Chapter 11.5, Problem 3P

To determine

How long it will take for both hoses to fill the pool together.

Answer to Problem 3P

  $4.8\text{ hours}$

Explanation of Solution

Given:

One hose can fill a pool in 12 h. Another hose can fill the same pool in 8 h.

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
1. Multiply both sides of equations by LCD(least common denominator) of all denominators.
2. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

In order tofind the time it will take for both hoses to fill the pool together, observe that the first hose can fill the pool in 12 h, so in 1 hour it can fill $\frac{1}{12}$ th part of the pool. And the second hose can fill the pool in 8 hours, so in 1 hour the second hose can fill $\frac{1}{8}$ th part of the pool.

Now if both the hoses are open, so in 1 hour they together can fill

  $\begin{array}{c}\frac{1}{12}+\frac{1}{8}=\frac{1}{12}\cdot \frac{2}{2}+\frac{1}{8}\cdot \frac{3}{3}\\ =\frac{2}{24}+\frac{3}{24}\\ =\frac{2+3}{24}\\ =\frac{5}{24}\end{array}$

So, they together will fill the pool in $\frac{24}{5}=4.8$ hours.

Thus, it will take $\frac{24}{5}=4.8$ hours for both hoses to fill the pool together.