Word Problems: Work and Workers
The shared work problem is a common type of word problem which requires you to solve a linear equation involving fractions. Here is an example:
James and Leon work at a park. Once a week, they have to rake up all the fallen leaves. If James does it, it takes him hours to finish. Leon works faster; if he does it, it takes him only hours to finish.
How long will it take them to rake the leaves in the park if they start at the same time and both work together?
Let be the number of hours it takes them if they both work together. This is the number we're trying to find.
Since James can rake the whole park in hours, he can rake of the park in hour. So if he rakes for hours, then he finishes or of the park.
Similarly, Leon finishes of the park.
We assumed that after hours, they are done... that is, they have raked "complete park". So:
Now solve for t . First, multiply through by .
Divide both sides by .
So, it takes them hours or hours working together. You can convert the fractional part to minutes by multiplying by minutes / hour.
So, it takes them hour and minutes working together.
A general strategy for this kind of problem: define a variable for the amount of time it takes everyone working together. Then write an equation using fraction of work completed by each person, and set it equal to complete job.
(Read carefully though: some problems might ask how long it takes to finish more than job.)