Questions: One pump can empty a pool in 7 days, whereas a second pump can empty the pool in 10 days. How long will it take the two pumps, working together, to empty the pool? (Fractional answers are OK.) The first pump's rate is per day. The second pump's rate is per day. The combined pumps rate is per day. It will take the two pumps days to empty the pool together.

Solution Steps

To solve this problem, we need to determine the rate at which each pump can empty the pool and then combine these rates to find the total rate at which both pumps working together can empty the pool. Finally, we will calculate the time it takes for the combined rate to empty the pool.

1. Calculate the rate of the first pump (pool per day).
2. Calculate the rate of the second pump (pool per day).
3. Add the two rates to get the combined rate.
4. Calculate the time it takes for the combined rate to empty the pool.

The first pump can empty the pool in 7 days. Therefore, its rate is: \[ \text{Rate of Pump 1} = \frac{1}{7} \approx 0.1429 \text{ pools per day} \]

The second pump can empty the pool in 10 days. Therefore, its rate is: \[ \text{Rate of Pump 2} = \frac{1}{10} = 0.1 \text{ pools per day} \]

The combined rate of both pumps working together is the sum of their individual rates: \[ \text{Combined Rate} = \text{Rate of Pump 1} + \text{Rate of Pump 2} = 0.1429 + 0.1 = 0.2429 \text{ pools per day} \]

The time it takes for both pumps to empty the pool together is the reciprocal of the combined rate: \[ \text{Time to Empty} = \frac{1}{\text{Combined Rate}} = \frac{1}{0.2429} \approx 4.1176 \text{ days} \]

Final Answer