Chapter 11, Problem 107RE

To determine

How long will it take each of them working alone to finish the job?

Answer to Problem 107RE

  $\boxed{4,2and8}$ hours respectively.

Explanation of Solution

Given information:

If Bruce and Bryce work together for $1$ hour and $20$ minutes, they will finish a certain job. If Bryce and Marty work together for $1$ hour and $36$ minutes, the same job can be finished. If Marty and Bruce work together, they can complete this job in $2$ hours and $40$ minutes. How long will it take each of them working alone to finish the job?

Calculation:

Let $x$ denote the work of Bruce, $y$ denote the work of Bryce and $z$ denote that of Marty.

The total time taken, in minutes, by Bruce and Bryce to finish the job is $80$ . So, the work done will be $\frac{1}{x}+\frac{1}{y}=\frac{1}{80}$ .

The total time taken, in minutes, by Bryce and Marty to finish the job is $96$ . So, the work done will be $\frac{1}{y}+\frac{1}{z}=\frac{1}{96}$ .

The total time taken, in minutes, by Marty and Bruce to finish the job is $160$ . So, the work done will be $\frac{1}{x}+\frac{1}{z}=\frac{1}{160}$ .

A system of three equations is formed.

  $\{\begin{array}{cccc}\frac{1}{x}+& \frac{1}{y}& =& \frac{1}{80}\\ & \frac{1}{y}+& \frac{1}{z}=& \frac{1}{96}\\ \frac{1}{x}+& & \frac{1}{z}=& \frac{1}{160}\end{array}$

Subtract the second equation from the first to eliminate $\frac{1}{y}$ .

  $\frac{1}{x}+\frac{1}{y}-\left(\frac{1}{y}+\frac{1}{z}\right)=\frac{1}{80}-\frac{1}{96}$

  $\frac{1}{x}-\frac{1}{z}=\frac{96-80}{7680}$

  $=\frac{16}{7680}$

  $=\frac{1}{480}$

Add the above equation to the third equation in the system.

  $\frac{1}{x}-\frac{1}{z}+\frac{1}{x}+\frac{1}{z}=\frac{1}{480}+\frac{1}{160}$

  $\frac{2}{x}=\frac{160+480}{480.160}$

  $=\frac{640}{76800}$

Simplify the expression to find $x$ .

  $2.76800=x.640$

  $153600=640x$

  $x=\frac{153600}{640}$

  $=240$

Divide by $60$ to find the time taken in hours.

  $x=\frac{240}{60}$

  $=4$

Substitute $240$ for $x$ in the first equation in the system to find $y$ .

  $\frac{1}{240}+\frac{1}{y}=\frac{1}{80}$

  $\frac{1}{y}=\frac{1}{80}-\frac{1}{240}$

  $=\frac{2}{240}$

Take the reciprocal to find $y$ .

  $y=\frac{240}{2}$

  $=120$

Divide by $60$ to find the time taken in hours.

  $y=\frac{120}{60}$

  $=2$

Substitute $120$ for $y$ in the second equation in the system to find $z$ .

  $\frac{1}{120}+\frac{1}{z}=\frac{1}{96}$

  $\frac{1}{z}=\frac{1}{96}-\frac{1}{120}$

  $=\frac{5-4}{480}$

  $=\frac{1}{480}$

Take the reciprocal to find $y$ .

  $z=480$

Divide by $60$ to find the time in hours.

  $z=\frac{480}{60}$

  $=8$

Hence, the times taken by Bruce, Bryce, and Marty to finish the job alone are $4,2and8$ hours respectively.