Work Problems

This lesson has been expanded and then broken into three lessons in
One Mathematical Cat, Please! A First Course in Algebra:

Work Problems: Basic Concepts
Quick-and-Easy Estimates for Two-Person Work Problems
Work Problems: Guided Practice

After you've mastered the ideas from this section, move on to:
Exponents

Introduction:

This section is an experiment with a totally different style.
Email me (OneMathematicalCat@outlook.com)
if you have a strong opinion one way or another!
Currently, the avatar only ‘fake-talks’;
the conversation goes in the avatar column.

Welcome to "Work Problems"!
Skip around if you want, or take this "thorough" approach:

0
To return to this introduction at any time, click Section 0 above.
Or, just scroll this column up!
1
Glance over a typical work problem in Section 1.
Don't worry if it looks a bit abstract—just focus on the general pattern.
2
Section 2 gives a sample problem and solution.
It may not make complete sense (yet)—just look it over to see what's in store.
3
All the important concepts are in Section 3.
Understand these ideas, and you'll be ready to solve lots of different work problems.
4
To get you started solving problems, Section 4 offers guided practice.
You can just "click" through each problem, but it's best to try each step on your own.
5
Practice a wide variety of questions in Section 5.
If you want, print a worksheet for practice away from the computer.

Want the avatar to talk to you?
If so, just click the "Talk to me!" button above.
Keep clicking the button to move through the text.

Enjoy learning about work problems!

ON TO: work problem "pattern"

WORK PROBLEM "PATTERN":

Don't worry if this section looks a bit abstract—just focus on the general pattern.
You'll see an example of this pattern in Section 2.

Person₁ can do a job in $\,t_1\,$ time.
Person₂ can do the same job in $\,t_2\,$ time.

How long will it take if they do the job together?

Assumptions:

Two People Can Do the Job at the Same Time:
Two people must be able to do the job at the same time,
each contributing (at their own rate) to the job's completion at every moment.
Individual Rates Stay the Same when People Work Together:
When the two people work together,
the rates at which they work must remain exactly the same as if they were working alone.

Prerequisites:

Rate Problems

ON TO: a sample problem and solution

A SAMPLE PROBLEM AND SOLUTION:

This is an example of the pattern illustrated in Section 1.
If you've done problems like this before,
then this section—the "in-a-nutshell" solution—might be all you need.
Need more explanation? Be sure to read Section 3.

Carol mows a lawn in 4 hours.
Julia mows the same lawn in only 2 hours.

How long will it take if they mow the lawn together?

Check Assumptions:

Two People Can Do the Job at the Same Time:
Two people can mow a lawn at the same time, providing:
-- they each have their own mower;
-- the lawn is big enough that they can be working on different sections at the same time.
Individual Rates Stay the Same when People Work Together:
Two people can mow a lawn together without changing their individual rates, providing:
-- they're using familiar equipment;
-- they can stay away from the other person (so they don't spend time chatting).

In order to proceed, we'll assume that these conditions hold.

Get an Estimate:

Julia is the faster worker.
If she mows the lawn all by herself in 2 hours,
then with anyone's help it will get done faster than 2 hours.
So, the answer must be less than 2 hours.
If Carol and Julia worked at exactly the same (faster) rate,
then the time would be cut in half (to just one hour).
However (sigh) Carol is older and slower.
So, the answer must be more than 1 hour.
Combining results, the answer must be between 1 and 2 hours.

Sample Solution:

STEP 1: Identify and rename the individual rates.

Carol's work rate is one job per 4 hours,
where the "job" is "mowing a (particular) lawn".

Rename this rate so the denominator is "1 hour" instead of "4 hours".
(Remember that dividing by 4 is the same as multiplying by $\frac14$.)

$$ \text{Carol's rate:}\qquad \frac{1\text{ job}}{4\text{ hr}}\ \ \ \ \ = \overset{\rm divide\ top\ and\ bottom\ by\ 4} { \overbrace { \frac{1\text{ job}}{4\text{ hr}}\cdot \frac{\frac1{4}}{\frac1{4}} } } = \ \ \ \ \ \ \frac{\frac1{4}\text{ job}}{1\text{ hr}} $$

So, Carol does $\frac14$ of a job in $1$ hour.

Similarly, rename Julia's rate:

$$ \text{Julia's rate:}\qquad \frac{1\text{ job}}{2\text{ hr}}\ \ \ \ \ = \overset{\rm divide\ top\ and\ bottom\ by\ 2} { \overbrace { \frac{1\text{ job}}{2\text{ hr}}\cdot \frac{\frac1{2}}{\frac1{2}} } } = \ \ \ \ \ \ \frac{\frac1{2}\text{ job}}{1\text{ hr}} $$

So, Julia does $\frac12$ of a job in $1$ hour.

STEP 2: Compute the combined rate.

Carol does $\frac14$ of a job in 1 hour; Julia does $\frac12$ of a job in 1 hour.
Together, they do $\frac14 + \frac12$ of a job in 1 hour.

Adding:

$$ \frac14 + \frac12 \ \ = \ \ \overset{\rm get\ common\ denominator}{\overbrace{\frac14 + \frac24}} \ \ =\ \ \frac34 $$

Putting together all our results so far, we have:

$$ \text{Combined rate:}\ \ \ \ \ \ \frac{1\text{ job}}{4\text{ hr}} + \frac{1\text{ job}}{2\text{ hr}} \ \ =\ \ \frac{\frac1{4}\text{ job}}{1\text{ hr}} + \frac{\frac1{2}\text{ job}}{1\text{ hr}} \ \ =\ \ \frac{\frac3{4}\text{ job}}{1\text{ hr}} $$

Together, they do $\frac34$ of a job in $1$ hour.

STEP 3: Rename the combined rate.

We want to know how long it takes to do 1 job, working together.
To find this, rename the combined rate so that the numerator is 1.
Remember: any number, multiplied by its reciprocal, gives 1.

$$ \text{Combined rate:}\ \ \frac{\frac34\text{ job}}{1\text{ hr}} \ \ \ \ = \overset{\rm multiply\ top\ and\ bottom} { \overset{\rm by\ the\ reciprocal\ of\ 3/4} { \overbrace { \frac{\frac34\text{ job}}{1\text{ hr}}\cdot \frac{\frac43}{\frac43} } } } =\ \ \ \ \frac{1\text{ job}}{\frac43\text{ hr}} $$

Together, they can do 1 job in $\frac 43$ hours.

STEP 4: State the conclusion in a desired way.

Note that $\frac 43$ hours is the exact solution.
No decimal approximation has been done up to this point.
Whenever possible, get an exact answer—do any needed approximation at the last step only.

Here, we'll choose to round the answer to two decimal places.
Using your calculator, 4/3 is approximately 1.33 .
Together, Carol and Julia can mow the lawn in about 1.33 hours.
(Note that this agrees with our estimate.)

ON TO: concepts/the lesson

CONCEPTS/THE LESSON:

This section thoroughly covers the concepts needed to understand a wide variety of work problems.
If you've done work problems before, then Section 2 might be all that you need.
If not, then stay right here—this is the section for you!

Choose Your Own Names!
(Make the practice problems more fun!)
What exactly IS a work problem?
(the Guessing Game)
What are "rates"?
(Quick Check: rates)
Rates involved in work problems
(Quick Check: individual versus combined rates)
Key Idea: the sum of the individual rates gives the combined rate
(Quick Check: the "individual rates" assumption)
Key Idea: rates have lots of different names
(Sentence Shuffle)

Choose Your Own Names!

The practice problems will be MORE FUN if they use people you know!
So... take a minute and put in some names!

Think of a name. Type it in the name box below.
Is the name you're thinking of male or female?
Click the appropriate male/female button.
Click the "Add this name!" button.
Put in as many or as few as you want.
(We may throw in some of our own, just to spice things up.)
Refresh this page if you want to throw everything away and start over.

What Exactly IS a "Work Problem"?

In a work problem, there is some "job" being done.

The job must meet the following requirements:
-- it could be done by one person working alone;
-- or, it could be done by two people working together.

For example, one person can mow a lawn.
Or, two people can mow a lawn together, providing there are two mowers,
and the lawn is big enough that they won't get in each other's way.

In a work problem, we know some information about how quickly the job can be done.
For example, we might know how fast each person could do it, if they work alone.
However, there's always something that we don't know, and want to figure out.

In a work problem, it might not even be people doing the work!
The job might be done by animals, or machines, or ... use your imagination!
However, in this discussion, we'll have people do the job, just to keep things simple.

Try the following "Guessing Game"!
It will introduce you to several types of work problems,
and develop your intuition for reasonable solutions.

the GUESSING GAME

What are "rates"?

Work problems involve rates.

For a good review of rates, study this Rates web exercise.
(It requires Internet Explorer, with MathPlayer installed.)
For completeness, however, the essential concepts are quickly reviewed here.

Definition: rate

A rate is a comparison of two quantities that are measured in different units.

For example, these are all rates:

5 dollars per hour, also written as $\ \frac{\$5}{\rm hr}$
1 job per 3 seconds, also written as 1 job/3 sec or $\ \frac{1{\rm\ job}}{3{\rm\ sec}}$
10 kilograms per cubic inch, also written as 10 kg/in³ or $\ \frac{10{\rm\ kg}}{{\rm in}^3}$

Work problems typically involve rates with a unit of time in the denominator,
like examples (1) and (2) above.
That is, you might see $\ \frac{\$5}{\rm hr}$ (denominator is "hours") or $\ \frac{1{\rm\ job}}{3{\rm\ sec}}$ (denominator is "seconds")
in a work problem, because these denominators are both units of time.
However, you don't typically see $\ \frac{10{\rm\ kg}}{{\rm in}^3}$ in a work problem,
because it doesn't have a unit of time in the denominator.

Quick Check: rates

Rates involved in work problems:
individual rates, combined rates

In a work problem, there are always three rates involved:

the rate the job is done when the first person works alone;
the rate the job is done when the second person works alone;
the rate the job is done when both people work together.

The first two rates are called the individual rates,
and the last one is called the combined rate.

Quick Check:
individual versus combined rates

Key Idea:
the sum of the individual rates gives the combined rate

Under certain conditions, there is a simple relationship between the individual rates and the combined rate:

The individual rates, when added together, give the combined rate.

Consider, for example, the following scenario:

Suppose Carol types 5 pages per hour, and Karl types 2 pages per hour.
Will they be able to type 7 pages together in one hour? (Note: 5 + 2 = 7)

Maybe. Maybe not.

If there's only one typewriter between the two of them, then one will have to wait while the other types.
They definitely won't be able to get 7 pages done in one hour.

Or, suppose Carol and Karl can't ever get together without chatting, chatting, chatting.
Then, they'll spend a lot of time talking, and not-so-much time typing.
They definitely won't be able to get 7 pages done in one hour.

So, under what circumstances will the following be true? $$ \overset{\rm individual\ rate} { \overbrace { \frac{5\rm\ pages}{\rm hour} } } + \overset{\rm individual\ rate} { \overbrace { \frac{2\rm\ pages}{\rm hour} } } = \overset{\rm combined\ rate} { \overbrace { \frac{7\rm\ pages}{\rm hour} } } $$

ANSWER:
When the two people work together, the rates at which they work
must remain exactly the same as if they were working alone.

If Carol can type 5 pages per hour when she's all by herself,
then she can still type 5 pages per hour when she's working with Karl.

If Karl can type 2 pages per hour when he's all by himself,
then he can still type 2 pages per hour when he's working with Carol.

In order to solve work problems, we have to assume that this relationship is true.

Quick Check:
the "individual rates" assumption

Key Idea:
rates have lots of different names

A rate is just an expression.
Like all mathematical expressions, rates have LOTS of different names.
The name you use for a rate depends on what you're doing with it.

Rates are easy to rename—just multiply by $1$ in an appropriate form!
(Remember: multiplying by $1$ doesn't change stuff!)
Here's an example, where we multiply by $1$ in the form of $\frac22$.

$$ \frac{5{\rm\ pages}}{\rm hr} = \overset {\rm "hr"\ is\ "1\ hr"} { \overbrace { \frac{5{\rm\ pages}}{1\rm\ hr} } } = \overset {\rm multiply\ by\ 1} { \overbrace { \frac{5{\rm\ pages}}{1\rm\ hr}\cdot 1 } } = \overset {\rm 1\ =\ 2/2} { \overbrace { \frac{5{\rm\ pages}}{1\rm\ hr}\cdot\frac22 } } = \overset {\rm multiply\ across} { \overbrace { \frac{10{\rm\ pages}}{2\rm\ hr} } } $$

Want to know how many pages are typed in one hour? Then $\ \frac{5{\rm\ pages}}{\rm hr}\ $ is the best name.
Want to know how many pages are typed in two hours? Then $\ \frac{10{\rm\ pages}}{2\rm\ hr}\ $ is the best name.

Sentence Shuffle

(THIS SECTION IS
NOT FINISHED)

ON TO: step-by-step practice

STEP-BY-STEP GUIDED PRACTICE:

Click "New Problem" to get started.
Want a hint for each step? Click the "Hint?" button.
Hints appear in the avatar talk section.
When you're ready to check your answer, click the "Show" button.

Back to the top to try another problem!

ON TO: more practice

MORE PRACTICE:

Online Practice
Practice with a wide variety of questions, all mixed up.
Input the answer yourself, then check to see if you're correct.
If you want, click a button to see the entire step-by-step solution.
Quick Worksheet Creator
Create a quick, randomly-generated worksheet for offline practice.
Custom Worksheet Creator
Put your own information (instructions, etc.) at the top of the worksheet.
Preview until you get it just right!
Pick-and-choose which problems you want.
Insert "point-values" next to each problem.
Learn some HTML along the way...
HTML—HyperText Markup Language—is the language of the World Wide Web!
(NOT YET FINISHED!)

ONLINE PRACTICE

QUICK WORKSHEET CREATOR

CUSTOM WORKSHEET CREATOR
(Not finished!)

Top of the Worksheet

What would you like at the top of your worksheet?
Type it below.
Click "PREVIEW" to check the way it looks.
Click "USE IT!" when you have it just the way you want.
You can use any HTML commands.
If you leave this part blank, then you'll get the Quick Worksheet top.

(Your PREVIEW will go here)

Close the worksheet
(Print as you would any web page; e.g., File-Print)

		To get a new problem, click the "New Problem" button at left!
Check the "Two People Can Do the Job at the Same Time" assumption:
Check the "Individual Rates Stay the Same when People Work Together" assumption:
Get an over-estimate for the answer:
Get an under-estimate for the answer:
Identify and rename the individual rates:
Compute the combined rate:
Rename the combined rate:
State your conclusion in a desired way:

How many problems would you like in your worksheet?
Would you like to include step-by-step solutions?	YES NO