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: QuickandEasy Estimates.
TABLE OF CONTENTS for this web page:
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.
Work problems always involve rates.
Recall from Rate Problems that:
For example, these are all rates:
Work problems typically involve rates with a unit of time in the denominator.
That is, you might see $\displaystyle\ \frac{\$5}{\rm hr}$ (denominator is ‘hours’)
or $\displaystyle\ \frac{1{\rm\ job}}{3{\rm\ sec}}$ (denominator unit is ‘seconds’) in a work problem,
because these denominators are both units of time.
However, you won't typically see $\displaystyle\ \frac{10{\rm\ kg}}{{\rm in}^3}$ in a work problem, because it doesn't have a unit of time in the denominator.
In a twoperson work problem, there are always three rates involved:
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: 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 notsomuch 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?
$$ \overset{\rm individual\ rate} { \overbrace { \frac{5\rm\ pages}{\rm hour} } } + \overset{\rm individual\ rate} { \overbrace { \frac{2\rm\ pages}{\rm hour} } } = \frac{7\rm\ pages}{\rm hour} \ \overset{?}{=\strut} \ \text{the combined rate} $$
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:
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 $\,\frac22\,$.
Want to know how many pages are typed in one hour?
Then $\displaystyle\ \frac{5{\rm\ pages}}{\rm hr}\ $ is the best name.
Want to know how many pages are typed in two hours?
Then $\displaystyle\ \frac{10{\rm\ pages}}{2\rm\ hr}\ $ is the best name.
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.However, you can check to see if your answer is correct. 
PROBLEM TYPES:
