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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 $\frac{1}{3}$ of the park in $1$ hour. So if he rakes for $t$ hours, then he finishes $\frac{1}{3}\cdot t$ or $\frac{t}{3}$ of the park.

Similarly, Leon finishes $\frac{t}{2}$ of the park.

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

$\frac{t}{3}+\frac{t}{2}=1$

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

$\begin{array}{l}6\cdot \frac{t}{3}+6\cdot \frac{t}{2}=6\cdot 1\\ 2t+3t=6\end{array}$

Combine like terms .

$5t=6$

Divide both sides by $5$ .

$t=\frac{6}{5}$

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

$\begin{array}{l}\frac{1}{5}\text{hr}\cdot \frac{60\text{min}}{1\text{hr}}=\frac{60}{5}\min \\ =12\text{min}\end{array}$

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.)