Work Problems: Basic Concepts
If you like to learn by example, jump right to Work Problems: Guided Practice.
For a more thorough approach, keep reading—this section covers all the concepts needed to successfully solve work problems.
After mastering this section, you'll be ready for
Work Problems: Quick-and-Easy Estimates.
A Typical Work Problem
Carol mows a lawn in 5 hours.
Julia mows the same lawn in only 3 hours.
How long will it take if they mow the lawn together?
Table of Contents for This Web Page
- Choose Your Own Names! (Personalize all the practice problems on this page)
- Different Types of Work Problems (Guessing Game)
- Rates in Work Problems (Quick Check)
- Individual Rates versus Combined Rates (Quick Check)
- When Does the Sum of the Individual Rates Give the Combined Rate? (Quick Check)
- Rates Have Lots of Different Names (Sentence Shuffle)
The practice problems in this section 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.
Different Types of Work Problems
In a work problem, we know some information about how quickly a 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.
The following ‘Guessing Game’ introduces you to several types of work problems, and develops your intuition for reasonable solutions.
Rates in Work Problems
Work problems always involve rates. Recall from Rate Problems that:
For example, these are all rates:
- 5 dollars per hour, also written as $\displaystyle\ \frac{\$5}{\rm hr}$
- 1 job per 3 seconds, also written as: $$\text{1 job/3 sec}\ \ \text{ or } \ \ \frac{1{\rm\ job}}{3{\rm\ sec}}\ \ \text{ or }\ \ \frac 13\ \frac{\rm\ job}{\rm\ sec}$$
- 10 kilograms per cubic inch, also written as: $$\text{10 kg/}\text{in}^3\ \ \text{ or }\ \ \frac{10{\rm\ kg}}{ { \rm in}^3}$$
Work problems typically involve rates with a unit of time in the denominator.
That is, you might see
$$\frac{\$5}{\rm hr}\ \ \text{(denominator is ‘hours’)}$$or
$$\frac{1{\rm\ job}}{3{\rm\ sec}}\ \ \text{(denominator unit is ‘seconds’)}$$in a work problem, because these denominators are both units of time.
However, you won't typically see $\ \frac{10\text{ kg}}{\text{in}^3}\ $ in a work problem, because it doesn't have a unit of time in the denominator.
Rates
Individual Rates versus Combined Rates
In a two-person 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.
Individual versus Combined Rates
When Does the Sum of the Individual Rates Give the Combined Rate?
Under certain conditions, there is a simple relationship between the individual rates and 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 that $\,5 + 2 = 7\,$.)
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. The combined rate will be less than $\,7\,$ pages/hour. The combined rate will be less than the sum of the individual rates.
So, under what circumstances will the sum of the individual rates equal the combined rate?
$$ \begin{align} &\overset{\rm individual\ rate} { \overbrace { \frac{5\rm\ pages}{\rm hour} } } + \overset{\rm individual\ rate} { \overbrace { \frac{2\rm\ pages}{\rm hour} } }\cr\cr &\qquad = \frac{7\rm\ pages}{\rm hour}\cr\cr &\qquad\overset{\bf\huge ?}{=\strut} \ \text{the combined rate} \end{align} $$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 still types $\,5\,$ 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 still types $\,2\,$ pages per hour when he's working with Carol.
In order to solve work problems, we have to assume that this relationship is true. Dr. Burns gives this assumption a special name—the Individual Rates Assumption:
Individual Rates Assumption
When two people work together, the rates at which they work remain exactly the same as if they were working alone.
(Under this assumption, the combined rate is the sum of the individual rates.)
Individual Rates Assumption
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! Here's an example, where we multiply by $\,1\,$ in the form of $\,\frac22\,$:
$$ \begin{align} \frac{5{\rm\ pages }}{\rm hr} &= \overset {\text{‘hr’ is ‘1 hr’}} { \overbrace { \frac{5{\rm\ pages}}{1\rm\ hr} } }\cr\cr &= \overset {\rm multiply\ by\ 1} { \overbrace { \frac{5{\rm\ pages}}{1\rm\ hr}\cdot 1 } }\cr\cr &= \overset {\rm 1\ =\ 2/2} { \overbrace { \frac{5{\rm\ pages}}{1\rm\ hr}\cdot\frac22 } }\cr\cr &= \overset {\rm multiply\ across} { \overbrace { \frac{10{\rm\ pages}}{2\rm\ hr} } } \end{align} $$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.