Chapter 1.6, Problem 64E

Sharing a Job Next-door neighbors Bob and Jim use hoses from both houses to fill Bob’s swimming pool. They know that it takes 18 h using both hoses. They also know that Bob’s hose, used alone, takes 20% less time than Jim’s hose alone. How much time is required to fill the pool by each hose alone?

To determine

To find: The time taken by Bob hose and Jim hose to fill the pool.

Answer to Problem 64E

The time taken by Bob hose is 32.4 hours and time taken by Jim hose is 40.5 hours.

Explanation of Solution

Given:

They both take 18 hours to fill the pool.

Bob’s hose takes 20% less time than Jim’s hose alone.

Calculation:

Let the time taken by Jim’s hose be x.

Time taken by Bob’s hose,

$\begin{array}{c}\text{Time}=x-20\%\text{of}x\\ =x-0.20x\\ =0.80x\end{array}$

Tabulate the given information into the language of algebra.

In words	In Algebra
Time taken by Jim’s	x hr
Time taken by Bob’s	0.80x hr
Work done by Jim’s hose	$\frac{1}{x}$
Work done by Bob’s hose	$\frac{1}{0.80x}$
Work done by both hoses	$\frac{1}{18}$

Model the equation for the above information.

$\begin{array}{c}\left(\begin{array}{l}\text{Fraction}\text{done}\text{by}\\ \text{Bob's}\text{hose}\end{array}\right)+\left(\begin{array}{l}\text{Fraction}\text{done}\text{by}\\ \text{Jim's}\text{hose}\end{array}\right)=\text{Fraction}\text{done}\text{by}\text{both}\\ \frac{1}{x}+\frac{1}{0.80x}=\frac{1}{18}\end{array}$

Simplify the above equation for x,

$\begin{array}{c}0.80+1=\frac{0.80x}{18}\left(\text{Multiply}\text{by}\text{LCD}0.80x\right)\\ 1.80=\frac{0.80x}{18}\\ 0.80x=32.4\\ x=40.5\end{array}$

The taken by Jim’s hose is 4.05 hours.

Time taken by Bob’s hose,

$\begin{array}{c}0.80x=0.80\times 4.05\\ =32.4\end{array}$

The time taken by Bob’s hose is 32.4 hours.

Thus, the time taken by Bob hose is 32.4 hours and time taken by Jim hose is 40.5 hours.