Work Word Problems Worksheets

If a man could finish a job in $t$ days, then in one day he could only finish $\frac{1}{t}$ of the job. Similarly, if it takes $a$ days for one man to finish a job alone while it takes $b$ days for another man to finish the same job by himself, then working together they could finish the job in $\frac{1}{a}+\frac{1}{b}=\frac{1}{t}$ days. These principles are very helpful in solving work word problems. To solve work problems, we simply define an equation congruent to the given definition of work above.

Examples

1. A man can finish a painting job in 4 days. Another man can finish the same job in 6 days. If both men plus a third man can finish the painting job in 2 days, how long will it take for the third man to finish the job alone?

2. A pump can drain a tank in 11 hours. Another pump can drain the same tank in 20 hours. How long will it take both pumps together to drain the tank?

3. If it takes James twice as much as it takes Danny to do a particular task. Working together they can do the task in 6 days. How long would it take James to do the task alone?

4. An electrician can finish the installation of an antenna tower in 200 man–hours while a second electrician can finish the same job in 300 man-hours. How long will it take both electricians to finish the same job, if they work together?

Solutions

1. Let $x$ be the number of hours it takes for the third man to finish the job.

$\begin{align*} \displaystyle \frac{1}{4}+\frac{1}{6}+\frac{1}{x}&=\frac{1}{2} \\12x\left (\frac{1}{4}+\frac{1}{6}+\frac{1}{x} \right )&=12x\left (\frac{1}{2} \right )\\3x+2x+12&=6x\\5x-6x&=-12\\\therefore x&=12 \end{align}$ Hence, it takes 12 days for the third man to finish the job alone.

2. Let $x$ be the number of hours it takes for the pump to drain the tank. $\begin{align*} \displaystyle \frac{1}{11}+\frac{1}{20}&=\frac{1}{x} \\220x\left (\frac{1}{11}+\frac{1}{20} \right )&=220x\left (\frac{1}{x} \right )\\22x+11x&=220\\33x&=220\\\therefore x&=6\frac{2}{3} \end{align}$ Therefore, it takes 6 hours and 40 minutes for the pump to drain the tank.

3. Let $x$ be the number of hours it takes Danny to finish the job alone. $\begin{align*} \displaystyle \frac{1}{x}+\frac{1}{2x}&=\frac{1}{6} \\6x\left (\frac{1}{x}+\frac{1}{2x} \right )&=6x\left (\frac{1}{6} \right )\\6+3&=x \\\therefore x&=9\end{align}$ Hence, it takes 18 hours for Danny to finish the job alone.

4. Let $x$ be the number of hours it takes both men to finish the job if they work together. $\begin{align*} \displaystyle \frac{1}{200}+\frac{1}{300}&=\frac{1}{x} \\600x\left (\frac{1}{200}+\frac{1}{300} \right )&=600x\left (\frac{1}{x} \right )\\3x+2x&=600\\5x&=600 \\\therefore x&=120\end{align}$ Ergo, it takes 120 hours for both electricians to finish the installation job.