• Pre-algebra lessons
• Pre-algebra word problems
• Algebra lessons
• Algebra word problems
• Algebra proofs
• Geometry lessons
• Geometry word problems
• Geometry proofs
• Trigonometry lessons
• Consumer math
• Baseball math
• Math for nurses
• High school physics
• Basic mathematics store
• SAT Math Prep
• Math skills by grade level
• Other websites
• K-12 worksheets
• Worksheets generator
• Algebra worksheets
• Geometry worksheets
• Free math problem solver
• Pre-algebra calculators
• Algebra Calculators
• Geometry Calculators
• Math puzzles
• Math tricks

Working together word problems

We will solve "working together word problems" with proportion. It may easier to tackle than algebra.

## Some working together word problems can be complicated. Problem #3 is a little challenging!

This means that it will take the friend 100 days to finish the house garage by himself.

## Working together in sales

Michael Scott, by himself, can close 8 sales in 4 hours. If he works with Dwight, they can close 8 sales in 2 hours. How many hours does it take Dwight to close 8 sales by himself?

For Michael, 8 sales in 4 hours means 2 sales per 1 hour or 2 / 1

Michael's sales / hour + Dwight's sales / hour =  together's sales per hour

Dwight's sales / hour is x / 1

Together, 8 sales in 2 hours means 4 sales per 1 hour or 4 / 1

2 / 1 + x / 1 = 4 / 1

2 - 2 + x = 4 - 2

So Dwight can do 2 sales per hour. It is basically the same as Michael. In this case, he can also do 8 sales in 4 hours.

Want more working together word problems ?

Ratio and proportion

## Recent Articles

Multiply a two digit number by 101 in one second.

Feb 16, 24 04:22 AM

## How to Solve a System of two Linear Equations Quickly

Feb 13, 24 11:26 AM

## How to Multiply 3-digit Numbers Having a Zero in the Tens Place

Feb 12, 24 11:58 AM

100 Tough Algebra Word Problems. If you can solve these problems with no help, you must be a genius!

Recommended

• EXPLORE Tech Help Pro About Us Random Article Quizzes Request a New Article Community Dashboard This Or That Game Popular Categories Arts and Entertainment Artwork Books Movies Computers and Electronics Computers Phone Skills Technology Hacks Health Men's Health Mental Health Women's Health Relationships Dating Love Relationship Issues Hobbies and Crafts Crafts Drawing Games Education & Communication Communication Skills Personal Development Studying Personal Care and Style Fashion Hair Care Personal Hygiene Youth Personal Care School Stuff Dating All Categories Arts and Entertainment Finance and Business Home and Garden Relationship Quizzes Cars & Other Vehicles Food and Entertaining Personal Care and Style Sports and Fitness Computers and Electronics Health Pets and Animals Travel Education & Communication Hobbies and Crafts Philosophy and Religion Work World Family Life Holidays and Traditions Relationships Youth
• Browse Articles
• Learn Something New
• Quizzes Hot
• This Or That Game New
• Explore More
• Support wikiHow
• Education and Communications
• Mathematics

## How to Solve Combined Labor Problems

Last Updated: November 3, 2023 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 77,701 times.

Combined labor problems, or work problems, are math problems involving rational equations. [1] X Research source These are equations that involve at least one fraction. The problems basically require finding unit rates, combining them, and setting them equal to an unknown rate. These problems require a lot of interpretive logic, but as long as you know how to work with fractions, solving them is fairly easy.

## Problems with Two People Working Together

• For example, the problem might ask, “If Tommy can paint a room in 3 hours, and Winnie can paint the same room in 4 hours, how long will it take them to paint the room together?

## Problems with Two People Working Against Each Other

• For example, the problem might ask, “If a hose can fill a pool 6 hours, and an open drain can empty it in 2 hours, how long will it take the open drain to empty the pool with the hose on?”

## Problems with Two People Working In Shifts

• For example, the problem might be: “Damarion can clean the cat shelter in 8 hours, and Cassandra can clean the shelter in 4 hours. They work together for 2 hours, but then Cassandra leaves to take some cats to the vet. How long will it take for Damarion to finish cleaning the shelter on his own?”

## Video . By using this service, some information may be shared with YouTube.

• If the problem involves more than two workers, simply add their individual work rates, then take the reciprocal of the sum to get the time taken working together. Thanks Helpful 2 Not Helpful 0
• Pay close attention to units. These methods will work for any unit of time, such as minutes or days. Some problems might state the rates in different units, and you will need to convert. Thanks Helpful 0 Not Helpful 0

• A calculator

## You Might Also Like

• ↑ http://www.mathguide.com/lessons/Word-Work.html
• ↑ http://www.algebralab.org/Word/Word.aspx?file=Algebra_WorkingTogether.xml
• ↑ https://www.mtsac.edu/marcs/worksheet/math51/course/10application_problems_rational_expressions.pdf
• ↑ http://purplemath.com/modules/workprob2.htm

• Send fan mail to authors

## Watch Articles

• Do Not Sell or Share My Info
• Not Selling Info

## 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:

$\text{rate}\times \text{time}=\text{work done}$

For this problem:

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

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

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

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:

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

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?

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

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

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

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?

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

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?

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

The apprentice can do the job in either $\dfrac{2}{5}$ 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.

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

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?

## Work Problems - Algebra

Related Pages Work Problems Math Work Problems Solving Work Word Problems Using Algebra More Algebra Lessons

In these lessons, we will learn how to solve algebra word problems that involve people working together at different rates.

## How To Solve Algebra Word Problems: Work

This lesson is appropriate for any Algebra 1 or Algebra 2 student.

Example: It takes Maria ten hours to pick forty bushels of apples. Kyla can pick the same amount in 12 hours. How long will it take take if they worked together?

## Algebra Word Problems: Work

• Working alone, Mofor can harvest a field in 12 hours. Carlos can harvest the same field in 14 hours. Find how long it would take them if they worked together.
• It takes Ted ten hours to harvest a field. Willie can harvest the same field in 15 hours. Find how ling it would take them if they worked together?
• Working alone, it takes Cody ten minutes to sweep a porch. Jacob can sweep the same porch in eight minutes. Find how long it would take them if they worked together?

## Work Problem

• If Tom takes 12 hours to paint a house, what is his rate per hour?
• Assume it takes Mary 2/3 of an hour to clean a room. What is her rate per hour?
• Tom takes 12 hours to paint a house alone, but Don takes 6 hours to paint it. How long would it take if they work together?

Example: It takes Jim 6 days to frame a house, but takes only 4 days if he works with his son. How long would it take his son, working alone, to frame the house?

• HW Guidelines
• Study Skills Quiz
• Find Local Tutors
• Demo MathHelp.com
• Join MathHelp.com

## Select a Course Below

• ACCUPLACER Math
• Math Placement Test
• PRAXIS Math
• + more tests
• Pre-Algebra
• College Pre-Algebra
• Introductory Algebra
• Intermediate Algebra
• College Algebra

## "Work" Word Problems

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

It's often easier to do these "work" word problems when you're going through the chapter, section by section, in the textbook. They'll cover a particular technique in a given section, and then ask only those types of questions in that section's homework. Then, in the next section, they'll cover another technique and you'll do exercises that use only that method. Then, when you get to the chapter review (or the next test), the questions aren't grouped according to the method that works best. And everything falls apart.

But all of these exercises have some basic commonalities. And, if you work logically and clearly, you can usually think your way to the answer.

Content Continues Below

## MathHelp.com

Below are a few examples, so you can see what that "thinking your way to the answer" process looks like in practice.

## Working together, Bill and Tom painted a fence in 8 hours. Last year, Tom painted the fence by himself. The year before, Bill painted it by himself, but took 12 hours less than Tom took. How long did Bill and Tom take, when each was painting alone?

Bill's time for completion is stated in terms of Tom's time for completion. So I'll pick a variable for Tom's time; I'll use T for the number of hours Tom needs. Then Bill's time will be T  − 12 . So I can set up the completion-time info as:

completion time, in hours:

Bill: T − 12

together: 8

Inverting, I can tabulate their per-hour completion rates:

completed per hour:

This works out to be a simple additive-labor exercise. So I'll add their individual completion rates, and set this equal to the "together" completion rate:

1/ T + 1/( T − 12) = 1/8

I can multiply through by the common denominator of 8 T ( T  − 12) :

8( T − 12) + 8 T = T ( T − 12)

8 T − 96 + 8 T = T  2 − 12 T

0 = T  2 − 28 T + 96

0 = ( T − 4)( T − 24)

(Review how to factor quadratics , if you're not sure how I just got to that last line above.)

I've derived two solution values. But if I say that Tom takes four hours to complete the job by himself, (1) this won't make any sense in light of the two of them together needing eight hours (it's not possible that one does it in half the time as the two of them together), and (2) this wouldn't make any sense in light of Bill's time being twelve hours less than Tom's (since Bill can't work for a negative number of hours). That means that this " T  = 4 " solution is "extraneous" (pronounced "ekk-STRAY-nee-uss"), meaning "valid mathematically, but pointless as far as our situation is concerned".

So I can ignore the T  = 4 solution. Instead, I'll use the other solution, which says that Tom takes twenty-four hours to paint the whole house himself. Then:

Tom takes twenty-four hours, and Bill takes twelve hours.

This next example is stated a bit differently:

## Ben takes 2 hours to wash 500 dishes, and Frank takes 3 hours to wash 450 dishes. How long will they take, working together, to wash 1000 dishes?

For this exercise, we are given how many items in a job can be done in one time unit, rather than how much of a job can be completed in one time unit. But the thinking process is otherwise the same.

Ben can wash 500 dishes in 2 hours, so, by dividing, I find that he can do 250 dishes per hour. Similarly, Frank can wash 450 dishes in 3 hours, so he can do 150 dishes per hour. Working together, they can do 250 + 150 = 400 dishes an hour. That is:

dishes per hour:

Ben: 500 ÷ 2 = 250

Frank: 450 ÷ 3 = 150

together: 250 + 150 = 400

They can do 400 dishes each hour. I need to find how many hours it takes for the two of them to wash 1000 dishes. To find out, I ask myself, how many sets of 400 dishes are there in 1000 dishes? I'll divide to get the value:

1000 ÷ 400 = 10 ÷ 4 = 2.5

In other words, in two hours, they'll wash two sets of 400 , or 800 dishes; in the additional half of an hour, they'll wash an additional half of another set of 400 dishes, which is 200 dishes. This gives me the required total of 1000 dishes.

They'll take 2.5 hours.

## If six men can do a job in fourteen days, how many fewer men would be needed if they were allowed twenty-one days for the job?

The men are being treated as interchangeable; there is no "faster" or "slower" here. So I'll start by converting this to man-hours — well, in this case, man-days. If it takes six guys fourteen days to finish the job, then my total is:

6 × 14 = 84

That is, the entire job requires 84 man-days.

This exercise is asking me to expand the time allowed from fourteen days to twenty-one days. Obviously, if they're giving my guys more time, then I'll need fewer guys. But how many fewer guys, exactly?

However many guys I end up needing, I'll still need them to provide me with 84 man-days of work; those man-days will be spread out over 21 working days. This means:

84 ÷ 21 = 4

I've divided the number of man-days by the number of days, which leaves me the number of men; it appears that I'll only need four guys. Does this check? Well, if I put four guys to work for each of the twenty-one days, then I'll end up with 4 × 21 = 84 man-days, which is exactly what I need.

Originally, I needed six guys. Now I'll need only four. So:

I'll need two fewer guys.

## If Bob can dig out twenty cubic feet of dirt in an hour, and Carl can dig out twenty-four cubic feet of dirt in an hour, how much dirt will be in the hole, if they dig together?

This is a trick question. If they've dug the dirt out of the hole then, by definition, there will be no dirt in the hole. Holes are empty. Har. De. Har.

## Jill, Karen, and Lisa are painting a house. Working together, they can paint the house is 6 hours. Working alone, Jill can paint the house in five hours faster than can either of Karen or Lisa. How long would it take Jill to paint the house working alone?

Okay, there are a lot of unknowns here. But I can start with the usual hourly-rate stuff, and I'll pick variables that make sense. First, I'll note that, since Jill takes five hours less than either of Karen or Lisa, then Karen and Lisa take the same amount of time. So I can set up the hours for completion for each:

Jill: K − 5

together: 6

Taking the reciprocals, I can find their hourly rates:

Since I'm assuming their labors are additive, I can add their individual per-hour accomplishments, and set this equal to their combined accomplishment:

1/ K + 1/( K − 5) + 1/ K = 1/6

Multiplying through by the common denominator of 6 K ( K  − 5) , I get:

6( K − 5) + 6 K + 6( K − 5) = K ( K − 5)

6 K − 30 + 6 K + 6 K − 30 = K 2 − 5 K

18 K − 60 = K  2 − 5 K

0 = K  2 − 23 K + 60

0 = ( K − 20)( K − 3)

K = 20, K = 3

But if Karen takes only three hours, then Jill would be into negative time, in order to be five hours faster. So I can discard the " K = 3 " solution as extraneous. The only valid solution is " K = 20 ". Since Karen takes twenty hours, and since Jill is five hours faster, then:

Jill can paint the house in fifteen hours.

You may have noticed that each of these problems used some form of the "how much can be done per time unit" construction, but other than that, each problem was done differently. That's how "work" problems often are. You'll have to be alert and clever to do these. But as you saw above, if you label things neatly and do your work clearly and logically, you should find your way to the solution.

By the way, I've mentioned once or twice that these exercises often assume that people are equal in their productivity, that chickens can come in fractional units, and that rates of completion (such as how much a person gets done in an hour) are additive. I'm sure you can think of somebody that isn't working as hard as you are in your class, and the chicken thing is kind of obvious. The classic counter-example for labor being additive is the idea that, if one woman takes nine months to have one baby, then nine women can have one baby in one month.

So definitely use the reasoning during this part of your course, but keep in mind that it isn't generally reflective of "real life".

URL: https://www.purplemath.com/modules/workprob4.htm

Page 1 Page 2 Page 3 Page 4

## Standardized Test Prep

• Tutoring from PM
• Site licencing

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

## What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

## Math Topics

More solvers.

• Simplify Fractions

• school Campus Bookshelves
• perm_media Learning Objects
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more

This action is not available.

## 9.3: Work-rate problems

• Last updated
• Save as PDF
• Page ID 45129

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

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?

## Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

## Most Used Actions

Number line.

• x^{2}-x-6=0
• -x+3\gt 2x+1
• line\:(1,\:2),\:(3,\:1)
• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• \frac{d}{dx}(\frac{3x+9}{2-x})
• (\sin^2(\theta))'
• \lim _{x\to 0}(x\ln (x))
• \int e^x\cos (x)dx
• \int_{0}^{\pi}\sin(x)dx
• \sum_{n=0}^{\infty}\frac{3}{2^n}
• Is there a step by step calculator for math?
• Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
• Is there a step by step calculator for physics?
• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• My Notebook, the Symbolab way Math notebooks have been around for hundreds of years. You write down problems, solutions and notes to go back... Read More

Message received. Thanks for the feedback.

• Testimonial
• Web Stories

## 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.

## Problems on Numbers System : Level 02

• Learn on-the-go
• Unlimited Prep Resources
• Better Learning Experience
• Personalized Guidance

Get More Out of Your Exam Preparation - Try Our App!

• Math for Kids
• Parenting Resources
• ELA for Kids
• Teaching Resources

## 10 Best Strategies for Solving Math Word Problems

5 Easy Tips & Tricks to Learn the 13 Time Table for Kids

How to Teach Number Sense to Kids: Step-by-Step Guide

How to Teach Decimals: A Step-by-Step Guide

How to Teach Fraction to Kids – 11 Best Activities

How to Choose Best School For Your Kid: 12 Best Tips

Why Kids Get Bored at School: 10 Tips to Keep Them Interested

11 Best Writing Apps for Kids

Homeschool vs Public School: 12 Tips on How to Choose One

15 Essential Life Skills Activities for Kids: Beyond ABCs

12 animal that starts with k.

60 Best Essay Topics for Kids: Nurturing Young Minds

How to Teach Sentence Structure to Kids: The Ultimate Guide

72 Best G words for Kids in 2024

## 11 Best Search Engine for Kids: Protecting Young Minds Online

How to Prevent Teacher’s Burnout – 12 Top Ways

46 Best Teaching Tools for Teachers in 2024

How to Teach 3rd Grade Kids: 25 Tips for a Successful Year

12 Best Online Tutoring Websites for All Grades

## 1. Understand the Problem by Paraphrasing

2. identify key information and variables, 3. translate words into mathematical symbols, 4. break down the problem into manageable parts, 5. draw diagrams or visual representations, 6. use estimation to predict answers, 7. apply logical reasoning for unknown variables, 8. leverage similar problems as templates, 9. check answers in the context of the problem, 10. reflect and learn from mistakes.

Have you ever observed the look of confusion on a student’s face when they encounter a math word problem ? It’s a common sight in classrooms worldwide, underscoring the need for effective strategies for solving math word problems . The main hurdle in solving math word problems is not just the math itself but understanding how to translate the words into mathematical equations that can be solved.

## SplashLearn: Most Comprehensive Learning Program for PreK-5

SplashLearn inspires lifelong curiosity with its game-based PreK-5 learning program loved by over 40 million children. With over 4,000 fun games and activities, it’s the perfect balance of learning and play for your little one.

Generic advice like “read the problem carefully” or “practice more” often falls short in addressing students’ specific difficulties with word problems. Students need targeted math word problem strategies that address the root of their struggles head-on.

## A Guide on Steps to Solving Word Problems: 10 Strategies

One of the first steps in tackling a math word problem is to make sure your students understand what the problem is asking. Encourage them to paraphrase the problem in their own words. This means they rewrite the problem using simpler language or break it down into more digestible parts. Paraphrasing helps students grasp the concept and focus on the problem’s core elements without getting lost in the complex wording.

Original Problem: “If a farmer has 15 apples and gives away 8, how many does he have left?”

Paraphrased: “A farmer had some apples. He gave some away. Now, how many apples does he have?”

This paraphrasing helps students identify the main action (giving away apples) and what they need to find out (how many apples are left).

Students often get overwhelmed by the details in word problems. Teach them to identify key information and variables essential for solving the problem. This includes numbers , operations ( addition , subtraction , multiplication , division ), and what the question is asking them to find. Highlighting or underlining can be very effective here. This visual differentiation can help students focus on what’s important, ignoring irrelevant details.

• Encourage students to underline numbers and circle keywords that indicate operations (like ‘total’ for addition and ‘left’ for subtraction).
• Teach them to write down what they’re solving for, such as “Find: Total apples left.”

Problem: “A classroom has 24 students. If 6 more students joined the class, how many students are there in total?”

Key Information:

• Original number of students (24)
• Students joined (6)
• Looking for the total number of students

The transition from the language of word problems to the language of mathematics is a critical skill. Teach your students to convert words into mathematical symbols and equations. This step is about recognizing keywords and phrases corresponding to mathematical operations and expressions .

Common Translations:

• “Total,” “sum,” “combined” → Addition (+)
• “Difference,” “less than,” “remain” → Subtraction (−)
• “Times,” “product of” → Multiplication (×)
• “Divided by,” “quotient of” → Division (÷)
• “Equals” → Equals sign (=)

Problem: “If one book costs $5, how much would 4 books cost?” Translation: The word “costs” indicates a multiplication operation because we find the total cost of multiple items. Therefore, the equation is 4 × 5 =$20

Complex math word problems can often overwhelm students. Incorporating math strategies for problem solving, such as teaching them to break down the problem into smaller, more manageable parts, is a powerful approach to overcome this challenge. This means looking at the problem step by step rather than simultaneously trying to solve it. Breaking it down helps students focus on one aspect of the problem at a time, making finding the solution more straightforward.

Problem: “John has twice as many apples as Sarah. If Sarah has 5 apples, how many apples do they have together?”

Steps to Break Down the Problem:

Find out how many apples John has: Since John has twice as many apples as Sarah, and Sarah has 5, John has 5 × 2 = 10

Calculate the total number of apples: Add Sarah’s apples to John’s to find the total,  5 + 10 = 15

By splitting the problem into two parts, students can solve it without getting confused by all the details at once.

Explore these fun multiplication word problem games:

Diagrams and visual representations can be incredibly helpful for students, especially when dealing with spatial or quantity relationships in word problems. Encourage students to draw simple sketches or diagrams to represent the problem visually. This can include drawing bars for comparison, shapes for geometry problems, or even a simple distribution to better understand division or multiplication problems .

Problem: “A garden is 3 times as long as it is wide. If the width is 4 meters, how long is the garden?”

Visual Representation: Draw a rectangle and label the width as 4 meters. Then, sketch the length to represent it as three times the width visually, helping students see that the length is 4 × 3 = 12

Estimation is a valuable skill in solving math word problems, as it allows students to predict the answer’s ballpark figure before solving it precisely. Teaching students to use estimation can help them check their answers for reasonableness and avoid common mistakes.

Problem: “If a book costs $4.95 and you buy 3 books, approximately how much will you spend?” Estimation Strategy: Round$4.95 to the nearest dollar ($5) and multiply by the number of books (3), so 5 × 3 = 15. Hence, the estimated total cost is about$15.

Estimation helps students understand whether their final answer is plausible, providing a quick way to check their work against a rough calculation.

Check out these fun estimation and prediction word problem worksheets that can be of great help:

When students encounter problems with unknown variables, it’s crucial to introduce them to logical reasoning. This strategy involves using the information in the problem to deduce the value of unknown variables logically. One of the most effective strategies for solving math word problems is working backward from the desired outcome. This means starting with the result and thinking about the steps leading to that result, which can be particularly useful in algebraic problems.

Problem: “A number added to three times itself equals 32. What is the number?”

Working Backward:

Let the unknown number be x.

The equation based on the problem is  x + 3x = 32

Solve for x by simplifying the equation to 4x=32, then dividing by 4 to find x=8.

By working backward, students can more easily connect the dots between the unknown variable and the information provided.

Practicing problems of similar structure can help students recognize patterns and apply known strategies to new situations. Encourage them to leverage similar problems as templates, analyzing how a solved problem’s strategy can apply to a new one. Creating a personal “problem bank”—a collection of solved problems—can be a valuable reference tool, helping students see the commonalities between different problems and reinforcing the strategies that work.

Suppose students have solved a problem about dividing a set of items among a group of people. In that case, they can use that strategy when encountering a similar problem, even if it’s about dividing money or sharing work equally.

It’s essential for students to learn the habit of checking their answers within the context of the problem to ensure their solutions make sense. This step involves going back to the original problem statement after solving it to verify that the answer fits logically with the given information. Providing a checklist for this process can help students systematically review their answers.

• Re-read the problem: Ensure the question was understood correctly.
• Compare with the original problem: Does the answer make sense given the scenario?
• Use estimation: Does the precise answer align with an earlier estimation?
• Substitute back: If applicable, plug the answer into the problem to see if it works.

Problem: “If you divide 24 apples among 4 children, how many apples does each child get?”

After solving, students should check that they understood the problem (dividing apples equally).

Their answer (6 apples per child) fits logically with the number of apples and children.

Their estimation aligns with the actual calculation.

Substituting back 4×6=24 confirms the answer is correct.

Teaching students to apply logical reasoning, leverage solved problems as templates, and check their answers in context equips them with a robust toolkit for tackling math word problems efficiently and effectively.

One of the most effective ways for students to improve their problem-solving skills is by reflecting on their errors, especially with math word problems. Using word problem worksheets is one of the most effective strategies for solving word problems, and practicing word problems as it fosters a more thoughtful and reflective approach to problem-solving

These worksheets can provide a variety of problems that challenge students in different ways, allowing them to encounter and work through common pitfalls in a controlled setting. After completing a worksheet, students can review their answers, identify any mistakes, and then reflect on them in their mistake journal. This practice reinforces mathematical concepts and improves their math problem solving strategies over time.

## 3 Additional Tips for Enhancing Word Problem-Solving Skills

Before we dive into the importance of reflecting on mistakes, here are a few impactful tips to enhance students’ word problem-solving skills further:

## 1. Utilize Online Word Problem Games

Incorporate online games that focus on math word problems into your teaching. These interactive platforms make learning fun and engaging, allowing students to practice in a dynamic environment. Games can offer instant feedback and adaptive challenges, catering to individual learning speeds and styles.

Here are some word problem games that you can use for free:

## 2. Practice Regularly with Diverse Problems

Consistent practice with a wide range of word problems helps students become familiar with different questions and mathematical concepts. This exposure is crucial for building confidence and proficiency.

Start Practicing Word Problems with these Printable Word Problem Worksheets:

## 3. Encourage Group Work

Solving word problems in groups allows students to share strategies and learn from each other. A collaborative approach is one of the best strategies for solving math word problems that can unveil multiple methods for tackling the same problem, enriching students’ problem-solving toolkit.

## Conclusion

Mastering math word problems is a journey of small steps. Encourage your students to practice regularly, stay curious, and learn from their mistakes. These strategies for solving math word problems are stepping stones to turning challenges into achievements. Keep it simple, and watch your students grow their confidence and skills, one problem at a time.

How can i help my students stay motivated when solving math word problems.

Encourage small victories and use engaging tools like online games to make practice fun and rewarding.

## What's the best way to teach beginners word problems?

Begin with simple problems that integrate everyday scenarios to make the connection between math and real-life clear and relatable.

## How often should students practice math word problems?

Regular, daily practice with various problems helps build confidence and problem-solving skills over time.

Recent posts.

## Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Watch your kids fall in love with math & reading through our scientifically designed curriculum.

• Games for Kids
• Worksheets for Kids
• Math Worksheets
• ELA Worksheets
• Math Vocabulary
• Number Games
• Subtraction Games
• Multiplication Games
• Division Games
• Subtraction Worksheets
• Multiplication Worksheets
• Division Worksheets
• Times Tables Worksheets
• Writing Games
• Phonics Games
• Sight Words Games
• Letter Tracing Games
• Writing Worksheets
• Phonics Worksheets
• Sight Words Worksheets
• Letter Tracing Worksheets
• Prime Number
• Order of Operations
• Long multiplication
• Place value
• Parallelogram
• SplashLearn Success Stories
• SplashLearn Apps
• [email protected]

• For a new problem, you will need to begin a new live expert session.
• You can contact support with any questions regarding your current subscription.
• You will be able to enter math problems once our session is over.
• I am only able to help with one math problem per session. Which problem would you like to work on?
• Does that make sense?
• I am currently working on this problem.
• Are you still there?
• It appears we may have a connection issue. I will end the session - please reconnect if you still need assistance.
• Let me take a look...
• Can you please send an image of the problem you are seeing in your book or homework?
• If you click on "Tap to view steps..." you will see the steps are now numbered. Which step # do you have a question on?
• Please make sure you are in the correct subject. To change subjects, please exit out of this live expert session and select the appropriate subject from the menu located in the upper left corner of the Mathway screen.
• What are you trying to do with this input?
• While we cover a very wide range of problems, we are currently unable to assist with this specific problem. I spoke with my team and we will make note of this for future training. Is there a different problem you would like further assistance with?
• Mathway currently does not support this subject. We are more than happy to answer any math specific question you may have about this problem.
• Mathway currently does not support Ask an Expert Live in Chemistry. If this is what you were looking for, please contact support.
• Mathway currently only computes linear regressions.
• We are here to assist you with your math questions. You will need to get assistance from your school if you are having problems entering the answers into your online assignment.
• Have a great day!
• Hope that helps!
• You're welcome!
• Per our terms of use, Mathway's live experts will not knowingly provide solutions to students while they are taking a test or quiz.

• a special character: @\$#!%*?&

#### IMAGES

1. Classroom Poster: 4 Steps to Solve Any Math Problem

2. 3 Ways to Solve Two Step Algebraic Equations

3. MATH 0310 WORKING TOGETHER PROBLEMS

4. Math Work Problems (video lessons, examples and solutions)

5. "Working Together" Problems

6. Algebra Working Together Rate Problem

#### VIDEO

1. Can you solve this grade 8 math problem? #math #mathproblems #motherson #grade8math

2. How To Solve Math Problems

3. how to solve maths problems quickly

4. International Math Olympiad Algebra Problem

6. How to Solve This Math Problem?

1. Working together word problems

Problem #1: Maria can clean all bathrooms in her house in 5 minutes. Her husband can do the same in 20 minutes. How long will it take them if they work together? One important observation is that an entire task or job can be represented with the number 1. If it takes Maria 5 minutes to do the entire job, then she can do 1 5 of the job in 1 minute

2. "Work" Word Problems

Purplemath "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.

3. 3 Ways to Solve Combined Labor Problems

1 Read the problem carefully. Use this method if the problem represents two or more people working together to complete a job. The problem should also give you the amount of time it would take each person to complete the job alone.

4. Math Work Problems (video lessons, examples and solutions)

Solution: Step 1: Assign variables: Let x = time to mow lawn together. Step 2: Use the formula: Step 3: Solve the equation The LCM of 40 and 60 is 120 Multiply both sides with 120 Answer: The time taken for both of them to mow the lawn together is 24 minutes. Work Problems With One Unknown Time Examples: Catherine can paint a house in 15 hours.

5. Work Word Problems (video lessons, examples, solutions)

Step 1: Assign variables: Let x = time to mow lawn together. Step 2: Use the formula: Step 3: Solve the equation The LCM of 40 and 60 is 120 Multiply both sides with 120 Answer: The time taken for both of them to mow the lawn together is 24 minutes. Example 2: It takes Maria 10 hours to pick forty bushels of apples.

6. 9.10 Rate Word Problems: Work and Time

The equation to solve is: 1 R + 1 2R = 1 10 h, where Doug's rate ( 1 D) = 1 2 × Becky's ( 1 R) rate. Sum the rates: 1 R + 1 2R = 2 2R + 1 2R = 3 2R Solve for R: 3 2R = 1 10 h which means 1 R = 1 10 × 2 3 h so 1 R = 2 30 or R = 30 2 1 R + 1 2 R = 1 10 h, where Doug's rate ( 1 D) = 1 2 × Becky's ( 1 R) rate.

7. Algebra Work Problems (solutions, examples, videos)

Step 1: Assign variables: Let x = time taken to fill up the tank. Step 2: Use the formula: Since pipe C drains the water it is subtracted. Step 3: Solve the equation The LCM of 3, 4 and 5 is 60 Multiply both sides with 60 Answer: The time taken to fill the tank is hours. Work Problem: Pumps draining a tank Example:

8. Work Word Problems

Solution: Step 1: Assign variables: Let x = time taken by Peter Step 2: Use the formula: Step 3: Solve the equation Multiply both sides with 30 x Answer: The time taken for Peter to paint the fence alone is hours. How To Solve Work Word Problems?

9. Work Problems

The next step to solving this problem demands that we find values for the last column, "Work." Imagine if Ben worked for 2 hours. We would multiply 1 ⁄ 4 times 2 and get 1 ⁄ 2, which would mean only 1 ⁄ 2 of the job was done. So, if we multiply "Work Per Time" times "Time," we will gain "Work" within our table.

10. Work Problems

Algebra Word Problems: Work. Examples: Working alone, Mofor can harvest a field in 12 hours. Carlos can harvest the same field in 14 hours. Find how long it would take them if they worked together. It takes Ted ten hours to harvest a field. Willie can harvest the same field in 15 hours. Find how ling it would take them if they worked together?

11. "Working Together" Problems

In this video, we work through an example of a "working together" word problem.

12. "Work" Word Problems

It's often easier to do these "work" word problems when you're going through the chapter, section by section, in the textbook. They'll cover a particular technique in a given section, and then ask only those types of questions in that section's homework. Then, in the next section, they'll cover another technique and you'll do exercises that use ...

13. Time and Work Problems

This math video tutorial focuses on solving work and time problems using simple tricks and shortcuts. It contains a simple formula that you can use with these problems. ...more ...more...

14. Combined Rate Example (2 People Working Together)

Learn how to find the time to complete a job when two or more people work together in a combined rate problem in this free math video tutorial by Mario's Mat...

15. Step-by-Step Math Problem Solver

What can QuickMath do? QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose.

16. Shared Work Word Problems

To solve the equation, let's start by simplifying the right-hand side. To simplify 1/5 + 1/7, we'll need to get a common denominator. Multiply the top and bottom of 1/5 by 7, and the top and bottom of 1/7 by 5, giving us 7/35 + 5/35. Once we have a common denominator, we add the numerators, so it's 12/35. So now we need to solve 1/x = 12/35 to ...

17. Rational equations word problem: combined rates (example 2)

It is mathematical. If 2 people are working together on one task, and they work at the same rate of speed, then each person completes 1/2 the work needed to complete the job and the whole job gets finished because 1/2+1/2 = 1 (the job is done).

18. Rational equations word problem: combined rates

Next lets multiply both sides by t (hours) to cancel the t (hours) on the right and get (8/15)t (hours)=1 (lawn). Next multiply both sides by 15/8 to cancel the 8/15 on the left and get t (hours)=15/8 (lawn). As a last step we could divide both sides by 1 (lawn) to get t (hours/lawn)=15/8. That is how he did it.

19. 9.3: Work-rate problems

So together, each hour they paint 1 4 + 1 12 of the room. Let's simplify this sum: 3 12 + 1 12 = 4 12 = 1 3. This means each hour, working together, they complete 1 3 of the room. If 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.

20. Step-by-Step Calculator

To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem. Show more; en. Related Symbolab blog posts.

21. Time and Work 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.

22. Rational equations word problem: eliminating solutions

When the two hoses are filling the pond together, all we know is that it takes a total of 12 minutes. We know that one hose can fill a pond 10 minutes faster than the other, but that's only when it's working on it's own. Your statement basically says that one hose puts in 11 times the work of the other hose when they're both filling a pond together, which is not a given in this problem.

23. 10 Best strategies for solving math word problems in 2024

2. Identify Key Information and Variables. Students often get overwhelmed by the details in word problems. Teach them to identify key information and variables essential for solving the problem. This includes numbers, operations (addition, subtraction, multiplication, division), and what the question is asking them to find.Highlighting or underlining can be very effective here.

24. Mathway

Algebra Free math problem solver answers your algebra homework questions with step-by-step explanations.