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9.3: Work-rate problems

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• 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?

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Chapter 9: Radicals

9.10 Rate Word Problems: Work and Time

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

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

For this problem:

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

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

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

Example 9.10.1

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

The equation to solve is:

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

Example 9.10.2

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

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

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

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

Example 9.10.3

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

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

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

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

Example 9.10.4

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

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

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

Example 9.10.5

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

The 7 minutes to drain will be subtracted.

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

17.5 min or 17 min 30 sec is the solution

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

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

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

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

Answer Key 9.10

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

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How to Master Work Problems: A Comprehensive Step-by-Step Guide

Understanding work problems in mathematics often involves dealing with scenarios where different people (or machines) contribute to completing a task. These problems can be solved by using the formula \(W=R×T\), where \(W\) is Work, \(R\) is Rate, and \(T\) is Time. Here is a step-by-step guide to help you understand and solve these problems:

Step-by-step Guide to Master Work Problems

Step 1: understand the problem.

• Read the Problem Carefully: Identify what the problem is asking. Pay attention to the time taken by each person or machine to complete the work independently.
• Define the Work: Usually, the total work done is considered as \(1\) unit. For example, mowing a lawn, painting a wall, or filling a tank is each considered as 1 unit of work.

Step 2: Determine the Rates

• Calculate Individual Rates: Determine the rate at which each person or machine completes the work. If a person completes the work in \(T\) hours, their rate is \(\frac{1}{T}\)​ units of work per hour.
• Combine Rates for Collaborative Work: If multiple people or machines work together, add their rates. For instance, if Person \(A\) has a rate of \(\frac{​1}{T_{A}}\)​ and Person \(B\) has a rate of \(\frac{1}{T_{B}}\), their combined rate is\(\frac{​1}{T_{A}}+\frac{1}{T_{B}}\) ​.

Step 3: Set Up the Equation

• Use the Work Formula: The formula \(W=R×T\) is pivotal. For combined work, set \(W=1\) and use the combined rate for \(R\).
• Formulate the Equation: The equation usually looks like \(1=(\frac{​1}{T_{A}}+\frac{1}{T_{B}})×T\), where \(T\) is the time taken for the combined work.

Step 4: Solve for the Unknown

• Rearrange the Equation: Isolate the variable you are solving for. This might involve algebraic manipulation.
• Solve Mathematically: Use algebra to find the value of the unknown variable. This might involve finding a common denominator, simplifying fractions, or solving linear equations.

Step 5: Check Your Solution

• Verify the Answer: Plug your answer back into the equation to see if it makes sense. Check if the units and the context align correctly.
• Consider Practical Implications: Ensure that the solution is practical and makes sense in the context of the problem.

Understanding each step and applying it to various problems will enhance your ability to tackle work problems effectively.

Sarah can clean a room in \(2\) hours. When she works with her friend Lisa, they can clean it in \(1.5\) hours. How long would it take Lisa to clean the room alone?

• Sarah’s rate \(=\frac{1}{2}\) room per hour.
• Together, their rate \(=\frac{1}{1.5}​\) room per hour.
• Let Lisa’s time be \(T\) hours, so her rate \(=\frac{1}{T}\)​.

Combine the rates: \(\frac{1}{2}+\frac{1}{T}=\frac{1}{1.5}​\)

Solve for \(T\): \(\frac{1}{T}=\frac{2}{3}-\frac{1}{2}=\frac{4-3}{6}=\frac{1}{6}\)​ So, \(T=6\) hours.

Kevin can type a document in \(5\) hours. Working together with Rachel, they can type it in \(4\) hours. How long would it take Rachel alone?

• Kevin’s rate \(=\frac{1}{5}\)​ document per hour.
• Together, their rate \(=\frac{1}{4}\) document per hour.
• Rachel’s rate \(=\frac{1}{T}\)​.

Combine the rates: \(\frac{1}{5}+\frac{1}{T}=\frac{1}{4}​\)

Solve for \(T\): \(\frac{1}{T}=\frac{1}{4}-\frac{1}{5}=\frac{5-4}{20}=\frac{1}{20}\)​ So, \(T=20\) hours.

by: Effortless Math Team about 5 months ago (category: Articles )

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"Work" Word Problems

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

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

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

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

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

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

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

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

I can solve by flipping the equation; I get:

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

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

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

hours to complete job:

first painter: 12

second painter: 8

together: t

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

completed per hour:

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

adding their labor:

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

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

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

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

slow pipe: s

together: 5

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

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

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

20 + 25 = 4 s

45/4 = 11.25 = s

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

The slower pipe takes 11.25 hours.

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

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Work-related Problems

Case 1: Workers have different rates

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

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

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

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

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

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

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

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

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

Case 2: Workers have equal rates

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

To finish the job

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

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

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

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

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

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Lesson Solving rate of work problem by reducing to a system of linear equations

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Do You Understand the Problem You’re Trying to Solve?

To solve tough problems at work, first ask these questions.

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Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to really understand the dilemma we face, according to Thomas Wedell-Wedellsborg , an expert in innovation and the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for just one root cause can be misleading.

Key episode topics include: leadership, decision making and problem solving, power and influence, business management.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

• Listen to the original HBR IdeaCast episode: The Secret to Better Problem Solving (2016)
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• Discover 100 years of Harvard Business Review articles, case studies, podcasts, and more at HBR.org .

HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Problem solving skills are invaluable in any job. But even the most experienced among us can fall into the trap of solving the wrong problem.

Thomas Wedell-Wedellsborg says that all too often, we jump to find solutions to a problem – without taking time to really understand what we’re facing.

He’s an expert in innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

  In this episode, you’ll learn how to reframe tough problems, by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for one root cause can be misleading. And you’ll learn how to use experimentation and rapid prototyping as problem-solving tools.

This episode originally aired on HBR IdeaCast in December 2016. Here it is.

SARAH GREEN CARMICHAEL: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Sarah Green Carmichael.

Problem solving is popular. People put it on their resumes. Managers believe they excel at it. Companies count it as a key proficiency. We solve customers’ problems.

The problem is we often solve the wrong problems. Albert Einstein and Peter Drucker alike have discussed the difficulty of effective diagnosis. There are great frameworks for getting teams to attack true problems, but they’re often hard to do daily and on the fly. That’s where our guest comes in.

Thomas Wedell-Wedellsborg is a consultant who helps companies and managers reframe their problems so they can come up with an effective solution faster. He asks the question “Are You Solving The Right Problems?” in the January-February 2017 issue of Harvard Business Review. Thomas, thank you so much for coming on the HBR IdeaCast .

THOMAS WEDELL-WEDELLSBORG: Thanks for inviting me.

SARAH GREEN CARMICHAEL: So, I thought maybe we could start by talking about the problem of talking about problem reframing. What is that exactly?

THOMAS WEDELL-WEDELLSBORG: Basically, when people face a problem, they tend to jump into solution mode to rapidly, and very often that means that they don’t really understand, necessarily, the problem they’re trying to solve. And so, reframing is really a– at heart, it’s a method that helps you avoid that by taking a second to go in and ask two questions, basically saying, first of all, wait. What is the problem we’re trying to solve? And then crucially asking, is there a different way to think about what the problem actually is?

SARAH GREEN CARMICHAEL: So, I feel like so often when this comes up in meetings, you know, someone says that, and maybe they throw out the Einstein quote about you spend an hour of problem solving, you spend 55 minutes to find the problem. And then everyone else in the room kind of gets irritated. So, maybe just give us an example of maybe how this would work in practice in a way that would not, sort of, set people’s teeth on edge, like oh, here Sarah goes again, reframing the whole problem instead of just solving it.

THOMAS WEDELL-WEDELLSBORG: I mean, you’re bringing up something that’s, I think is crucial, which is to create legitimacy for the method. So, one of the reasons why I put out the article is to give people a tool to say actually, this thing is still important, and we need to do it. But I think the really critical thing in order to make this work in a meeting is actually to learn how to do it fast, because if you have the idea that you need to spend 30 minutes in a meeting delving deeply into the problem, I mean, that’s going to be uphill for most problems. So, the critical thing here is really to try to make it a practice you can implement very, very rapidly.

There’s an example that I would suggest memorizing. This is the example that I use to explain very rapidly what it is. And it’s basically, I call it the slow elevator problem. You imagine that you are the owner of an office building, and that your tenants are complaining that the elevator’s slow.

Now, if you take that problem framing for granted, you’re going to start thinking creatively around how do we make the elevator faster. Do we install a new motor? Do we have to buy a new lift somewhere?

The thing is, though, if you ask people who actually work with facilities management, well, they’re going to have a different solution for you, which is put up a mirror next to the elevator. That’s what happens is, of course, that people go oh, I’m busy. I’m busy. I’m– oh, a mirror. Oh, that’s beautiful.

And then they forget time. What’s interesting about that example is that the idea with a mirror is actually a solution to a different problem than the one you first proposed. And so, the whole idea here is once you get good at using reframing, you can quickly identify other aspects of the problem that might be much better to try to solve than the original one you found. It’s not necessarily that the first one is wrong. It’s just that there might be better problems out there to attack that we can, means we can do things much faster, cheaper, or better.

SARAH GREEN CARMICHAEL: So, in that example, I can understand how A, it’s probably expensive to make the elevator faster, so it’s much cheaper just to put up a mirror. And B, maybe the real problem people are actually feeling, even though they’re not articulating it right, is like, I hate waiting for the elevator. But if you let them sort of fix their hair or check their teeth, they’re suddenly distracted and don’t notice.

But if you have, this is sort of a pedestrian example, but say you have a roommate or a spouse who doesn’t clean up the kitchen. Facing that problem and not having your elegant solution already there to highlight the contrast between the perceived problem and the real problem, how would you take a problem like that and attack it using this method so that you can see what some of the other options might be?

THOMAS WEDELL-WEDELLSBORG: Right. So, I mean, let’s say it’s you who have that problem. I would go in and say, first of all, what would you say the problem is? Like, if you were to describe your view of the problem, what would that be?

SARAH GREEN CARMICHAEL: I hate cleaning the kitchen, and I want someone else to clean it up.

THOMAS WEDELL-WEDELLSBORG: OK. So, my first observation, you know, that somebody else might not necessarily be your spouse. So, already there, there’s an inbuilt assumption in your question around oh, it has to be my husband who does the cleaning. So, it might actually be worth, already there to say, is that really the only problem you have? That you hate cleaning the kitchen, and you want to avoid it? Or might there be something around, as well, getting a better relationship in terms of how you solve problems in general or establishing a better way to handle small problems when dealing with your spouse?

SARAH GREEN CARMICHAEL: Or maybe, now that I’m thinking that, maybe the problem is that you just can’t find the stuff in the kitchen when you need to find it.

THOMAS WEDELL-WEDELLSBORG: Right, and so that’s an example of a reframing, that actually why is it a problem that the kitchen is not clean? Is it only because you hate the act of cleaning, or does it actually mean that it just takes you a lot longer and gets a lot messier to actually use the kitchen, which is a different problem. The way you describe this problem now, is there anything that’s missing from that description?

SARAH GREEN CARMICHAEL: That is a really good question.

THOMAS WEDELL-WEDELLSBORG: Other, basically asking other factors that we are not talking about right now, and I say those because people tend to, when given a problem, they tend to delve deeper into the detail. What often is missing is actually an element outside of the initial description of the problem that might be really relevant to what’s going on. Like, why does the kitchen get messy in the first place? Is it something about the way you use it or your cooking habits? Is it because the neighbor’s kids, kind of, use it all the time?

There might, very often, there might be issues that you’re not really thinking about when you first describe the problem that actually has a big effect on it.

SARAH GREEN CARMICHAEL: I think at this point it would be helpful to maybe get another business example, and I’m wondering if you could tell us the story of the dog adoption problem.

THOMAS WEDELL-WEDELLSBORG: Yeah. This is a big problem in the US. If you work in the shelter industry, basically because dogs are so popular, more than 3 million dogs every year enter a shelter, and currently only about half of those actually find a new home and get adopted. And so, this is a problem that has persisted. It’s been, like, a structural problem for decades in this space. In the last three years, where people found new ways to address it.

So a woman called Lori Weise who runs a rescue organization in South LA, and she actually went in and challenged the very idea of what we were trying to do. She said, no, no. The problem we’re trying to solve is not about how to get more people to adopt dogs. It is about keeping the dogs with their first family so they never enter the shelter system in the first place.

In 2013, she started what’s called a Shelter Intervention Program that basically works like this. If a family comes and wants to hand over their dog, these are called owner surrenders. It’s about 30% of all dogs that come into a shelter. All they would do is go up and ask, if you could, would you like to keep your animal? And if they said yes, they would try to fix whatever helped them fix the problem, but that made them turn over this.

And sometimes that might be that they moved into a new building. The landlord required a deposit, and they simply didn’t have the money to put down a deposit. Or the dog might need a $10 rabies shot, but they didn’t know how to get access to a vet.

And so, by instigating that program, just in the first year, she took her, basically the amount of dollars they spent per animal they helped went from something like $85 down to around $60. Just an immediate impact, and her program now is being rolled out, is being supported by the ASPCA, which is one of the big animal welfare stations, and it’s being rolled out to various other places.

And I think what really struck me with that example was this was not dependent on having the internet. This was not, oh, we needed to have everybody mobile before we could come up with this. This, conceivably, we could have done 20 years ago. Only, it only happened when somebody, like in this case Lori, went in and actually rethought what the problem they were trying to solve was in the first place.

SARAH GREEN CARMICHAEL: So, what I also think is so interesting about that example is that when you talk about it, it doesn’t sound like the kind of thing that would have been thought of through other kinds of problem solving methods. There wasn’t necessarily an After Action Review or a 5 Whys exercise or a Six Sigma type intervention. I don’t want to throw those other methods under the bus, but how can you get such powerful results with such a very simple way of thinking about something?

THOMAS WEDELL-WEDELLSBORG: That was something that struck me as well. This, in a way, reframing and the idea of the problem diagnosis is important is something we’ve known for a long, long time. And we’ve actually have built some tools to help out. If you worked with us professionally, you are familiar with, like, Six Sigma, TRIZ, and so on. You mentioned 5 Whys. A root cause analysis is another one that a lot of people are familiar with.

Those are our good tools, and they’re definitely better than nothing. But what I notice when I work with the companies applying those was those tools tend to make you dig deeper into the first understanding of the problem we have. If it’s the elevator example, people start asking, well, is that the cable strength, or is the capacity of the elevator? That they kind of get caught by the details.

That, in a way, is a bad way to work on problems because it really assumes that there’s like a, you can almost hear it, a root cause. That you have to dig down and find the one true problem, and everything else was just symptoms. That’s a bad way to think about problems because problems tend to be multicausal.

There tend to be lots of causes or levers you can potentially press to address a problem. And if you think there’s only one, if that’s the right problem, that’s actually a dangerous way. And so I think that’s why, that this is a method I’ve worked with over the last five years, trying to basically refine how to make people better at this, and the key tends to be this thing about shifting out and saying, is there a totally different way of thinking about the problem versus getting too caught up in the mechanistic details of what happens.

SARAH GREEN CARMICHAEL: What about experimentation? Because that’s another method that’s become really popular with the rise of Lean Startup and lots of other innovation methodologies. Why wouldn’t it have worked to, say, experiment with many different types of fixing the dog adoption problem, and then just pick the one that works the best?

THOMAS WEDELL-WEDELLSBORG: You could say in the dog space, that’s what’s been going on. I mean, there is, in this industry and a lot of, it’s largely volunteer driven. People have experimented, and they found different ways of trying to cope. And that has definitely made the problem better. So, I wouldn’t say that experimentation is bad, quite the contrary. Rapid prototyping, quickly putting something out into the world and learning from it, that’s a fantastic way to learn more and to move forward.

My point is, though, that I feel we’ve come to rely too much on that. There’s like, if you look at the start up space, the wisdom is now just to put something quickly into the market, and then if it doesn’t work, pivot and just do more stuff. What reframing really is, I think of it as the cognitive counterpoint to prototyping. So, this is really a way of seeing very quickly, like not just working on the solution, but also working on our understanding of the problem and trying to see is there a different way to think about that.

If you only stick with experimentation, again, you tend to sometimes stay too much in the same space trying minute variations of something instead of taking a step back and saying, wait a minute. What is this telling us about what the real issue is?

SARAH GREEN CARMICHAEL: So, to go back to something that we touched on earlier, when we were talking about the completely hypothetical example of a spouse who does not clean the kitchen–

THOMAS WEDELL-WEDELLSBORG: Completely, completely hypothetical.

SARAH GREEN CARMICHAEL: Yes. For the record, my husband is a great kitchen cleaner.

You started asking me some questions that I could see immediately were helping me rethink that problem. Is that kind of the key, just having a checklist of questions to ask yourself? How do you really start to put this into practice?

THOMAS WEDELL-WEDELLSBORG: I think there are two steps in that. The first one is just to make yourself better at the method. Yes, you should kind of work with a checklist. In the article, I kind of outlined seven practices that you can use to do this.

But importantly, I would say you have to consider that as, basically, a set of training wheels. I think there’s a big, big danger in getting caught in a checklist. This is something I work with.

My co-author Paddy Miller, it’s one of his insights. That if you start giving people a checklist for things like this, they start following it. And that’s actually a problem, because what you really want them to do is start challenging their thinking.

So the way to handle this is to get some practice using it. Do use the checklist initially, but then try to step away from it and try to see if you can organically make– it’s almost a habit of mind. When you run into a colleague in the hallway and she has a problem and you have five minutes, like, delving in and just starting asking some of those questions and using your intuition to say, wait, how is she talking about this problem? And is there a question or two I can ask her about the problem that can help her rethink it?

SARAH GREEN CARMICHAEL: Well, that is also just a very different approach, because I think in that situation, most of us can’t go 30 seconds without jumping in and offering solutions.

THOMAS WEDELL-WEDELLSBORG: Very true. The drive toward solutions is very strong. And to be clear, I mean, there’s nothing wrong with that if the solutions work. So, many problems are just solved by oh, you know, oh, here’s the way to do that. Great.

But this is really a powerful method for those problems where either it’s something we’ve been banging our heads against tons of times without making progress, or when you need to come up with a really creative solution. When you’re facing a competitor with a much bigger budget, and you know, if you solve the same problem later, you’re not going to win. So, that basic idea of taking that approach to problems can often help you move forward in a different way than just like, oh, I have a solution.

I would say there’s also, there’s some interesting psychological stuff going on, right? Where you may have tried this, but if somebody tries to serve up a solution to a problem I have, I’m often resistant towards them. Kind if like, no, no, no, no, no, no. That solution is not going to work in my world. Whereas if you get them to discuss and analyze what the problem really is, you might actually dig something up.

Let’s go back to the kitchen example. One powerful question is just to say, what’s your own part in creating this problem? It’s very often, like, people, they describe problems as if it’s something that’s inflicted upon them from the external world, and they are innocent bystanders in that.

SARAH GREEN CARMICHAEL: Right, or crazy customers with unreasonable demands.

THOMAS WEDELL-WEDELLSBORG: Exactly, right. I don’t think I’ve ever met an agency or consultancy that didn’t, like, gossip about their customers. Oh, my god, they’re horrible. That, you know, classic thing, why don’t they want to take more risk? Well, risk is bad.

It’s their business that’s on the line, not the consultancy’s, right? So, absolutely, that’s one of the things when you step into a different mindset and kind of, wait. Oh yeah, maybe I actually am part of creating this problem in a sense, as well. That tends to open some new doors for you to move forward, in a way, with stuff that you may have been struggling with for years.

SARAH GREEN CARMICHAEL: So, we’ve surfaced a couple of questions that are useful. I’m curious to know, what are some of the other questions that you find yourself asking in these situations, given that you have made this sort of mental habit that you do? What are the questions that people seem to find really useful?

THOMAS WEDELL-WEDELLSBORG: One easy one is just to ask if there are any positive exceptions to the problem. So, was there day where your kitchen was actually spotlessly clean? And then asking, what was different about that day? Like, what happened there that didn’t happen the other days? That can very often point people towards a factor that they hadn’t considered previously.

SARAH GREEN CARMICHAEL: We got take-out.

THOMAS WEDELL-WEDELLSBORG: S,o that is your solution. Take-out from [INAUDIBLE]. That might have other problems.

Another good question, and this is a little bit more high level. It’s actually more making an observation about labeling how that person thinks about the problem. And what I mean with that is, we have problem categories in our head. So, if I say, let’s say that you describe a problem to me and say, well, we have a really great product and are, it’s much better than our previous product, but people aren’t buying it. I think we need to put more marketing dollars into this.

Now you can go in and say, that’s interesting. This sounds like you’re thinking of this as a communications problem. Is there a different way of thinking about that? Because you can almost tell how, when the second you say communications, there are some ideas about how do you solve a communications problem. Typically with more communication.

And what you might do is go in and suggest, well, have you considered that it might be, say, an incentive problem? Are there incentives on behalf of the purchasing manager at your clients that are obstructing you? Might there be incentive issues with your own sales force that makes them want to sell the old product instead of the new one?

So literally, just identifying what type of problem does this person think about, and is there different potential way of thinking about it? Might it be an emotional problem, a timing problem, an expectations management problem? Thinking about what label of what type of problem that person is kind of thinking as it of.

SARAH GREEN CARMICHAEL: That’s really interesting, too, because I think so many of us get requests for advice that we’re really not qualified to give. So, maybe the next time that happens, instead of muddying my way through, I will just ask some of those questions that we talked about instead.

THOMAS WEDELL-WEDELLSBORG: That sounds like a good idea.

SARAH GREEN CARMICHAEL: So, Thomas, this has really helped me reframe the way I think about a couple of problems in my own life, and I’m just wondering. I know you do this professionally, but is there a problem in your life that thinking this way has helped you solve?

THOMAS WEDELL-WEDELLSBORG: I’ve, of course, I’ve been swallowing my own medicine on this, too, and I think I have, well, maybe two different examples, and in one case somebody else did the reframing for me. But in one case, when I was younger, I often kind of struggled a little bit. I mean, this is my teenage years, kind of hanging out with my parents. I thought they were pretty annoying people. That’s not really fair, because they’re quite wonderful, but that’s what life is when you’re a teenager.

And one of the things that struck me, suddenly, and this was kind of the positive exception was, there was actually an evening where we really had a good time, and there wasn’t a conflict. And the core thing was, I wasn’t just seeing them in their old house where I grew up. It was, actually, we were at a restaurant. And it suddenly struck me that so much of the sometimes, kind of, a little bit, you love them but they’re annoying kind of dynamic, is tied to the place, is tied to the setting you are in.

And of course, if– you know, I live abroad now, if I visit my parents and I stay in my old bedroom, you know, my mother comes in and wants to wake me up in the morning. Stuff like that, right? And it just struck me so, so clearly that it’s– when I change this setting, if I go out and have dinner with them at a different place, that the dynamic, just that dynamic disappears.

SARAH GREEN CARMICHAEL: Well, Thomas, this has been really, really helpful. Thank you for talking with me today.

THOMAS WEDELL-WEDELLSBORG: Thank you, Sarah.  

HANNAH BATES: That was Thomas Wedell-Wedellsborg in conversation with Sarah Green Carmichael on the HBR IdeaCast. He’s an expert in problem solving and innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

We’ll be back next Wednesday with another hand-picked conversation about leadership from the Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

We’re a production of Harvard Business Review. If you want more podcasts, articles, case studies, books, and videos like this, find it all at HBR dot org.

This episode was produced by Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Music by Coma Media. Special thanks to Maureen Hoch, Adi Ignatius, Karen Player, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener.

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Rate Work Problems

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

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

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

Download Grade 6, Module 1, Lesson 23 Worksheets

Lesson 23 Outcome

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

Lesson 23 Summary

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

NYS Math Module 1 Grade 6 Lesson 23 Classwork

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

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

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

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

Problem Set

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

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'ZDNET Recommends': What exactly does it mean?

ZDNET's recommendations are based on many hours of testing, research, and comparison shopping. We gather data from the best available sources, including vendor and retailer listings as well as other relevant and independent reviews sites. And we pore over customer reviews to find out what matters to real people who already own and use the products and services we’re assessing.

When you click through from our site to a retailer and buy a product or service, we may earn affiliate commissions. This helps support our work, but does not affect what we cover or how, and it does not affect the price you pay. Neither ZDNET nor the author are compensated for these independent reviews. Indeed, we follow strict guidelines that ensure our editorial content is never influenced by advertisers.

ZDNET's editorial team writes on behalf of you, our reader. Our goal is to deliver the most accurate information and the most knowledgeable advice possible in order to help you make smarter buying decisions on tech gear and a wide array of products and services. Our editors thoroughly review and fact-check every article to ensure that our content meets the highest standards. If we have made an error or published misleading information, we will correct or clarify the article. If you see inaccuracies in our content, please report the mistake via this form .

Sennheiser's new earbuds solve one of my biggest problems as a runner

ZDNET's key takeaways

• The Sennheiser Momentum Sport earbuds are available now in three colors for $329.95.
• The earbuds provide accurate heart rate and temperature sensors, sports watch connectivity, amazing sound, and a secure fit.
• They're on the expensive side, and the six-hour battery life is not the best on the market.

Wrist-based heart rate sensors have advanced significantly over the past few years, but active movement of your arms, cold weather, and other factors can reduce their accuracy in reporting biological metrics. Chest straps and arm bands offer an improved level of performance for heart rate monitoring, but also require that you wear an extra piece of gear and keep it charged up and connected.

Also:  The best noise-canceling earbuds you can buy

Sennheiser has created a third option with the Momentum Sport earbuds , offering a stunning audio experience while enhancing your workout with integrated heart rate and body temperature sensors, both of which are positioned in the left earbud.

The inner ear is an optimal location for the heart rate sensor with minimal light pollution and good stability. It is very important to also ensure a secure fit and Sennheiser provides that with silicone tips and silicone wings in several sizes. I tried the earbuds out with just the silicone tips and they were not secure enough for me to run with because the earbuds are rather large in size.

Positioning the large-size wings over the earbuds was the key to success for me, though, and now the earbuds fit snugly without moving at all while running, jumping, and tossing my head around. I also have two different-sized eartips for my left and right ears so make sure to try out the available options and perform the fit test in the Sennheiser Smart Control smartphone application.

Sennheiser also provides an in-ear thermometer and states that these sensors are more accurate than wrist-based sensors within 0.3 degrees Celsius accuracy. Heat-related fatigue can be managed if an athlete knows their body temperature and these earbuds help provide a viable source of that data.

While Polar powers the heart rate and temperature sensors, you can also add the earbuds as a heart rate monitor with your Garmin, Apple Watch, Suunto, or other device/service for an accurate ear-based heart rate reading. In order to get the optimal experience out of the  Sennheiser Momentum Sport earbuds , it's recommended to use the Polar Flow smartphone application. With this app, you can access important data such as performance tracking, training analytics, smart coaching, as well as voice guidance. Also, keep in mind that while the Sennheiser Smart Control app displays your heart rate and body temperature, you have to have a compatible Polar GPS sports watch to track the history of that data. At this time, the Vantage V3 and new Grit X2 Pro are supported.

I was able to test out the Sennheiser Momentum Sport earbuds with a Polar Vantage V3 that supports full integration with these earbuds and sports sensors. Real-time data was streamed directly to the watch and captured for analysis in the Polar Flow app and website. Make sure to connect the earbuds to your watch as a sensor and after that initial set up the heart rate sensor in the earbuds will be used for capturing this data while you exercise rather than using the heart rate data from the sensor mounted on the watch.

Also: The best bone conduction headphones you can buy: Expert tested

The Momentum Sport earbuds are available in Polar Black, Burned Olive, and Metallic Graphite for $329.95. I tested the Burned Olive model, and it looks great with orange highlights on the earbuds and charging case. The earbuds should provide audio for up to 5.5 hours with active noise cancellation and 6 hours without it enabled. The case can provide another 18 hours (three charges) of battery life with support for USB-C charging and wireless Qi charging of the case.

Thankfully, I can use these earbuds in Washington state where I often run into rain, since they have IP55 sweat and water resistance. The earbuds themselves and case are also covered in a soft-touch silicone material that provides a solid grip. Sennheiser also designed these earbuds with an acoustic relief channel for air ventilation so that the noise from your footsteps, and breathing, for example, are minimized while they're in.

Also:  Sennheiser Momentum True Wireless 4 review

Bluetooth 5.2 provides the connection with support for aptX Adaptive, AAC, and SBC codecs. I was thinking these might just be another typical earbud, but I was blown away by the crystal-clear audio with excellent bass and wonderful volume. During my testing, I ran with them and then mowed the lawn for an hour and the earbuds easily beat out every other earbud I've used while up against my noisy gas-powered lawn mower.

The earbuds support various modes of operation, including transparency, anti-wind, and adaptive noise cancellation. These modes can be changed with touches on the earbuds and through the Smart Control smartphone software. The Smart Control software also provides settings for the equalizer, a sound check, sound zones, and customizing the double tap, triple tap, and tap-hold settings of each earbud.

For the past couple of years, I've been using bone conduction headsets for running, but the Sennheiser Momentum Sport earbuds provide an impressive audio experience and stay securely in place, while also providing a reliable and accurate heart rate sensor without a chest or arm strap, so these will definitely be my preferred earbuds going forward.

ZDNET's buying advice

There seems to be an unlimited number of earbuds on the market today, and it's tough to distinguish one from the other. Sennheiser, however, stands out from the pack with the heart rate and temperature sensors that will appeal to athletes and casual runners alike looking for a way to capture accurate heart rate and body temperature readings. With the Momentum Sport earbuds , Polar and Sennheiser partnered up to create an amazing pair of earbuds that sound fantastic, integrate well with other wearables, and provide hours of media support.

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I tested sennheiser's new mid-range headphones and they're so close to perfect, the best bone conduction headphones you can buy: expert tested, the best noise-canceling earbuds you can buy: expert tested.

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How One Family Lost $900,000 in a Timeshare Scam

A mexican drug cartel is targeting seniors and their timeshares..

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A massive scam targeting older Americans who own timeshare properties has resulted in hundreds of millions of dollars sent to Mexico.

Maria Abi-Habib, an investigative correspondent for The Times, tells the story of a victim who lost everything, and of the criminal group making the scam calls — Jalisco New Generation, one of Mexico’s most violent cartels.

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Fortune’s Best Companies to Work For credit trust and culture—not pay and benefits—for high employee satisfaction

Good morning.  As the U.S. election heats up, many corporate leaders are backing down from their commitments to so-called “ESG” topics—climate, diversity, speaking out on social issues. They want to avoid being branded a “woke” CEO or getting caught in political crossfire.

But for the best companies, this discussion has never been about politics or adhering to some alphabet mandate. It’s about how you lead a great company in an economy where people are your most important asset and where engaging and inspiring people is the route to business success.

I spent an hour earlier this week with three of the best of this breed, in a webinar Fortune held in partnership with UKG , timed to the release of Fortune ’s 27th annual 100 Best Companies to Work For list. (You can find the list here .) Three CEOs ran companies on the list— Hilton (No. 1), Edward Jones (No. 31) and Delta (No. 94). But what I found interesting was that none of them focused on a splashy pay and benefit package, and indeed two of them achieved their success while recovering from the devastating effects of the pandemic.

How did they do it? Some excerpts from the conversation:

“ We used to write a lot about the perks. But what we know is that a great workplace is defined by trust— the trust that we have for the people that we work with. We want to be respected. We want to be communicated with in an honest way. We want to be treated in a fair and equitable way. We want to work with people that we enjoy working with and we expect leaders to add people to teams that are going to add to that enjoyment. And we want our work to have special meaning, which means, when we go to work, our personal purpose is somehow fulfilled by the work that we do. ”

—Michael Bush, CEO, Great Place to Work ( Fortune ‘s partner on the Best Companies list)

“ We’re in a service industry, and our job is delivering exceptional experiences to our customers all the time … If I want to do that, and I want to do it better than our competitors and deliver that  alpha . My team has to be inspired. They have to feel part of something bigger than them. They have to understand what the purpose is and understand how they fit into it and trust it. They have to feel good about it. And when they do feel good about it, they feel empowered. They give more effort, which translates into  alpha  and delivering better experiences for customers. ”

—Chris Nassetta, CEO, Hilton

“ Why are we here? Why do we get up every morning? Why do we continue to be resilient in the face of so many things that are changing for our clients and for us as professionals. Well, it’s because it really matters. It really matters in the lives of millions and millions of people that we are entrusted with their assets, with their hopes and dreams, with their financial plans. And so that trust, that human connection is very purposeful.  “

—Penny Pennington, Managing Partner, Edward Jones

“ Trust is foundational, as both Chris and Penny  expressed well. It sits at the core of culture. It sits at the core of our business. Our customers trust us with respect to their lives, you know. Every day we put 500,000 people on our planes, and we take the very best care of them to get them safely and comfortably to their destination, and we do it 5,000 flights a day, with over 100,000 employees scattered around the world. Our employees have to know we have their back .”

—Ed Bastian, CEO, Delta.

More news below.

Alan Murray @alansmurray [email protected]

South Korea’s elections

South Korea voters rejected President Yoon Suk Yeol’s conservative party in legislative elections on Wednesday. Korea’s opposition Democratic Party is on track to win over 170 seats in the legislature, just short of a super majority that can override presidential vetoes. Yoon has tried to deepen Korea’s security relationships with partners like the U.S. and Japan, but has stumbled in implementing domestic policy changes, like tax cuts and eased business regulations. Reuters

Simon & Schuster turns 100

Simon & Schuster celebrated its 100th anniversary on Tuesday, as the publisher's new owner KKR invests in new imprints and hires more editors. Fears that KKR would gut the publisher haven’t been born out: “That’s not how you deliver a return on investment in the current world,” says Richard Sarnoff, KKR’s chairman of media and chair of Simon & Schuster’s board. The New York Times

McKinsey layoffs

McKinsey will cut around 360 jobs globally, primarily in departments like design, data engineering, and software. Traditional consultants aren’t affected by the layoffs. Consulting firms are warning that their usual corporate clients are starting to pull back on long-term spending. Bloomberg

AROUND THE WATERCOOLER

Exclusive: Wiz acquires Gem Security by Allie Garfinkel 

SEC moves to sue Uniswap in bid to hobble fast-growing DeFi sector by Jeff John Roberts

A German Rust Belt? As Chinese EVs like BYD swarm Europe’s key markets, historic examples of deindustrialization pose a warning to the continent’s carmakers by Ryan Hogg

Engineers at Baltimore’s fallen port are working so diligently, they expect to restore service months earlier than expected by Dylan Sloan

French IT firm Atos was once a crown jewel valued at $15 billion. Now, it’s drowning in debt, and the government is helping it stay afloat by Prarthana Prakash

How a ‘rebel’ hire at Autodesk ascended to the company’s top job by Fortune Editors

T his edition of CEO Daily was curated by Nicholas Gordon. 

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