Chapter 7, Problem 78RE

Solution

To calculate: The distance would take each of them to paint the room alone.

The obtained time is 9 hr 20 min per room, 12 hr per room, and 16 hr 48 min per room.

Given Information:

Sanchez Remodelling has three painters: Sue. Esther, and Murphy. Working together they can paint a large room in 4 hr. Sue and Murphy can paint the same size room in 6 hr. Esther and Murphy can paint the same size room in 7 hr

Calculation:

Consider the given information,

Suppose that s ,  e , and m  be the portions of a room Sue, Esther, and Murphy respectively can each do in 1 hour. (Rooms per hour).

  $\begin{array}{l}1\text{ room}=4\left(s+e+m\right)\\ 1\text{ room}=6\left(s+m\right)\\ 1\text{ room}=7\left(e+m\right)\end{array}$

Rewrite the given above equations.

  $\begin{array}{c}s+e+m=\frac{1}{4}\\ s+m=\frac{1}{6}\\ e+m=\frac{1}{7}\end{array}$

Solve the third equation for m .

  $m=\frac{1}{7}-e$

Substitute into the 1st and 2nd equations.

  $\begin{array}{c}s+e+\left(\frac{1}{7}-e\right)=\frac{1}{4}\\ s+\frac{1}{7}=\frac{1}{4}\\ s=\frac{1}{4}-\frac{1}{7}\\ =\frac{3}{28}\end{array}$

And,

  $\begin{array}{c}s+\frac{1}{7}-e=\frac{1}{6}\\ \frac{3}{28}+\frac{1}{7}-e=\frac{1}{6}\\ e=\frac{3}{28}+\frac{1}{7}-\frac{1}{6}\\ =\frac{1}{12}\end{array}$

Now, find the value of m .

  $\begin{array}{c}\frac{1}{12}+m=\frac{1}{7}\\ m=\frac{1}{7}-\frac{1}{12}\\ =\frac{5}{84}\end{array}$

The rates are in rooms per hour, so take the reciprocal to find the number of hours per room.

So you get a system of 3 equations we can solve by back substitution.

  $s=\frac{3}{28}\to \frac{28}{3}$

So, the time is 9 hr 20 min per room.

  $e=\frac{1}{12}\to 12$

So, the time is 12 hr per room.

  $m=\frac{5}{84}\to \frac{84}{5}$

So, the time is 16 hr 48 min per room.

Therefore, the obtained time is 9 hr 20 min per room, 12 hr per room, and 16 hr 48 min per room.