The first step is to define the daily rates of each pump
From the information given,
First pump can empty the pool in 7 days. This means that
Daily rate of first pump = 1/7
The first pump's rate is 1/7 per day
Second pump can empty the pool in 14 days. This means that
Daily rate of second pump = 1/14
The second pump's rate is 1/14 per day
Let t be the number of days it will take both pumps, working together to empty the pool. Thus,
combined daily rate of both pumps = 1/t
The rates are additive. It means that
1/7 + 1/14 = 1/t
Simplifying the left side, we have
3/14 = 1/t
The combined pumps rate is 3/14 per day
By taking reciprocal of both sides,
t = 14/3 = 4.67
It will take the two pumps 4.67 days to empty the pool together