Math Homework. Do It Faster, Learn It Better.

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 3 hours to finish. Leon works faster; if he does it, it takes him only 2 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 t 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 3 hours, he can rake 1 3 of the park in 1 hour. So if he rakes for t hours, then he finishes 1 3 t or t 3 of the park.

Similarly, Leon finishes t 2 of the park.

We assumed that after t hours, they are done... that is, they have raked 1 "complete park". So:

t 3 + t 2 = 1

Now solve for t . First, multiply through by 6 .

6 t 3 + 6 t 2 = 6 1 2 t + 3 t = 6

Combine like terms .

5 t = 6

Divide both sides by 5 .

t = 6 5

So, it takes them 6 5 hours or 1 1 5 hours working together. You can convert the fractional part to minutes by multiplying by 60 minutes / 1 hour.

1 5 hr 60 min 1 hr = 60 5 min = 12 min

So, it takes them 1 hour and 12 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 1 complete job.

(Read carefully though: some problems might ask how long it takes to finish more than 1 job.)