Question

working together, two pumps can drain a certain pool in 4 hours. if it takes the older pump 9 hours to drain the pool by itself, how long will it take the newer pump to drain the pool on its own? do not do any rounding.

Explanation:

Step1: Define rates

Let $t$ = time (hours) for newer pump alone.
Rate of older pump: $\frac{1}{9}$ pool/hour
Rate of newer pump: $\frac{1}{t}$ pool/hour
Combined rate: $\frac{1}{4}$ pool/hour

Step2: Set up combined rate equation

Sum of individual rates = combined rate
$$\frac{1}{9} + \frac{1}{t} = \frac{1}{4}$$

Step3: Solve for $t$

Subtract $\frac{1}{9}$ from both sides:
$$\frac{1}{t} = \frac{1}{4} - \frac{1}{9}$$
Find common denominator (36):
$$\frac{1}{t} = \frac{9}{36} - \frac{4}{36} = \frac{5}{36}$$
Take reciprocal of both sides:
$$t = \frac{36}{5}$$

Answer:

$\frac{36}{5}$ hours (or 7.2 hours, but as a fraction, $\frac{36}{5}$)