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Module 5.2 Work Problems

ByDr. Marcia L. Tharp

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Introduction

– In today’s world of work people are asked to work in

groups to complete projects or a task. For this reason

it is important to know how long it will take a group of

people working together to complete a job or task such

as taking an inventory, xeroxing a report, painting

walls or cleaning a building. These type of problems in

which two or more people or sometimes machines

work together to complete a job or task are calledwork problems. To solve them we need to understand

some key ideas.

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Big Idea #1

– Work problems are represented

using fractional equations. The

part of each job or task done byeach worker is usually totaled to

equal one whole job. The number

1 represents one whole job.

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For example:

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Check your understanding:

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Solution:

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You can check your work by placing your

answer in to the fraction equation.

Joe’s part of job done + Mary’s part of job done = 1 whole job.

+ = 1

or = 1

or 1 = 1Since both sides are equal our answer is correct.

4

1

4

3

4

4

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Big Idea #2

The rate at which a person or machine

works to complete a task is based on

the time it takes them to complete the task.In general if a person can complete a task

in t hours, then the rate they work at is

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For example:

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Julia can complete a task in 4 hours.

What is her rate per hour?

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Solution:

Because

4

1+

4

1+

4

1+

4

1=1 whole job

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Big Idea #3

To determine the part of a job completed by a

person or machine multiply the rate a person

works by the time they spend doing the work.

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For example:

What part of the job can Bonita complete

if she can complete the whole job in

4 hours and she only works on it for3 hours?

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Solution:

Using our formula we know:

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Megan can complete a job in 12

hours. How much of the job will

she complete if she only workson it for 5 hours?

It’s your turn now! Think aboutthis before you answer.

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Solution:

Using the formula we have:

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Now lets put these Big Ideas

together to solve some work

problems. But first here is thepattern we will use to solve these

problems.

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A Pattern For Problem Solving

1) List the important information.

2) Determine what the problem asks you to look for.

3) Assign a variable to what you are looking for.

4) Guess the answer using a table so that you

become familiar with how the problem works.

5) Set up an equation.

6) Solve the equation.

7) Check the result by replacing your number in the

equation you set up.

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Example 1

Curtis and Nadine work as a teamat a car wash. Working alone ittakes Curtis 3 hours to wash and

detail a van. If Nadine workedalone it would take her 4 hours towash and detail a van. Their bosswants to know how long will it takethen to wash and detail a vantogether?

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1) List the important information.

Curtis works 3 hours to wash and detail the van.

Nadine works 4 hours to wash and detail the van.

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2) Determine what the problem

asks you to look for.

– We need the time it takes Curtis and Nadine to

wash and detail the van together.

– (Hint: look at the question “?” sentence.)

3) Assign a variable to what you

are looking for. Let t= the time it takes Nadine and Curtis

to wash & detail the van together

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4) Guess the answer using a table below to

become familiar with the problem.

Click here to check your guess by

changing the number in the firstcolumn on the spreadsheet.

http://localhost/var/www/apps/conversion/tmp/scratch_7/Work1.xls



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5) Set up an equation.

We first need to have the part of the job

Curtis and Nadine each did separately.

We can put this in a table to get it.

Person Rate x Time = Part of jobcompleted

Curtis

3

1 t

3

t

Nadine

4

1 t

4

t

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Now complete the equation below by filling in the part of

the job completed by each person who worked.

Part of jobCompleted by +Curtis

Part of jobCompleted byNadine

= 1 whole job

3

t +

4

t = 1 whole job

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6) Solve the equation.

Multiply each side of the equation by the LCD, 12.

12

3

t + 12

4

t = 1 ( 12 ) whole job

4t + 3t = 12

7t = 12

t =

7

12or

7

51 hours

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Answer:

So the job will take

7

12hours or

7

5

1 hours to complete

when Curtis and Nadine work together.

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Is that your final answer?(Step 8 Check the result.)

Well to be sure we will check by putting

our answer in to the fraction equation.

3

t +

4

t = 1 whole job

3

7

12

+

4

7

12

= 1 whole job

7

12

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We finish simplifying it.

Since both sides are equal the answer is correct.

We have no doubt that our final answer is correct.

14

1

7

12

3

1

7

12 whole job

17

3

7

4 whole job

1=1 whole job

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Example 2

Kareem and Jody are in a hurry to fill

their Jacuzzi. So they use both the hot

and cold-water faucets. With the hot

water on the Jacuzzi can be filled in 2

hours. If the cold water is used alone it

will take 3 hours to fill the Jacuzzi. How

long will it take to fill a Jacuzzi usingboth faucets together?

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Step 1 List the important information.

It takes 2 hours to fill the Jacuzzi

using the hot water faucet alone.

It takes 3 hours to fill the Jacuzzi

using the cold-water faucet alone.

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Step 2Determine what the problem asks you to look for.

We need the time it takes to fill the Jacuzzi using

both the hot and cold water faucets.

Step 3

Assign a variable to what you are looking for.

Let t = the time it takes to fill the Jacuzzi using

both the hot and cold water faucets.

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Step 4 Guess the answer using a table to

become familiar with problem.

To make your guess click on the link

to the spreadsheet end change the

first column.


http://a/Work1.xls

http://a/Work1.xls


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Step 5 Set up the equation

Hint: Make a table to show the part the job each

faucet completes in one hour.

See if you can fill in the table below.

Faucet Rate Time Part of JobCompleted

Hot

Cold

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Here is what to put in the table.

Faucet Rate Time Part of Job

CompletedHot

2

1 t

2

t

Cold

3

1 t

3

t

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Step 5 Set up the equation.

Part of the job +

done by the hot

water faucet

Part of the job

done with the cold

water faucet

= 1 whole job

2

t

+ 3

t = 1 whole job

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Step 6-Solve the equation.

Multiply both sides by the LCD, 6.

6 t

2 + 6

t

3= 6 1

3 t + 2 t = 65t = 6

t =6

5hours or 1

1

5hours

or 1 hour +1

5x 6o minutes = 1 hr. and 12 min.

So it takes 1 hour and 12 minutes to fill the Jacuzzi.

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Step 7 - Check your answer by replacing it in

the equation you set up in step 5.

t

2 +

t

3= 1

6

5

2

+

6

5

3

= 1

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Simplify this equation.

This is true because both sides are equal.11

15

5

15

2

5

3

131

56

21

56

13

5

6

2

5

6

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Example 3

Two computers are available to process

a batch of data. The slower computer

can process it in 30 minutes. If both

computers are used at the same time

they can process it in 10 minutes. How

long will it take the faster computer to

do the whole job alone?

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Step 1 List important information.

The slower computer completes the job

in 30 minutes.

The 2 computers working together complete

the job in 10 minutes.

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Step 2 Determine what we are looking for.

How long will it take the faster computer

to complete the job alone?

Step 3 Assign a variable to it.

Let t = the time it takes the faster

computer to complete the job.

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Step 3 Put what you are looking for in

variable form.

Let t = how long it will take the faster

computer to complete the job alone.

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Step 4 Guess the answer using a table to

become familiar with the problem.

Click on this link to make your guess by

using the spreadsheet.



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First fill in a rate time chart.

Computer Rate Time Part of job

completed

Slower

Faster

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Remember rate = 1/time it takes to do the whole job


completed

Slower

Faster

1

30

1

t

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Next fill in the time.


completed

Slower

10 min.

Faster

10 min.

1

30

1

t

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Now multiply rate x time to find the part of the job completed .


completed

Slower

10 min.

Faster

10 min.

1

30

1

t

10

30

1

3or

10

t

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1

30

10

30

1

3or

1

t

10

t


completed

Slower

10 min.

Faster

10 min.

Use the last column to build your equation.

Slower Computer + Faster Computer = 1 whole job

+ = 13

1 10

t

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Step 6 Solve the equation.

Multiply by the LCD, 3t

1)

3t + 3t = 3t 11

3

10

t

t + 30 = 3t

30 = 2t 15 = t

It takes 15 minutes for the faster computer

to complete the job.

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Step 7 Check your answer by replacing it in

the equation you set up in step 5.

.

110

3

1

t

11

115

15

115

10

15

5

115

10

3

1

Both sides are the same. Sowe have the correct solution.

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Now it is your turn!

Go on to the practice problems

in the next file by clicking the

back button.

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