Question

one pump can empty a pool in 4 days, whereas a second pump can empty the pool in 14 days. how long will it take the two pumps, working together, to empty the pool? (fractional answers are ok.)
the first pumps rate is \boxed{} of a pool per day.
the second pumps rate is \boxed{} of a pool per day.
the combined pump rate is \boxed{} of a pool per day.
it will take the two pumps \boxed{} days to empty the pool together.

Explanation:

Step1: Find first pump's rate

The first pump empties 1 pool in 4 days. Rate = work / time, so rate is $1/4$ pool per day.

Step2: Find second pump's rate

The second pump empties 1 pool in 14 days. So rate is $1/14$ pool per day.

Step3: Find combined rate

Add the two rates: $\frac{1}{4} + \frac{1}{14}$. Find a common denominator (28): $\frac{7}{28} + \frac{2}{28} = \frac{9}{28}$ pool per day.

Step4: Find time to empty together

Time = work / rate. Work is 1 pool, rate is $\frac{9}{28}$. So time is $1 \div \frac{9}{28} = \frac{28}{9}$ days.

Answer:

The first pump's rate is $\frac{1}{4}$ of a pool per day.
The second pump's rate is $\frac{1}{14}$ of a pool per day.
The combined pump rate is $\frac{9}{28}$ of a pool per day.
It will take the two pumps $\frac{28}{9}$ days to empty the pool together.