York received 31 inches of snow over a 15 hour period. Which city had a heavier snowfall rate? Round your answers to the nearest hundredth.
• A striped marlin can swim at a rate of 70 miles per hour. Is this a faster or slower rate than a sailfish, which takes 30 minutes to swim 40 miles?
• One math student, John, can solve these 6 math problems in 20 minutes while another student, Juaquine, can solve them at a rate of 1 problem per 4 minutes. Who works faster?

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Course: 6th grade   >   Unit 3

• Intro to rates
• Solving unit rate problem
• Solving unit price problem

Rate problems

• Comparing rates example
• Comparing rates
• Rate review
• Your answer should be
• an integer, like 6 ‍  
• an exact decimal, like 0.75 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  

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Stormy Daniels Takes the Stand

The porn star testified for eight hours at donald trump’s hush-money trial. this is how it went..

Hosted by Michael Barbaro

Featuring Jonah E. Bromwich

Produced by Olivia Natt and Michael Simon Johnson

Edited by Lexie Diao

With Paige Cowett

Original music by Will Reid and Marion Lozano

Engineered by Alyssa Moxley

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music | YouTube

This episode contains descriptions of an alleged sexual liaison.

What happened when Stormy Daniels took the stand for eight hours in the first criminal trial of former President Donald J. Trump?

Jonah Bromwich, one of the lead reporters covering the trial for The Times, was in the room.

On today’s episode

Jonah E. Bromwich , who covers criminal justice in New York for The New York Times.

Background reading

In a second day of cross-examination, Stormy Daniels resisted the implication she had tried to shake down Donald J. Trump by selling her story of a sexual liaison.

Here are six takeaways from Ms. Daniels’s earlier testimony.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

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Jonah E. Bromwich covers criminal justice in New York, with a focus on the Manhattan district attorney’s office and state criminal courts in Manhattan. More about Jonah E. Bromwich

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The Worsening Prescription Drug Shortage and How to Cope With It

Many or all of the products featured here are from our partners who compensate us. This influences which products we write about and where and how the product appears on a page. However, this does not influence our evaluations. Our opinions are our own. Here is a list of our partners and here's how we make money .

Drug shortages are an ongoing fact of life, and many Americans have known the inconvenience of having to wait to fill a prescription. But shortages have mounted in recent years, alarming observers and generating headlines.

The number of prescription drugs in shortage climbed to a 10-year high of 323 in the first quarter of 2024 — but it’s not just about the numbers. Patients and their health care providers are also being hit with much more consequential shortages, some with potentially life-altering results.

Hospital crash carts, used to provide emergency treatment to patients with critical conditions like cardiac arrest, have run short of premeasured packages of drugs like epinephrine, forcing clinicians to spend extra time measuring a dose from a vial, and increasing the odds of a medication error.

Some cancer patients, if their first-choice chemotherapy is unavailable, have had to take another drug that has much worse side effects.

Shortages can affect everyone connected with health care in any setting. “Drug shortages impact patients, families, caregivers, pharmacists, hospitals, nursing homes, hospices, and other individuals and entities across the health care system,” according to an April 2024 white paper from the U.S. Department of Health and Human Services.

To make matters worse, over the past 12 months, there have been shortages of medicines for which there is no alternative, says Erin Fox, senior pharmacy director at University of Utah Health. Fox gives one example: oxytocin, a drug that’s ordered by obstetricians to induce labor or, after delivery, to help stop maternal bleeding.

Which drugs are in short supply?

Hundreds of prescription medications are difficult or impossible to obtain as of April 2024. Among them are quinapril, which is primarily used to treat hypertension; injectable acyclovir, an antiviral; and propofol, for general anesthesia.

You can check the availability of a prescription drug by searching the U.S. Food and Drug Administration's FDA Drug Shortages database or the American Society of Health-System Pharmacists' ASHP Drug Shortages List . Click on any drug name to learn more about that specific shortage.

The effects of shortages on patients and their providers can be profound. Scott Matsuda, a member of the Patient and Family Advisory Council at the nonprofit Patient Access Network (PAN) Foundation, has worked with cancer patients “who, all of a sudden, find their chemotherapy drugs aren’t available.” Sometimes, patients report that symptoms of their cancers return soon after they stop taking their first-choice chemo, he adds.

No health care setting has been spared, not even hospital operating rooms. Dr. Jesse Ehrenfeld, an anesthesiologist and the president of the American Medical Association, at times has had to use second-choice drugs to induce anesthesia. These medications may have additional side effects, such as lasting longer than necessary, he said.

“Complex workarounds also have the potential to introduce errors,” says Ehrenfeld. “And I’ve had colleagues who had to delay or cancel care due to a shortage.”

Why are so many drugs in short supply?

What causes shortages? About 12% of them are due to manufacturing problems, as when a tornado hit a Pfizer plant in North Carolina, according to drugmakers’ reports compiled by the ASHP. Another 14% of shortages happen when demand suddenly outruns supply; for example, that occurs when demand for the antiviral Tamiflu spikes during a severe flu season. And 12% are due to a business decision, like when a drug manufacturer decides they can reap greater profits by making a different drug.

But the biggest category of reasons that manufacturers give for a shortage, at 60%, is “unknown/would not provide.” Ehrenfeld says, “That’s called flying blind. In most cases, we really don’t have an attributable cause, which makes it hard to pinpoint sustainable solutions.”

What’s being done to solve this multifaceted problem?

Drug shortages are a tough problem because they have such diverse causes. Stakeholders have been seeking solutions for decades, but shortages have continued throughout health care.

ASHP has recommended that, for starters, Congress give the FDA the power to levy “meaningful penalties” on drugmakers that fail to report manufacturing and supply chain problems. Failure to report is rampant in the industry.

“We think FDA requiring greater transparency in manufacturing and distribution could help us understand the causes and mitigate the challenges,” says Ehrenfeld.

The pharmacist group also recommends that to improve the profitability of generics, the FDA waive some fees on manufacturers who promptly bring these lower-priced drugs to market.

Some observers are hopeful that momentum is gathering for impactful action. “I’ve been working on drug shortages since 2001, and we now have the most interest in Congress and elsewhere that I’ve ever seen,” says Fox, who has testified about the problem at Senate committee hearings.

How can you work around a shortage of your drug?

The bottom line for health care consumers is that systemic problems with prescription drug supplies will likely persist. If you’re faced with a shortage of a drug you need, these steps will improve your chances of getting hold of it more quickly.

Refill prescriptions as early as your insurer will let you. Ordering prescriptions ahead can give you more options for obtaining a refill before you run out.

Contact your prescriber. Let them know that your prescription isn’t available at your pharmacy or will be delayed. Solicit their suggestions for other ways to get the drug, a generic or brand-name equivalent, or an alternative medication that would work for you.

Ask your insurer to cover the brand-name version if the generic drug is unavailable. If your insurer resists paying for the more expensive brand-name drug, ask your prescriber to go to bat for you.

Develop a long-term working relationship with your pharmacist. 

If the pharmacist knows you’re a regular customer, they may be more likely to hustle to find a supplier that has your medication.

Try other pharmacies. Drugs in shortage are often distributed unevenly among pharmacies. Call around and see if another retailer in your area has a supply. If your insurer has a mail-order prescription option, consider trying it.

(Photo by Sean Rayford/Getty Images )

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