how to solve work done problems

  • 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
  • 5th Grade Math
  • 6th Grade Math
  • Pre-Algebra
  • College Pre-Algebra
  • Introductory Algebra
  • Intermediate Algebra
  • College Algebra

"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:

Content Continues Below

MathHelp.com

Need a personal math teacher?

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.

Advertisement

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  ".

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

Page 1 Page 2 Page 3 Page 4

Standardized Test Prep

College math, homeschool math, share this page.

  • Terms of Use
  • About Purplemath
  • About the Author
  • Tutoring from PM
  • Advertising
  • Linking to PM
  • Site licencing

Visit Our Profiles

how to solve work done problems

  • PRO Courses Guides New Tech Help Pro Expert Videos About wikiHow Pro Upgrade Sign In
  • EDIT Edit this Article
  • 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
  • Train Your Brain
  • Explore More
  • Support wikiHow
  • About wikiHow
  • Log in / Sign up
  • 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 77,597 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

Step 1 Read the problem carefully.

  • 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?

Step 2 Determine the hourly rate of each individual.

Problems with Two People Working Against Each Other

Step 1 Read the problem carefully.

  • 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?”

Step 2 Determine the hourly rate of the individual completing the job.

Problems with Two People Working In Shifts

Step 1 Read the problem carefully.

  • 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?”

Step 2 Determine the hourly rate of each individual.

Community Q&A

Community Answer

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

how to solve work done problems

Things You'll Need

  • A calculator

You Might Also Like

Write Numbers in Words

  • ↑ 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

About This Article

Grace Imson, MA

  • Send fan mail to authors

Did this article help you?

how to solve work done problems

Featured Articles

How to Make Friends in Your 20s after College

Trending Articles

Everything You Need to Know to Rock the Corporate Goth Aesthetic

Watch Articles

Cook Fresh Cauliflower

  • Terms of Use
  • Privacy Policy
  • Do Not Sell or Share My Info
  • Not Selling Info

Don’t miss out! Sign up for

wikiHow’s newsletter

Logo for BCcampus Open Publishing

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Chapter 9: Radicals

9.10 Rate Word Problems: Work and Time

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

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

For this problem:

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

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

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

Example 9.10.1

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

The equation to solve is:

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

Example 9.10.2

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

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

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

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

Example 9.10.3

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

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

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

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

Example 9.10.4

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

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

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

Example 9.10.5

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

The 7 minutes to drain will be subtracted.

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

17.5 min or 17 min 30 sec is the solution

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

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

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

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

Answer Key 9.10

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

Share This Book

how to solve work done problems

Talk to our experts

1800-120-456-456

  • Work Done Problems

ffImage

A Brief Explanation of Work Done Problems

Work problems typically occur when two people are painting a house together. You are typically asked how long it takes each individual to paint a house of a comparable size and how long it will take the two of them to paint the house when they collaborate. So, here the concept of work done will be used. In this article, we will learn how we can solve the work done problems.

Work Done Related Important Formula

In time and work, we will learn to calculate and determine the number of hours needed to complete a task as well as the amount of work completed in a specific period of time. We are aware that a person's productivity is closely correlated with the time it takes him to do a task.

For Example: Suppose Shyam can finish a work in 7 days.

Then, work done by Shyam in 1 day $=\dfrac{1}{7}$

Suppose if a person A can finish a work in $\mathrm{n}$ days.

Then, work done by $A$ in 1 day $=1 / n^{\text {th }}$ part of the work.

Suppose that the work done by $\mathrm{A}$ in 1 day is $\dfrac{1}{n}$

Then, time taken by $\mathrm{A}$ to finish the whole work $=\mathrm{n}$ days.

Solved Problems on Work Done

Here are some solved problems on work done , through which it can be understood in a better way:

Q1. Piyush and Rahul together can complete a work in 18 days. Piyush alone can do the same work in 24 days. What will be the number of days Rahul alone can complete the whole work?

Ans: Piyush and Rahul can complete the work in 18 days

Piyush alone can complete the work in 24 days

Taking the L.C.M of 18 and 24

L.C.M of 18 and 24 is 72

$\Rightarrow$ One day work of Piyush and Rahul $=\dfrac{72}{18}=4$

$\Rightarrow$ One day work of Piyush $=\dfrac{72}{24}=3$

$\Rightarrow$ One day work of Rahul $=4-3=1$

$\Rightarrow$ Number of days Rahul alone takes to complete the work $=\dfrac{72}{1}=72$

$\therefore$ The number of days Rahul takes to complete the whole work is 72.

Q2. A and $B$ together can do a piece of work in 15 days, while $B$ alone can finish it in 20 days. In how many days can $A$ alone finish the work?

Ans: Time taken by $(A+B)$ to finish the work $=15$ days.

Time taken by B alone to finish the work is 20 days.

$(A+B)$ 's 1 day's work $=\dfrac{1}{15}$

and $B^{\prime}$ s 1 day's work $=\dfrac{1}{20}$

A's 1 day's work $=\left\{(A+B)^{\prime}\right.$ s 1 day's work $\}-\left\{B^{\prime}\right.$ s 1 day's work $\}$

$=(\dfrac{1}{15}-\dfrac{1}{20})=\dfrac{4-3}{60}=\dfrac{1}{60}$

Therefore, A alone can finish the work in 60 days.

Q 3. A can do a piece of work in 25 days and $B$ can finish it in 20 days. They work together for 5 days and then A leaves. In how many days will $B$ finish the remaining work?

Ans: Time taken by $\mathrm{A}$ to finish the work $=25$ days.

A's 1 day's work $=\dfrac{1}{25}$

Time taken by $B$ to finish the work $=20$ days.

B's 1 day's work $=\dfrac{1}{20}$

$(A+B)$ 's 1 day's work $=(\dfrac{1}{25}+\dfrac{1}{20})=\dfrac{9}{100}$

$(A+B)$ 's 5 day's work $(5 \times \dfrac{9}{100})=\dfrac{45}{100}=\dfrac{9}{20}$

Remaining work $(1-\dfrac{9}{20})=\dfrac{11}{20}$

Now, $\dfrac{11}{20}$ work is done by $B$ in 1 day

Therefore, $\dfrac{11}{20}$ work will be done by $B$ in $(\dfrac{11}{20} \times 20)$ days $=11$ days.

Hence, the remaining work is done by $B$ in 11 days.

Practice Questions

Here are practice questions related to work done, through which it can be made in a better way:

Q1. In 300 days, Sanjay finished the school project. If Piyush is 50% more productive than Sanjay, how many days will it take him to finish the identical task?

Q2. A task can be completed by Sourav and Anshu in 18 days. Anshu and Himanshu can do it in 24 and Sourav and Himanshu can do it in 36 days, respectively. How many days will it take Sourav, Himanshu, and Anshu to complete the task if they collaborate?

Ans . 16 Days

Q3. In 600 days, Sanjay finished the school project. If Piyush is 20% more productive than Sanjay, how many days will it take him to finish the identical task?

Q4. Piyush, Santosh, and Ramesh are hired as construction workers by a builder on one of his projects. They finish a piece of work in 20, 30, and 60 days, respectively. If Santosh and Ramesh help Piyush every third day, how many days will it take him to do the entire task?

Ans . 15 Days

Q5. A project that Santosh and Prajapati are working on can be finished in 30 days. Santosh put in 16 days of labour, and Prajapati took 44 days to finish it all. How many days would it have taken Prajapati to do the entire project on her own?

Ans . 60 Days

In this article, we discussed the topic of time and work. Time and work are related concepts. Time is a unit of time, work is an activity done in a given time. Time and work are very important in the field of mathematics. This is because it helps them understand the concept of time and how it can be used to solve mathematical equations. The amount of work you do is related to the amount of time you spend on it. We have understood the topic of time and work perfectly by using some solved problems on work done and time spent.

arrow-right

FAQs on Work Done Problems

1. What does the term "word problems" mean?

Word problems are defined as mathematical problems that are written in an ordinary language rather than mathematical terms and symbols.

2. What are some of the different work done problems?

Problems based on Painting and Pipes, Tubs and Man-Hours, Unequal Times Etc. are some of the different work done problems.

3. Are time and work directly or inversely correlated?

The relationship between time and work is directly correlated. As a result, more work may be done in more time, and vice versa.

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

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?

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?

  • Introduction
  • 1.1 The Scope and Scale of Physics
  • 1.2 Units and Standards
  • 1.3 Unit Conversion
  • 1.4 Dimensional Analysis
  • 1.5 Estimates and Fermi Calculations
  • 1.6 Significant Figures
  • 1.7 Solving Problems in Physics
  • Key Equations
  • Conceptual Questions
  • Additional Problems
  • Challenge Problems
  • 2.1 Scalars and Vectors
  • 2.2 Coordinate Systems and Components of a Vector
  • 2.3 Algebra of Vectors
  • 2.4 Products of Vectors
  • 3.1 Position, Displacement, and Average Velocity
  • 3.2 Instantaneous Velocity and Speed
  • 3.3 Average and Instantaneous Acceleration
  • 3.4 Motion with Constant Acceleration
  • 3.5 Free Fall
  • 3.6 Finding Velocity and Displacement from Acceleration
  • 4.1 Displacement and Velocity Vectors
  • 4.2 Acceleration Vector
  • 4.3 Projectile Motion
  • 4.4 Uniform Circular Motion
  • 4.5 Relative Motion in One and Two Dimensions
  • 5.2 Newton's First Law
  • 5.3 Newton's Second Law
  • 5.4 Mass and Weight
  • 5.5 Newton’s Third Law
  • 5.6 Common Forces
  • 5.7 Drawing Free-Body Diagrams
  • 6.1 Solving Problems with Newton’s Laws
  • 6.2 Friction
  • 6.3 Centripetal Force
  • 6.4 Drag Force and Terminal Speed
  • 7.2 Kinetic Energy
  • 7.3 Work-Energy Theorem
  • 8.1 Potential Energy of a System
  • 8.2 Conservative and Non-Conservative Forces
  • 8.3 Conservation of Energy
  • 8.4 Potential Energy Diagrams and Stability
  • 8.5 Sources of Energy
  • 9.1 Linear Momentum
  • 9.2 Impulse and Collisions
  • 9.3 Conservation of Linear Momentum
  • 9.4 Types of Collisions
  • 9.5 Collisions in Multiple Dimensions
  • 9.6 Center of Mass
  • 9.7 Rocket Propulsion
  • 10.1 Rotational Variables
  • 10.2 Rotation with Constant Angular Acceleration
  • 10.3 Relating Angular and Translational Quantities
  • 10.4 Moment of Inertia and Rotational Kinetic Energy
  • 10.5 Calculating Moments of Inertia
  • 10.6 Torque
  • 10.7 Newton’s Second Law for Rotation
  • 10.8 Work and Power for Rotational Motion
  • 11.1 Rolling Motion
  • 11.2 Angular Momentum
  • 11.3 Conservation of Angular Momentum
  • 11.4 Precession of a Gyroscope
  • 12.1 Conditions for Static Equilibrium
  • 12.2 Examples of Static Equilibrium
  • 12.3 Stress, Strain, and Elastic Modulus
  • 12.4 Elasticity and Plasticity
  • 13.1 Newton's Law of Universal Gravitation
  • 13.2 Gravitation Near Earth's Surface
  • 13.3 Gravitational Potential Energy and Total Energy
  • 13.4 Satellite Orbits and Energy
  • 13.5 Kepler's Laws of Planetary Motion
  • 13.6 Tidal Forces
  • 13.7 Einstein's Theory of Gravity
  • 14.1 Fluids, Density, and Pressure
  • 14.2 Measuring Pressure
  • 14.3 Pascal's Principle and Hydraulics
  • 14.4 Archimedes’ Principle and Buoyancy
  • 14.5 Fluid Dynamics
  • 14.6 Bernoulli’s Equation
  • 14.7 Viscosity and Turbulence
  • 15.1 Simple Harmonic Motion
  • 15.2 Energy in Simple Harmonic Motion
  • 15.3 Comparing Simple Harmonic Motion and Circular Motion
  • 15.4 Pendulums
  • 15.5 Damped Oscillations
  • 15.6 Forced Oscillations
  • 16.1 Traveling Waves
  • 16.2 Mathematics of Waves
  • 16.3 Wave Speed on a Stretched String
  • 16.4 Energy and Power of a Wave
  • 16.5 Interference of Waves
  • 16.6 Standing Waves and Resonance
  • 17.1 Sound Waves
  • 17.2 Speed of Sound
  • 17.3 Sound Intensity
  • 17.4 Normal Modes of a Standing Sound Wave
  • 17.5 Sources of Musical Sound
  • 17.7 The Doppler Effect
  • 17.8 Shock Waves
  • B | Conversion Factors
  • C | Fundamental Constants
  • D | Astronomical Data
  • E | Mathematical Formulas
  • F | Chemistry
  • G | The Greek Alphabet

Learning Objectives

By the end of this section, you will be able to:

  • Represent the work done by any force
  • Evaluate the work done for various forces

In physics, work is done on an object when energy is transferred to the object. In other words, work is done when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. We first define the increment of work dW done by a force F → F → acting through an infinitesimal displacement d r → d r → as the dot product of these two vectors:

Then, we can add up the contributions for infinitesimal displacements, along a path between two positions, to get the total work.

Work Done by a Force

The work done by a force is the integral of the force with respect to displacement along the path of the displacement:

The vectors involved in the definition of the work done by a force acting on a particle are illustrated in Figure 7.2 . While in general, Equation 7.2 requires mathematics beyond the scope of this text, in many simple situations this integral becomes a familiar integral in one variable. We will examine several such examples and restrict our discussion to these cases.

We choose to express the dot product in terms of the magnitudes of the vectors and the cosine of the angle between them, because the meaning of the dot product for work can be put into words more directly in terms of magnitudes and angles. We could equally well have expressed the dot product in terms of the various components introduced in Vectors . In two dimensions, these were the x - and y -components in Cartesian coordinates, or the r - and φ φ -components in polar coordinates; in three dimensions, it was just x -, y -, and z -components. Which choice is more convenient depends on the situation. In words, you can express Equation 7.1 for the work done by a force acting over a displacement as a product of one component acting parallel to the other component. From the properties of vectors, it doesn’t matter if you take the component of the force parallel to the displacement or the component of the displacement parallel to the force—you get the same result either way.

Recall that the magnitude of a force times the cosine of the angle the force makes with a given direction is the component of the force in the given direction. The components of a vector can be positive, negative, or zero, depending on whether the angle between the vector and the component-direction is between 0 ° 0 ° and 90 ° 90 ° or 90 ° 90 ° and 180 ° 180 ° , or is equal to 90 ° 90 ° . As a result, the work done by a force can be positive, negative, or zero, depending on whether the force is generally in the direction of the displacement, generally opposite to the displacement, or perpendicular to the displacement. The maximum work is done by a given force when it is along the direction of the displacement ( cos θ = ± 1 cos θ = ± 1 ), and zero work is done when the force is perpendicular to the displacement ( cos θ = 0 cos θ = 0 ).

The units of work are units of force multiplied by units of length, which in the SI system is newtons times meters, N · m. N · m. This combination is called a joule , for historical reasons that we will mention later, and is abbreviated as J. In the English system, still used in the United States, the unit of force is the pound (lb) and the unit of distance is the foot (ft), so the unit of work is the foot-pound ( ft · lb ) . ( ft · lb ) .

Work Done by Constant Forces and Contact Forces

The simplest work to evaluate is that done by a force that is constant in magnitude and direction. In this case, we can factor out the force; the remaining integral is just the total displacement, which only depends on the end points A and B , but not on the path between them:

Figure 7.3 (a) shows a person exerting a constant force F → F → along the handle of a lawn mower, which makes an angle θ θ with the horizontal. The horizontal displacement of the lawn mower, over which the force acts, is d → . d → . The work done on the lawn mower is W = F → · d → = F d cos θ W = F → · d → = F d cos θ , which the figure also illustrates as the horizontal component of the force times the magnitude of the displacement.

Figure 7.3 (b) shows a person holding a briefcase. The person must exert an upward force, equal in magnitude to the weight of the briefcase, but this force does no work, because the displacement over which it acts is zero.

In Figure 7.3 (c), where the person in (b) is walking horizontally with constant speed, the work done by the person on the briefcase is still zero, but now because the angle between the force exerted and the displacement is 90 ° 90 ° ( F → F → perpendicular to d → d → ) and cos 90 ° = 0 cos 90 ° = 0 .

Example 7.1

Calculating the work you do to push a lawn mower.

Substituting the known values gives

Significance

When you mow the grass, other forces act on the lawn mower besides the force you exert—namely, the contact force of the ground and the gravitational force of Earth. Let’s consider the work done by these forces in general. For an object moving on a surface, the displacement d r → d r → is tangent to the surface. The part of the contact force on the object that is perpendicular to the surface is the normal force N → . N → . Since the cosine of the angle between the normal and the tangent to a surface is zero, we have

The normal force never does work under these circumstances. (Note that if the displacement d r → d r → did have a relative component perpendicular to the surface, the object would either leave the surface or break through it, and there would no longer be any normal contact force. However, if the object is more than a particle, and has an internal structure, the normal contact force can do work on it, for example, by displacing it or deforming its shape. This will be mentioned in the next chapter.)

The part of the contact force on the object that is parallel to the surface is friction, f → . f → . For this object sliding along the surface, kinetic friction f → k f → k is opposite to d r → , d r → , relative to the surface, so the work done by kinetic friction is negative. If the magnitude of f → k f → k is constant (as it would be if all the other forces on the object were constant), then the work done by friction is

where | l A B | | l A B | is the path length on the surface. The force of static friction does no work in the reference frame between two surfaces because there is never displacement between the surfaces. As an external force, static friction can do work. Static friction can keep someone from sliding off a sled when the sled is moving and perform positive work on the person. If you’re driving your car at the speed limit on a straight, level stretch of highway, the negative work done by air resistance is balanced by the positive work done by the static friction of the road on the drive wheels. You can pull the rug out from under an object in such a way that it slides backward relative to the rug, but forward relative to the floor. In this case, kinetic friction exerted by the rug on the object could be in the same direction as the displacement of the object, relative to the floor, and do positive work. The bottom line is that you need to analyze each particular case to determine the work done by the forces, whether positive, negative or zero.

Example 7.2

Moving a couch.

  • The work done by friction i W = − ( 0.6 ) ( 1 kN ) ( 3 m + 1 m ) = − 2.4 kJ . W = − ( 0.6 ) ( 1 kN ) ( 3 m + 1 m ) = − 2.4 kJ .
  • The length of the path along the hypotenuse is 10 m 10 m , so the total work done against friction is W = ( 0.6 ) ( 1 kN ) ( 3 m + 1 m + 1 0 m ) = 4.3 kJ . W = ( 0.6 ) ( 1 kN ) ( 3 m + 1 m + 1 0 m ) = 4.3 kJ .

Check Your Understanding 7.1

Can kinetic friction ever be a constant force for all paths?

The other force on the lawn mower mentioned above was Earth’s gravitational force, or the weight of the mower. Near the surface of Earth, the gravitational force on an object of mass m has a constant magnitude, mg , and constant direction, vertically down. Therefore, the work done by gravity on an object is the dot product of its weight and its displacement. In many cases, it is convenient to express the dot product for gravitational work in terms of the x -, y -, and z -components of the vectors. A typical coordinate system has the x -axis horizontal and the y -axis vertically up. Then the gravitational force is − m g j ^ , − m g j ^ , so the work done by gravity, over any path from A to B , is

The work done by a constant force of gravity on an object depends only on the object’s weight and the difference in height through which the object is displaced. Gravity does negative work on an object that moves upward ( y B > y A y B > y A ), or, in other words, you must do positive work against gravity to lift an object upward. Alternately, gravity does positive work on an object that moves downward ( y B < y A y B < y A ), or you do negative work against gravity to “lift” an object downward, controlling its descent so it doesn’t drop to the ground. (“Lift” is used as opposed to “drop”.)

Example 7.3

Shelving a book.

  • Since the book starts on the shelf and is lifted down y B − y A = − 1 m y B − y A = − 1 m , we have W = − ( 20 N ) ( − 1 m ) = 20 J . W = − ( 20 N ) ( − 1 m ) = 20 J .
  • There is zero difference in height for any path that begins and ends at the same place on the shelf, so W = 0 . W = 0 .

Check Your Understanding 7.2

Can Earth’s gravity ever be a constant force for all paths?

Work Done by Forces that Vary

In general, forces may vary in magnitude and direction at points in space, and paths between two points may be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path,

Here, the components of the force are functions of position along the path, and the displacements depend on the equations of the path. (Although we chose to illustrate dW in Cartesian coordinates, other coordinates are better suited to some situations.) Equation 7.2 defines the total work as a line integral, or the limit of a sum of infinitesimal amounts of work. The physical concept of work is straightforward: you calculate the work for tiny displacements and add them up. Sometimes the mathematics can seem complicated, but the following example demonstrates how cleanly they can operate.

Example 7.4

Work done by a variable force over a curved path.

Then, the integral for the work is just a definite integral of a function of x .

The integral of x 2 x 2 is x 3 / 3 , x 3 / 3 , so

Check Your Understanding 7.3

Find the work done by the same force in Example 7.4 over a cubic path, y = ( 0.25 m −2 ) x 3 y = ( 0.25 m −2 ) x 3 , between the same points A = ( 0 , 0 ) A = ( 0 , 0 ) and B = ( 2 m, 2 m ) . B = ( 2 m, 2 m ) .

One very important and widely applicable variable force is the force exerted by a perfectly elastic spring, which satisfies Hooke’s law F → = − k Δ x → , F → = − k Δ x → , where k is the spring constant, and Δ x → = x → − x → eq Δ x → = x → − x → eq is the displacement from the spring’s unstretched (equilibrium) position ( Newton’s Laws of Motion ). Note that the unstretched position is only the same as the equilibrium position if no other forces are acting (or, if they are, they cancel one another). Forces between molecules, or in any system undergoing small displacements from a stable equilibrium, behave approximately like a spring force.

To calculate the work done by a spring force, we can choose the x -axis along the length of the spring, in the direction of increasing length, as in Figure 7.7 , with the origin at the equilibrium position x eq = 0 . x eq = 0 . (Then positive x corresponds to a stretch and negative x to a compression.) With this choice of coordinates, the spring force has only an x -component, F x = − k x F x = − k x , and the work done when x changes from x A x A to x B x B is

Notice that W A B W A B depends only on the starting and ending points, A and B , and is independent of the actual path between them, as long as it starts at A and ends at B. That is, the actual path could involve going back and forth before ending.

Another interesting thing to notice about Equation 7.5 is that, for this one-dimensional case, you can readily see the correspondence between the work done by a force and the area under the curve of the force versus its displacement. Recall that, in general, a one-dimensional integral is the limit of the sum of infinitesimals, f ( x ) d x f ( x ) d x , representing the area of strips, as shown in Figure 7.8 . In Equation 7.5 , since F = − k x F = − k x is a straight line with slope − k − k , when plotted versus x , the “area” under the line is just an algebraic combination of triangular “areas,” where “areas” above the x -axis are positive and those below are negative, as shown in Figure 7.9 . The magnitude of one of these “areas” is just one-half the triangle’s base, along the x -axis, times the triangle’s height, along the force axis. (There are quotation marks around “area” because this base-height product has the units of work, rather than square meters.)

Example 7.5

Work done by a spring force.

For part (a), x A = 0 x A = 0 and x B = 6 cm x B = 6 cm ; for part (b), x B = 6 cm x B = 6 cm and x B = 12 cm x B = 12 cm . In part (a), the work is given and you can solve for the spring constant; in part (b), you can use the value of k , from part (a), to solve for the work.

  • W = 0.54 J = 1 2 k [ ( 6 cm ) 2 − 0 ] W = 0.54 J = 1 2 k [ ( 6 cm ) 2 − 0 ] , so k = 3 N/cm . k = 3 N/cm .
  • W = 1 2 ( 3 N/cm ) [ ( 12 cm ) 2 − ( 6 cm ) 2 ] = 1.62 J . W = 1 2 ( 3 N/cm ) [ ( 12 cm ) 2 − ( 6 cm ) 2 ] = 1.62 J .

Solver Title

Practice

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 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

Click to reveal more operations

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

Please add a message.

Message received. Thanks for the feedback.

how to solve work done problems

  • Testimonial
  • Web Stories

Hitbullseye Logo

Learning Home

how to solve work done problems

Not Now! Will rate later

how to solve work done problems

Time and Work Formula and Solved Problems

how to solve work done 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.

Time and Work Concepts

Problems on Ages Practice Problems : Level 02

Chain Rule : Theory & Concepts

Chain Rule : Theory & Concepts

Chain Rule Solved Examples

Chain Rule Solved Examples

Chain Rule Practice Problems

Chain Rule Practice Problems: Level 01

Chain Rule Practice Problems

Chain Rule Practice Problems : Level 02

Problems on Numbers System : Level 02

Problems on Numbers System : Level 02

Download our app.

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

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

Download on App Store

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

Download on App Store

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.

  • Add Fractions
  • Simplify Fractions
  • Government Exam Articles
  • Time and Work - Concept, Formula & Aptitude Questions

Time and Work - Concept, Formula & Aptitude Questions

Time and Work is one of the most common quantitative aptitude topics which is asked in the Government exams . This is one of those topics which candidates are familiar with even before they start their competitive exam preparation.

The concept of time and work remains the same, however, the type of questions asked may have a bit of variety. Mostly, 1-2 words problems are asked from this topic but candidates must also keep themselves prepared to have questions in data sufficiency and data interpretation to be picked up from time and work.

Candidates can check the detailed syllabus for the quantitative aptitude section along with the list of exams, which include this subject as a part of its syllabus and can visit the linked article.

For aspirants who shall be appearing for the competitive exams for the first time or the ones who are preparing for the exams for a while must know that sufficient time must be devoted to each subject so that your concept and basics are clear. This will help you cope with the tough competition for each exam.

In this article, we bring to you the concept of time and work along with the formulas which shall help you answer the questions easily. Also, further below in the article, we have a few sample time and work questions with solutions for your reference.

Interested aspirants can also check the links given below and strengthen their preparation for the quantitative aptitude section:

Time and Work Questions PDF:- Download PDF Here

Time and Work – Introduction and Concept

Before we move on to the questions and important formulas, it is important that a candidate is well aware of the concept and the types of questions which may be asked in the exam. 

Time and work deals with the time taken by an individual or a group of individuals to complete a piece of work and the efficiency of the work done by each of them.

Time and Work - Quantitative Aptitude for Government Exams

Given below are the basic type of questions which may be asked in the exam with respect to the time and work topic:

  • To find the efficiency of a person
  • To find the time taken by an individual to do a piece of work
  • To find the time taken by a group of individuals to complete a piece of work
  • Work done by an individual in a certain time duration
  • Work done by a group of individuals in a certain time duration

Mostly the questions asked may involve one of these things to find and candidates can use the related formulas to easily get through the answers for the same.

A few other quantitative aptitude related links have been given below. It is suggested that candidates go through each of these topics carefully to excel in the upcoming Government exams:

Important Time and Work Formula

Knowing the formulas can completely link you to a solution as soon as you read the question. Thus, knowing the formula for any numerical ability topic make the solution and the related calculations simpler.

Given below are a few such important time and work formulas for your reference:

  • Work Done = Time Taken × Rate of Work
  • Rate of Work = 1 / Time Taken
  • Time Taken = 1 / Rate of Work
  • If a piece of work is done in x number of days, then the work done in one day = 1/x
  • Total Wok Done = Number of Days × Efficiency
  • Efficiency and Time are inversely proportional to each other
  • X:y is the ratio of the number of men which are required to complete a piece of work, then the ratio of the time taken by them to complete the work will be y:x
  • If x number of people can do W1 work, in D1 days, working T1 hours each day and the number of people can do W2 work, in D2 days, working T2 hours each day, then the relation between them will be

Time and Work Formula

Aspirants for the various Government exams must start their preparation now to ensure they give their 100 per cent and complete hard work and dedication to their preparation.

Online Quiz 2023

Time & Work – Sample Questions

All the hard work can go in vain if the candidate does not solve questions based on time and work on a regular basis and try using the different formulas to crack the solution for each question in an even shorter time span.

So, discussed below are a few time and work questions to give an idea as to what type of questions are asked in the competitive exam and what format and pattern is used for the same. 

Q 1. A builder appoints three construction workers Akash, Sunil and Rakesh on one of his sites. They take 20, 30 and 60 days respectively to do a piece of work. How many days will it take Akash to complete the entire work if he is assisted by Sunil and Rakesh every third day?

Answer: (2) 15 days

Total work done by Akash, Sunil and Rakesh in 1 day = {(1/20) + (1/30) + (1/60)} = 1/10

Work done along by Akash in 2 days = (1/20) × 2 = 1/10

Work Done in 3 days (1 day of all three together + 2 days of Akash’s work) = (1/10) + (1/10) = 1/5

So, work done in 3 days = 1/5

Time taken to complete the work = 5×3 = 15 days

Q 2. To complete a piece of work, Samir takes 6 days and Tanvir takes 8 days alone respectively. Samir and Tanvir took Rs.2400 to do this work. When Amir joined them, the work was done in 3 days. What amount was paid to Amir?

Answer: (1) Rs.300

Total work done by Samir and Tanvir = {(1/6) + (1/8)} = 7/24

Work done by Amir in 1 day = (1/3) – (7/24) = 1/24

Amount distributed between each of them =  (1/6) : (1/8) : (1/24) = 4:3:1

Amount paid to Amir = (1/24) × 3 × 2400 = Rs.300

Q 3. Dev completed the school project in 20 days. How many days will Arun take to complete the same work if he is 25% more efficient than Dev? 

Answer: (3) 16 days

Let the days taken by Arun to complete the work be x

The ratio of time taken by Arun and Dev = 125:100 = 5:4

5:4 :: 20:x

⇒ x = {(4×20) / 5}

Q 4. Time taken by A to finish a piece of work is twice the time taken B and thrice the time taken by C. If all three of them work together, it takes them 2 days to complete the entire work. How much work was done by B alone?

  • Cannot be determined

Answer: (2) 6 days

Time taken by A  = x days

Time taken by B = x/2 days

Time Taken by C = x/3 days

⇒ {(1/x) + (2/x) + (3/x) = 1/2

⇒ 6/x = 1/2

Time taken by B = x/2 = 12/2 = 6 days

Q 5. Sonal and Preeti started working on a project and they can complete the project in 30 days. Sonal worked for 16 days and Preeti completed the remaining work in 44 days. How many days would Preeti have taken to complete the entire project all by herself? 

Answer: (5) 60 days

Let the work done by Sonal in 1 day be x

Let the work done by Preeti in 1 day be y

Then, x+y = 1/30 ——— (1)

⇒ 16x + 44y = 1  ——— (2)

Solving equation (1) and (2), 

Thus, Preeti can complete the entire work in 60 days

Candidates must solve more such questions to understand the concept better and analyse the previous year papers to know more about the pattern of questions asked.

Other Related Links:

Candidates interested in applying for the upcoming Bank, Insurance, SSC, RRB and other Government sector exams must start their preparation now.

For any help and assistance regarding the exam preparation, candidates can turn to BYJU’S.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

how to solve work done problems

Connect with us for Free Preparation

Get access to free crash courses & video lectures for all government exams..

  • Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Live Support

TIME AND WORK PROBLEMS

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

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

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

4. Time and work are always in direct proportion.

more work ----> more time

less work ----> less time

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

Problem 1 :

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

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

Problem 2 :

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

From the given information, we can have

(A + B)'s 1 day work = ⅙

(B + C)'s 1 day work = ⅛

(A + C)'s 1 day work = ¹⁄₁₂

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

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

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

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

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

2 (A + B + C)'s 1 day work = ⁹⁄₂₄

2 (A + B + C)'s 1 day work = ⅜

(A + B + C)'s 1 day work = ³⁄₁₆

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

= 5 ⅓  days

Problem 3 :

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

(A + B)'s 1 day work = ¹⁄₁₅

(B + C)'s 1 day work = ¹⁄₃₀

(A + C)'s 1 day work = ¹⁄₁₈

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

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

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

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

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

2 (A + B + C)'s 1 day work = ¹⁴⁄₉₀

2 (A + B + C)'s 1 day work = ⁷⁄₄₅

(A + B + C)'s 1 day work = ⁷⁄₉₀

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

= 9  ⋅  ⁷⁄₉₀

=  ⁷⁄₁₀

Amount of work left for B and C to complete is

=  ³⁄₁₀

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

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

= ³⁄₁₀   ÷  ¹⁄₃₀

= ³⁄₁₀   ⋅   ³⁰⁄₁

Problem 4 :

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

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

Total no. of men appointed initially = 50.

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

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

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

For the remaining work,

70 men ----> 70 days

50 men -----> x days

The above one is a inverse variation.

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

By inverse variation, we have

70  ⋅ 70 = 50 ⋅ x

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

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

Problem 5 :

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

A's 1 hour work = ⅒

B's 1 hour work = ¹⁄₁₅

C's 1 hour work = ¹⁄₂₀

In the first hour, we have

(A + B)'s work = ⅒  + ¹⁄₁₅

(A + B)'s work = ³⁄₃₀ + ²⁄₃₀

(A + B)'s work = ⁵⁄₃₀

(A + B)'s work = ⅙

In the second hour, we have

(A + C)'s work = ⅒  +  ¹⁄₂₀

(A + C)'s work = ²⁄₂₀ +  ¹⁄₂₀

(A + C)'s work = ³⁄₂₀

Amount of work done in each two hours is

= ¹⁰⁄₆₀ +  ⁹⁄₆₀

Amount of work done :

In the first 2 hours :  ¹⁹⁄₆₀

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

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

After 6 hours, the remaining work will be

=  ¹⁄₂₀

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

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

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

= ¹⁄₂₀  ÷  ⅒

=  ¹⁄₂₀ ⋅ ¹⁰⁄₁

=  ½ hours

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

So, total time taken to complete the work is

= 6 hours + 30  minutes

= 6 ½  hours

Kindly mail your feedback to   [email protected]

We always appreciate your feedback.

© All rights reserved. onlinemath4all.com

  • Sat Math Practice
  • SAT Math Worksheets
  • PEMDAS Rule
  • BODMAS rule
  • GEMDAS Order of Operations
  • Math Calculators
  • Transformations of Functions
  • Order of rotational symmetry
  • Lines of symmetry
  • Compound Angles
  • Quantitative Aptitude Tricks
  • Trigonometric ratio table
  • Word Problems
  • Times Table Shortcuts
  • 10th CBSE solution
  • PSAT Math Preparation
  • Privacy Policy
  • Laws of Exponents

Recent Articles

RSS

Eliminating the Parameter in Parametric Equations

Feb 16, 24 11:02 AM

Easy Way to Find Square Root of a Number

Feb 16, 24 01:36 AM

sqrt1p2.png

Percent Proportion Worksheet

Feb 14, 24 09:47 PM

tutoring.png

Algebra Work Problems

Related Pages Work Problems that involve two persons Work Problems that involve more than two persons More Algebra Word Problems

How to solve algebra work problems?

Work Problems are word problems that involve different people or entities doing work together but at different rates. If the people or entities were working at the same rate then we would use the Inversely Proportional Method .

In these lessons, we will learn work problems with pipes filling up a tank and work problems with pumps draining a tank. Look at the related pages above for work problems that involve people.

The following diagram shows the formula for Work Word Problems. Scroll down the page for more examples and solutions for work word problems.

Work Word Problems

“Work” Problems: Pipes Filling up a Tank

Example 1: A tank can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the tank is full, it can be drained by pipe C in 4 hours. if the tank is initially empty and all three pipes are open, how many hours will it take to fill up the tank?

Solution: 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

Work Problem: Pumps draining a tank

Example: A swimming pool can be emptied in 6 hours using a 10-horsepower pump along with a 6-horsepower pump. The 6-horsepower pump requires 5 hours more than the 10-horsepower pump to empty the pool when working by itself. How long will it take to empty the pool using just the 10-horsepower pump?

Cooperative Work Word Problems (Time to Finish)

Pump A can empty a pool in 20 hours and pump B can empty it in 24 hours. Working together, how long will it take to empty the pool?

A painter can paint a building in 15 days and a coworker can do the same job in 10 days. If the first painter starts and 3 days later the coworker joins in to help finish the job, how many days doe it take to paint the building?

Rates of Performing Work Problems

Example: It takes 12 hours to fill a water tank. It takes 16 hours to drain the same water tank. How long will it take to fill the tank if the drain is left open?

Mathway Calculator Widget

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Microsoft

Game Central

how to solve work done problems

Get step-by-step explanations

Graph your math problems

Graph your math problems

Practice, practice, practice

Practice, practice, practice

Get math help in your language

Get math help in your language

CNBC

How to solve problems at work, says Deloitte exec: 'My best idea didn't come when I was sitting tied to a desk'

D eborah Golden has a pretty unique job: As consulting firm Deloitte's chief innovation officer , she gets to "solve really hard problems," she says. Often this can mean leaning into emerging technologies to evolve offerings and products for clients.

"I'm very good at seeing A to Z," she says.

While Golden's job calls for solving big problems on an organizational level, every worker comes across snags in their day-to-day, even if they only have to deal with their specific duties. And with so much going on, it can be easy to get stuck on how to solve those problems.

If you can't figure out how to trim a growing bottom line, how to train your employees in the best way, how to write the next slide in your presentation — and so on, here are a few ways to clear your head and hit refresh on your brain to come up with some solutions.

'It's beautiful outside, you want to go for a walk?'

One good way to get the juices flowing is to step away from your screen. "Have a walking meeting," suggests Golden. She gives the example of a recent meeting she had in which she and a colleague were tackling problems in the pharmaceutical industry.

"I looked at my colleague and I said, 'it's beautiful outside, you want to go for a walk?'" she says. "That was my walking meeting. We spent 30 minutes outside, no laptops, no phones." When they got back, they rushed to their computers to write down the three points they'd come up with.

There's evidence this kind of practice helps rejuvenate your mind. Fresh air and exercise can stimulate blood flow to the brain, which ultimately helps with focus and clarity, certified health and wellness coach Marissa Vicario previously told NBC News .

'You have 30 seconds, go grab something'

If you're a manager and are about to have a meeting with your reports, one good way to help clear their minds is to start the meeting with an exercise.

"I always do an icebreaker at every start of every meeting," says Golden, adding that, "an easy one is, 'you have 30 seconds, go grab something, no matter where you are, and you have to come back and in less than 30 seconds, you have to tell us why you grabbed it.'"

This also helps people disconnect and get some blood flowing in their brains and bodies, and puts "an ease to solving some of the challenges that you're solving in that 30 minutes," she says.

Hit refresh in 'one minute, five minute and 15-minute increments'

"We tend to say we're going to refresh our brain when we take a vacation or at the weekend," says Golden. But you can give your brain refreshers throughout the day.

"What I encourage everyone to very simply do," she says, is "write down on a piece of paper what are the ways that you can refresh your brain in one minute, five minute and 15-minute increments." Call someone you love and tell them you love them. Do a crossword puzzle. Go for a jog. Do jumping jacks in the room — whatever is going to get you to disconnect for that period of time and come back to your task anew.

"My best idea didn't come when I was sitting tied to a desk," she says. "It came when I was outside, when I was in the shower, when I was on a plane, when I was listening to music, when I was running … That's when the ideas come."

Create those opportunities for yourself even during the workday and see what you can accomplish.

Want to earn more and land your dream job?  Join the free CNBC Make It: Your Money virtual event  on Oct. 17 at 1 p.m. ET to learn how to level up your interview and negotiating skills, build your ideal career, boost your income and grow your wealth.  Register  for free today.

DON'T MISS:  Want to be smarter and more successful with your money, work & life?  Sign up for our new newsletter !

How to get over writer's block, unlock your creativity, and brainstorm great ideas

3 ways to use AI right now to get ahead—if you do, you're 'really going to succeed,' says expert

4 sure-fire ways to stimulate creativity

How to solve problems at work, says Deloitte exec: 'My best idea didn't come when I was sitting tied to a desk'

right-icon

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Vivamus convallis sem tellus, vitae egestas felis vestibule ut.

Error message details.

Reuse Permissions

Request permission to republish or redistribute SHRM content and materials.

The Performance Review Problem

As the arcane annual assessment earns a failing grade, employers struggle to create a better system to measure and motivate their workers.

​After an annual review that lasted about 10 minutes, a New Jersey-based account coordinator knew it was time to leave the public relations agency where he had worked for almost a year. 

The 25-year-old, who requested anonymity, asked for the meeting because his boss had not mentioned any formal assessment process, nor had his manager ever critiqued his work. The coordinator says he sat with a trio of senior executives who did not ask him any questions beyond how he would rate himself. He says they ignored his requests for guidance on how to advance at the agency. 

Screen Shot 2023-03-15 at 85749 AM.png

This example also illustrates one of the common failures in performance management: limiting reviews to once or twice a year without having any other meaningful career discussions in between. Nearly half (49 percent) of companies give annual or semiannual reviews, according to a study of 1,000 full-time U.S. employees released late last year by software company Workhuman. 

The only situation that is worse than doing one review per year is doing none at all, experts say. The good news is that only 7 percent of companies are keeping employees in the dark about their performance, and 28 percent of organizations are conducting assessments quarterly, the Workhuman study found.  

A Pervasive Problem

Reviews generally do not work.

That doesn’t mean that more-frequent formal meetings or casual sit-downs between supervisors and their direct reports are solving the performance review quandary, either. Only about 1 in 4 companies in North America (26 percent) said their performance management systems were effective, according to a survey of 837 companies conducted last fall by consulting firm WTW. And only one-third of the organizations said employees felt their efforts were evaluated fairly. 

Meanwhile, a Gallup survey conducted last year found that 95 percent of managers are dissatisfied with their organization’s review system.

The problem is not new, though it is taking on greater importance, experts say. Millennials and members of Generation Z crave feedback and are focused on career development. Meanwhile, the tight labor market has companies searching for ways to keep high-performing employees in the fold. Fewer than 20 percent of employees feel inspired by their reviews, and disengaged employees cost U.S. companies a collective $1.6 trillion a year, according to Gallup.

Lesli Jennings, a senior director at WTW, says part of the issue is that reviews are now so much more than a discussion of past performance. They include conversations about career development, employee experience and compensation. 

“The performance management design itself is not evolving as quickly as the objectives and the purpose that we have set out for what we want it to do,” Jennings says. 

Screen Shot 2023-03-15 at 84340 AM.png

Poor Review Practices

Some argue that means it’s time to completely scrap annual reviews and stop using scales composed of numbers or adjectives to rate employees. 

“Every single human alive today is a horribly unreliable rater of other human beings,” says Marcus Buckingham, head of people and performance research at the Roseland, N.J.-based ADP Research Institute. He says people bring their own backgrounds and personalities to bear in the reviews in what is called the “idiosyncratic rating effect.” He says the ratings managers bestow on others are more a reflection of themselves than of those they’re reviewing.

Buckingham adds that very few positions have quantifiable outcomes that can be considered a measure of competence, talent or success. It’s possible to tally a salesperson’s results or test someone’s knowledge of a computer program, he says, but he’s baffled by attempts to measure attributes such as “leadership potential.”

“I’m going to rate you on a theoretical construct like ‘strategic thinking’? Everybody knows that’s rubbish,” Buckingham says. He adds that performance reviews that offer rankings give “data that’s just bad” and insists that companies rely on data analytics because they don’t trust their managers’ judgment. But instead of working on improving their managers’ skills, he says, they put data systems in place. 

“Because we don’t educate our managers on how to have some of these conversations, we’ve decided that the solution is to give them really bad ratings systems or really bad categorization systems,” Buckingham says. 

R eviewing the Data

A mong North American employers:

  • More than 9 in 10 (93 percent) cited driving organizational performance as a key objective for performance management, yet less than half (44 percent) said their performance management program is ­meeting that objective.
  • Nearly 3 in 4 (72 percent) said ­supporting the career development of their employees is a primary objective, but only 31 percent said their performance management program was meeting that objective.
  • Less than half (49 percent) agreed that managers at their organization are ­effective at assessing the performance of their direct reports. 
  • Only 1 in 3 indicated that employees feel their performance is evaluated fairly. 
  • Just 1 in 6 (16 percent) reported having altered their performance management approach to align with remote and hybrid work models, which are rapidly becoming more prevalent.

Source: WTW 2022 Performance Reset Survey of 837 organizations worldwide, including 150 North American employers.

Data Lovers

Ratings aren’t likely to disappear anytime soon, however. “Data-driven” has become a rallying cry for companies as they seek to operate more efficiently. Organizations are trying to measure everything from sales to productivity, though such efforts can cause turmoil and hurt some individuals’ careers.

A June 2022 study of nearly 30,000 workers at an unnamed North American retail chain found that women were more likely to receive higher overall ratings than men, though women were ranked lower on “potential.” 

In that study, women were 12 percent more likely to be given the lowest rating for potential, as well as 15 percent and 28 percent less likely to receive the middle and highest potential ratings, respectively, according to the professors who conducted the study, Alan Benson of the University of Minnesota, Danielle Li of MIT and Kelly Shue of Yale. The authors also said women were 14 percent less likely to get promoted than men. “Because potential is not directly observed,” they noted, “these assessments can be highly subjective, leaving room for bias.” 

Screen Shot 2023-03-15 at 85749 AM.png

Birmingham left abruptly one afternoon and did not go in to work the next day, which he says Blizzard interpreted as his resignation. Blizzard did not respond to requests for comment.

Stack ranking became popular in the 1980s after it was embraced by General Electric. Its adoption has waned, though several tech companies continue to use it. Google and Twitter relied on stack ranking to decide who to let go in their recent rounds of layoffs, according to published reports.

Birmingham says that the system can cause anxiety and competition, which can kill team cohesion, and that arbitrary lower ratings adversely affect compensation and promotion potential. These systems can also suggest that a manager is ineffective, he says. “It implies that as managers, we basically have not done our job to hire them and train them appropriately or terminate them if they really aren’t working out.”

Birmingham says he is not opposed to ranking systems but doesn’t think they’re necessary. “I feel like the conversation about how to improve your career, what the expectations are for your job and what it will take to get to the next level are all things you can do without a rating,” he says.

Measurements Matter

Grant Pruitt, president and co-founder of Whitebox Real Estate, does not give any type of rating in his performance reviews, though he believes in using data to track his employees’ performance. “What isn’t measured can’t be managed,” says Pruitt, whose company has about 20 employees in several offices across Texas. 

At the beginning of the year, Whitebox employees set goals with their managers. Discussions are held about what benchmarks are reasonable, and these targets can be changed if there is a meaningful shift in business conditions. Team leaders hold weekly department meetings with their direct reports to discuss what’s happening and track progress. Managers hold quarterly private reviews with individuals to dig deeper into whether they’re meeting their goals and if not, why.

“Was it an achievable goal? Realistic? If it was, then what do we need to do to make sure we don’t miss it the next time?” Pruitt says. Whitebox switched to quarterly reviews about four years ago to address problems earlier and avoid having issues fester, Pruitt adds.

It’s easier to set goals for people in sales than for those in other departments, Pruitt concedes. However, he adds that executives need to brainstorm about targets they can use for other roles. For example, administrative employees can be rated on how quickly and efficiently they handle requests.

Pruitt maintains that the goal system makes it easier to respond when an employee disagrees with their manager about their performance review because there are quantitative measures to examine. The data also helps eliminate any unconscious bias a manager may have and helps ensure that a leader isn’t just giving an employee a good rating because they work out at the same gym or their children go to school together.

“I think that’s really where the numbers and the data are important,” Pruitt says. “The data doesn’t know whose kids play on the same sports team.”

Whitebox employees are also judged on how well they embrace the company’s core values, such as integrity, tenacity and coachability. Some of those values may require more-subjective judgments that can be more important than hitting quantifiable goals. 

Pruitt admits that there were occasions when he looked the other way with a few individuals who were “hitting it out of the park,” even though he believed they lacked integrity. But eventually, he had to let them go and the company lost money.

“They really came back to bite me,” Pruitt says.

Screen Shot 2023-03-15 at 84352 AM.png

Grades Are Good

Diane Dooley, CHRO of Iselin, N.J.-based World Insurance Associates LLC, also believes establishing quantitative methods to gauge employees’ performance is essential. “We are living in a world of data analytics,” she says. The broker’s roughly 2,000 employees are rated on a scale of 1 to 5.

World Insurance has taken numerous steps to remove bias from reviews. For example, last year the company conducted unconscious-bias training to help managers separate personal feelings from performance reviews. And all people managers convene to go over the reviews they’ve conducted. Dooley says that process gives everyone a chance to discuss why an employee was given a certain rank and to question some decisions. “We want to make sure we’re using the same standards,” she explains.

Currently, World Insurance conducts reviews only once a year because it has been on an acquisition binge and there hasn’t been time to institute a more frequent schedule. That will change eventually, says Dooley, who adds that she wants to introduce department grids that show how an employee’s rank compares to others’ on the team. 

“It’s just a tool that helps the department or the division understand where their people are and how we can help them collectively,” says Dooley, who has used the system at other companies. 

Dooley says she isn’t worried about World Insurance holding reviews only annually, because good managers regularly check in with their employees regardless of how frequently reviews are mandated.

Such conversations can easily fall through the cracks, however. “Managers want to manage the employees, but they get so caught up in the company’s KPIs [key performance indicators] and making sure that they’re doing everything that they need to do,” says Jennifer Currence, SHRM-SCP, CEO of WithIn Leadership, a leadership development and coaching firm in Tampa, Fla. “It’s hard to set aside the time.” 

WTW’s Jennings adds that managers sometimes avoid initiating conversations with employees who are not performing well. Such discussions are often difficult, and managers may not feel equipped to conduct them. 

“Having to address underperformers is hard work,” Jennings says. 

Additionally, experts say, coaching managers to engage in such sensitive discourse can be expensive and time-consuming.

Improve Your Performance Reviews

H ere’s how to make the review process more ­palatable for both managers and their direct reports:

  • Don’t limit conversations to once or twice per year. Every team is different, so leaders should decide what schedule is most appropriate for their departments. However, it’s important to deal with any problems as they arise; don’t let them fester.
  • Set performance goals and expectations at the beginning of the year so employees understand their responsibilities. This helps lend objectivity to the process by introducing measurable targets. However, the goals should be adjusted if there are major changes to the business or an employee’s circumstances. 
  • Explain how each employee’s position, as well as each department, fits into the company’s overall ­strategy. This will help employees understand why their job matters and why it’s important.
  • Simplify the process. There’s no need for a ­double-digit number of steps or numerous
  • questions that require long-winded answers. 
  • Consider a 360-degree approach. Input from employees’ colleagues or from other managers can help give a fuller picture of employees’ capabilities and contributions.
  • Eliminate proximity bias. You may not see some employees as often as others, especially if they work remotely, but that doesn’t mean they’re not working hard. 
  • End recency bias, which is basing a review on an employee’s most recent performance while ignoring earlier efforts. Don’t let recent mistakes overshadow the employee’s other impressive accomplishments.
  • Solicit feedback from employees. Reviews should be a two-way conversation, not a lecture.
  • Train managers to give advice calmly and helpfully. This is especially important when leaders must call out an employee’s subpar performance. 
  • Don’t discuss compensation during reviews. Employees are likely to be so focused on learning about a raise or bonus that they won’t pay much attention to anything else.

Increase Conversations

Finding the right formula for performance reviews is tricky. The company’s size, values, industry and age all play a role. Currence says businesses need to think about the frequency and purpose of these meetings. Some managers may have weekly discussions with their direct reports, but the conversations might center on status updates as opposed to performance. 

“We need to have more regular conversations,” Currence says. “There has to be a happy balance.”

San Jose, Calif.-based software maker Adobe Inc. was a pioneer when it eliminated annual reviews in 2012 after employees said assessments that look backward weren’t useful and managers lamented how time-consuming they were. Instead, Adobe introduced quarterly check-ins and did away with its numerical ratings system, even though the company is “data-driven,” according to Arden Madsen, senior director of talent management.

Screen Shot 2023-03-15 at 85749 AM.png

Adobe’s system has changed over the years as the company grew from about 11,000 employees in 2012 to around 28,000 today. In the beginning, employees were not asked a universal set of questions and the information gathered was not stored in a central place accessible to all. In 2020, Adobe instituted three or four questions that must be asked at each quarterly meeting, one of which is whether the employee has feedback for the manager. Other topics covered depend on the employee, their role and their goals.

Madsen says asking consistent questions and making reviews easily accessible are important, as internal mobility within the company has grown. 

Adobe, like many businesses, separates conversations about performance from discussions about raises and bonuses, even though they’re intertwined. 

“Money is so emotionally charged,” says WithIn Leadership’s Currence. “When we tie performance review conversations with money, we as human beings do not hear anything about performance. We only focus on the money.”    

Theresa Agovino is the workplace editor for SHRM.

Illustrations by Neil Jamieson.

Related Articles

how to solve work done problems

Rising Demand for Workforce AI Skills Leads to Calls for Upskilling

As artificial intelligence technology continues to develop, the demand for workers with the ability to work alongside and manage AI systems will increase. This means that workers who are not able to adapt and learn these new skills will be left behind in the job market.

HR Daily Newsletter

New, trends and analysis, as well as breaking news alerts, to help HR professionals do their jobs better each business day.

Success title

Success caption

Cart

  • SUGGESTED TOPICS
  • The Magazine
  • Newsletters
  • Managing Yourself
  • Managing Teams
  • Work-life Balance
  • The Big Idea
  • Data & Visuals
  • Reading Lists
  • Case Selections
  • HBR Learning
  • Topic Feeds
  • Account Settings
  • Email Preferences

Find the AI Approach That Fits the Problem You’re Trying to Solve

  • George Westerman,
  • Sam Ransbotham,
  • Chiara Farronato

how to solve work done problems

Five questions to help leaders discover the right analytics tool for the job.

AI moves quickly, but organizations change much more slowly. What works in a lab may be wrong for your company right now. If you know the right questions to ask, you can make better decisions, regardless of how fast technology changes. You can work with your technical experts to use the right tool for the right job. Then each solution today becomes a foundation to build further innovations tomorrow. But without the right questions, you’ll be starting your journey in the wrong place.

Leaders everywhere are rightly asking about how Generative AI can benefit their businesses. However, as impressive as generative AI is, it’s only one of many advanced data science and analytics techniques. While the world is focusing on generative AI, a better approach is to understand how to use the range of available analytics tools to address your company’s needs. Which analytics tool fits the problem you’re trying to solve? And how do you avoid choosing the wrong one? You don’t need to know deep details about each analytics tool at your disposal, but you do need to know enough to envision what’s possible and to ask technical experts the right questions.

  • George Westerman is a Senior Lecturer in MIT Sloan School of Management and founder of the Global Opportunity Forum  in MIT’s Office of Open Learning.
  • SR Sam Ransbotham is a Professor of Business Analytics at the Boston College Carroll School of Management. He co-hosts the “Me, Myself, and AI” podcast.
  • Chiara Farronato is the Glenn and Mary Jane Creamer Associate Professor of Business Administration at Harvard Business School and co-principal investigator at the Platform Lab at Harvard’s Digital Design Institute (D^3). She is also a fellow at the National Bureau of Economic Research (NBER) and the Center for Economic Policy Research (CEPR).

Partner Center

U.S. flag

An official website of the United States government

Here’s how you know

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Consumer Alerts

Your right to get information about funeral services by phone.

View all Consumer Alerts

Get Consumer Alerts

Credit, Loans, and Debt

Learn about getting and using credit, borrowing money, and managing debt.

View Credit, Loans, and Debt

What to do if you can’t make car payments

Jobs and making money.

What to know when you're looking for a job or more education, or considering a money-making opportunity or investment.

View Jobs and Making Money

Job scams targeting college students are getting personal

Unwanted calls, emails, and texts.

What to do about unwanted calls, emails, and text messages that can be annoying, might be illegal, and are probably scams.

View Unwanted Calls, Emails, and Texts

Fake shipping notification emails and text messages: What you need to know this holiday season

Identity theft and online security.

How to protect your personal information and privacy, stay safe online, and help your kids do the same.

View Identity Theft and Online Security

  • Search Show/hide Search menu items Items per page 20 50 100 Filters Fulltext search

Solving Problems With a Business: Returns, Refunds, and Other Resolutions

Facebook

Go Back to the Store or Website

Write a letter, get outside help, post an online review, consider dispute resolution alternatives.

Before you go back to the store or website, learn about the company’s return policies and collect documents related to your purchase.

  • Look for return policies, deadlines, customer service numbers and other important information on receipts, product packaging, warranties , or the seller’s website.
  • Check deadlines. Many stores will not take returns or exchanges after a certain time, like 30 or 90 days. Promptly return your undamaged item to have the best chance of a refund, exchange, or store credit. You might have to contact the manufacturer to return a defective or damaged product after a seller’s return deadline.
  • Collect key documents. Gather your receipts, warranties, canceled checks, credit card statements, invoices, contracts, or other documents. Make copies of documents to give the business and keep the originals. If you go to a store, bring any tags and original packaging you have.

Tell the business what happened. Give details about the problem, and about the resolution you want.

  • Explain the problem. Is the product defective or damaged? Did you get poor service or repairs that didn’t fix the problem? Reputable businesses want to know about their customers’ problems so they can act and avoid future complaints.
  • Be clear about what you want. Say if you want a full refund, an exchange, a store credit, a markdown on the item you bought, or a percentage discount on a future purchase. Explain why you want that result. Sellers are often more willing to offer a store credit than a refund. It’s less expensive for them and keeps you as a customer.
  • Ask to speak with a manager. If a customer representative doesn’t offer the result you want, be calm, polite, and persistent. Ask for a manager or supervisor. A manager will likely have more flexibility and authority to resolve the issue. Explain the problem to each person you talk with and describe what you want them to do.
  • Keep notes about what you did to solve the problem. List the people you talked to, the date of your conversation, and what they promised to do. If you chat online or send forms to customer service, save a copy of what you wrote, or take a picture of the screen before you exit, and note the date.

If you can’t resolve the problem by going back to the store or website, use this sample letter to write an effective complaint. When you write

Give your name, mailing address, and email address. Include your phone number too, if you want the business to contact you that way. Include your account number if you have an account with the business, and the related order or transaction number.

Give the basics. Describe the product or service you bought and important details of the transaction. Give the product’s name, its serial or model number, and the date and place you bought it or had it repaired or serviced.

Explain the problem. For example, say the product doesn’t work, you were billed incorrectly, something wasn’t disclosed clearly, or a product’s features were misrepresented.

Tell the business what you want. For example, say you want a refund, repair, exchange, or store credit.

Include copies of relevant documents , like receipts, repair orders, and warranties. Keep the originals.

Describe your next steps. Say how long you’ll wait for the business to answer. Give time for the business to act, and let it know you’ll report the matter to your state attorney general or state consumer protection office if you don’t hear by your deadline.

Make a copy of your letter to keep. Send your letter by certified mail and ask for a return receipt. If you send your complaint letter online, print the screen that shows your letter or take a screenshot of your letter before you click “submit.”

If you’re not satisfied with a business’s response to your complaint

Contact your state attorney general or state consumer protection office . These government agencies might mediate complaints, conduct investigations, and take other action against those who break consumer protection laws.

Contact a national consumer organization. Groups like Call for Action and Consumer Action try to help people with complaints.

Contact your local Better Business Bureau The Better Business Bureau is made up of organizations supported by local businesses. Local Better Business Bureaus try to resolve customer complaints.

File a report with the FTC. The FTC doesn’t resolve individual complaints, but your report helps law enforcement detect patterns and might lead to an investigation. Tell your story at ReportFraud.ftc.gov .

Visit USA.gov/complaints  to get information about filing complaints about specific types of products, steps to filing a complaint with a seller or manufacturer, links to product recall information, and more.

If you can’t resolve the problem and feel the business has been unfair, write an online review. The Consumer Review Fairness Act protects your ability to share your honest opinions about a business’s products, services, or conduct, in any forum, including social media.

It’s illegal for companies to threaten or penalize you for posting honest reviews. Many companies check social media and might reply if they see you’re dissatisfied with their response to your complaint.

Many consumers and businesses use dispute resolution programs instead of going to court.

  • In mediation , a neutral third party helps you and the other party try to resolve the problem. However, it's up to you and the other party to reach an agreement.
  • In arbitration you and the other party might appear at hearings, present evidence, or question each other’s witnesses, although the setting is less formal than court. An arbitrator or panel makes a decision or award after you present your case. The decision might be legally binding.

Many dispute resolution programs are voluntary, so you decide whether to use them. But in some states, a court might order you to try mediation or arbitration. Some companies require you to use arbitration for disputes and give up your right to go to court. Check your contract or product packaging to see what a business requires.

Your state consumer protection office or bar association might be able to suggest alternative dispute resolution programs in your area.

Small claims courts can resolve many financial disputes. The dollar limits on claims vary by state, but some states set the limit as high as $25,000. The costs of using small claims courts is relatively low, the procedures are simple, and you usually don’t need a lawyer. Check with your local small claims court for information about how to file your lawsuit.

If all else fails, consider a lawsuit. You’ll be able to sue for damages or any other type of relief the court awards, including legal fees. A lawyer can advise you about your options.

IMAGES

  1. The 5 Steps of Problem Solving

    how to solve work done problems

  2. sample problem solving for work

    how to solve work done problems

  3. 10 Problem Solving Skills Examples: How To Improve

    how to solve work done problems

  4. Work Done by a Force (examples, solutions, videos, notes)

    how to solve work done problems

  5. Problem solving infographic 10 steps concept Vector Image

    how to solve work done problems

  6. What are the problems at workplace and h

    how to solve work done problems

VIDEO

  1. 5 Powerful Problem Solving Techniques

  2. Refrigerant Leaking Point Finally Detected

  3. How to solve a problem

  4. Which Problems Do You Want To Solve? 👀💪

  5. Difficult Problem that can be solved in less than a minute

  6. Problem solved by itself?!

COMMENTS

  1. 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 How...

  2. "Work" Word Problems

    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: MathHelp.com Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours to paint a similarly-sized house.

  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. 9.10 Rate Word Problems: Work and Time

    The equation used to solve problems of this type is one of reciprocals. It is derived as follows: rate ×time = work done rate × time = work done. For this problem: Felicia's rate: F rate × 4 h = 1 room Katy's rate: Krate × 12 h = 1 room Isolating for their rates: F = 1 4 h and K = 1 12 h Felicia's rate: F rate × 4 h = 1 room Katy's rate: K ...

  5. Work Problems

    Introduction In this lesson, you will learn how to solve work problems. Work problems are those that involve the speeds of people and/or machines. We will teach you how to calculate how long it will take for multiple workers to perform a job. This lesson is broken down into three sections: Rationale for Learning Work Problems Work Basics

  6. Work example problems (video)

    Work example problems Google Classroom About Transcript David goes through some example problems on the concept of work. By reviewing these, you'll have a better knowledge of how to calculate work done by individual forces on an object in motion.

  7. Work Done Problems

    Here are some solved problems on work done, through which it can be understood in a better way: Q1. Piyush and Rahul together can complete a work in 18 days. Piyush alone can do the same work in 24 days. What will be the number of days Rahul alone can complete the whole work? Ans: Piyush and Rahul can complete the work in 18 days

  8. 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.

  9. 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.

  10. 7.1 Work

    We can solve this problem by substituting the given values into the definition of work done on an object by a constant force, stated in the equation W = F d cos θ W = F d cos θ. The force, angle, and displacement are given, so that only the work W is unknown. Solution The equation for the work is

  11. How to Solve Problems at Work: A Step-by-Step Guide

    Step 3: Identify the source of the issue. Beyond defining the problem that you're faced with, you may also need to identify the root of the problem. This will guide you towards a solution that not only fixes the problem that lies at the surface but also resolve a far deeper issue that could cause more problems to arise in the future.

  12. How to Solve Problems

    Teams today aren't just asked to execute tasks: They're called upon to solve problems. You'd think that many brains working together would mean better solutions, but the reality is that too ...

  13. How to Develop Problem Solving Skills: 4 Tips

    How to Develop Problem Solving Skills: 4 Tips. Learning problem-solving techniques is a must for working professionals in any field. No matter your title or job description, the ability to find the root cause of a difficult problem and formulate viable solutions is a skill that employers value. Learning the soft skills and critical thinking ...

  14. 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.

  15. 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.

  16. 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.

  17. Work Problems

    This calculus video tutorial explains how to solve work problems. It explains how to calculate the work required to lift an object against gravity or the wo...

  18. Solve

    Example: 2x-1=y,2y+3=x 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.

  19. Time and Work

    Given below are a few such important time and work formulas for your reference: Work Done = Time Taken × Rate of Work. Rate of Work = 1 / Time Taken. Time Taken = 1 / Rate of Work. If a piece of work is done in x number of days, then the work done in one day = 1/x. Total Wok Done = Number of Days × Efficiency.

  20. Time and Work Problems

    TIME AND WORK PROBLEMS. 1. If a person can do a piece of work in 'm' days, he can do ¹⁄m part of the work in 1 day. 2. If the number of persons engaged to do a piece of work be increased (or decreased) in a certain ratio the time required to do the same work will be decreased (or increased) in the same ratio. 3.

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

    Solution: 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.

  22. Microsoft Math Solver

    Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

  23. 12 Problems at Work and How to Solve Them (With Examples)

    3. Lack of work-life balance for employees. A strong work-life balance can help employees stay focused on the quality of their work and avoid excess stress. Some employees might experience a poor work-life balance, spending too much time at work and not enough time attending to personal issues with family and friends.

  24. Your Work Can Only Be As Good As Your Problems Are Meaningful

    Ask yourself what problem(s) you're trying to solve in your work. If they're not big, interesting, or important enough (in your own mind), ... To get maximum happiness from your work, find the most important problems to work on. It won't just improve the impact of your work, but will likely make you better at doing the work as well.

  25. How to solve problems at work, says Deloitte exec: 'My best idea didn't

    D eborah Golden has a pretty unique job: As consulting firm Deloitte's chief innovation officer, she gets to "solve really hard problems," she says.Often this can mean leaning into emerging ...

  26. The Performance Review Problem

    Reviews generally do not work. That doesn't mean that more-frequent formal meetings or casual sit-downs between supervisors and their direct reports are solving the performance review quandary ...

  27. Find the AI Approach That Fits the Problem You're Trying to Solve

    Summary. AI moves quickly, but organizations change much more slowly. What works in a lab may be wrong for your company right now. If you know the right questions to ask, you can make better ...

  28. Solving Problems With a Business: Returns, Refunds, and Other

    Ask for a manager or supervisor. A manager will likely have more flexibility and authority to resolve the issue. Explain the problem to each person you talk with and describe what you want them to do. Keep notes about what you did to solve the problem. List the people you talked to, the date of your conversation, and what they promised to do.

  29. Artificial Intelligence In The Classroom: What Do Educators Think

    AI-powered programs and devices can make decisions, solve problems, understand and mimic natural language and learn from unstructured data. ... While we work hard to provide accurate and up to ...