Q. 60 - Precalculus Enhanced with Graphing Utilities

Question

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?

Step-by-Step Solution

Answer

The amount of time taken by Bruce to finish the job is $4$ hours, the time taken by Bryce is $2$ hours and the time taken by Marty is $8$ hours.

Step 1. Given

Time taken by Bruce and Bryce to complete the work is $1 hour 20 minutes.$

Time taken by Marty and Bryce to complete the work is $1 hour and 36 minutes$

Time taken by Bruce and Marty to complete the work is $2 hours and 40 minutes$

Step 2. Assume the variables.

Let the time taken for Bruce, Bryce and Marty be denoted by $x,y,z$ respectively.

Convert the time in hours to minutes,

$1$hour $20$ minutes$=60+20$

                                $=80$ minutes

$1$ hour $36$  minutes $=60+36$

$=96$ minutes

$2$hour $40$  minutes $=120+40$

                                  $=160$ minutes.

Step 3. Form a system of equations

The system of linear equation from the given information is formatted as:

$\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y}=\frac{1}{80}\\ \frac{1}{y}+\frac{1}{z}=\frac{1}{96}\\ \frac{1}{x}+\frac{1}{z}=\frac{1}{160}\end{array}\right.$

Step 4. Solve the equations

Subtract second equation from first equation,

$$\begin{gathered} \frac{1}{x}-\frac{1}{z}=\frac{1}{80}-\frac{1}{96} \\ \frac{1}{x}-\frac{1}{z}=\frac{16}{7680} \\ \frac{1}{x}-\frac{1}{z}=\frac{1}{480} \end{gathered}$$

Step 5. Find x

Add above equation with the third equation, we get,

$$\begin{gathered} \frac{1}{x}-\frac{1}{z}+\frac{1}{x}+\frac{1}{z}=\frac{1}{480}+\frac{1}{160} \\ \frac{2}{x}=\frac{640}{76800} \\ \frac{2}{x}=\frac{1}{120} \\ \frac{1}{x}=\frac{1}{240} \end{gathered}$$

Step 6. Find y

Substitute $x=240$ in the first equation,

$$\begin{gathered} \frac{1}{240}+\frac{1}{y}=\frac{1}{80} \\ \frac{1}{y}=\frac{1}{80}-\frac{1}{240} \\ =\frac{2}{240} \\ =\frac{1}{120} \end{gathered}$$

Step 7. Find z

Substitute $y=120$ in second equation,

$$\begin{gathered} \frac{1}{y}+\frac{1}{z}=\frac{1}{160} \\ \frac{1}{120}+\frac{1}{z}=\frac{1}{160} \\ \frac{1}{z}=\frac{1}{160}-\frac{1}{120} \\ =\frac{1}{480} \end{gathered}$$

Step 8. Convert the minutes into hours

$x=240$ minutes

$$\begin{gathered} =\frac{240}{60} \\ =4 hours \end{gathered}$$

$y=120 minutes$

  $$\begin{gathered} =\frac{120}{60} \\ =2 hours \end{gathered}$$

$z=480 minutes$

  $$\begin{gathered} =\frac{480}{60} \\ =8 hours \end{gathered}$$