One pump can drain a pool in 10 minutes. When the other pump is also used, the pool only takes 8 minutes to drain. How long would it take the second pump to drain the pool if it were the only pump in use?

This is a rate of work problem, where the rate of work for each pump is as follows:

rate of work of pump 1 = drains 1 pool per 10 minutes = 1 pool/8 minutes = 1/10

rate of work of pump 2 = drains 1 pool per t minutes = 1 pool/t minutes = 1/t

The rate of work of the 2 pumps combined is given by the following:

rate of work of pump 1 and 2 = drain 1 pool per 8 minutes = 1 pool/8 minutes = 1/8

Therefore, combining the rate of work of each pump yields the rate of work of both pumps combined:

rate of work Pump 1 + rate of work Pump 2 = rate of work of Pump 1 and Pump 2

1/10 + 1/t = 1/8

Solving this equation for t gives you the time it takes pump 2 to drain the pool by itself:

1/10 + 1/t = 1/8

To combine the fractions on the left hand side of the equation, the fractions must share a common denominator. Since the first fraction has a denominator of 10 and the second fraction a denominator of t, then the common denominator among both fractions is the product of 10 and t (i.e., 10t). Multiplying the numerator and denominator of the 1st fraction by t and multiplying the numerator and denominator of the 2nd fraction by 10, we arrive at the following:

1(t)/10(t) + 1(10)/t(10) = 1/8

t/10t + 10/10t = 1/8

Now that the fractions on the left hand side of the equation have a common denominator, we can combine them by adding their numerators and keeping the common denominator:

t/10t + 10/10t = 1/8

(t + 10)/10t = 1/8

Cross-multiplying, we get:

10t · 1 = 8 · (t + 10)

10t = 8·t + 8·10

10t = 8t + 80

We want all terms with the variable t on one side of the equation, so we subtract 8t from both sides of the equation and combine the like terms:

10t **- 8t** = 8t **- 8t** + 80

2t = 80

Divide both sides of the equation by 2 to solve for t:

2t **/ 2** = 80 **
/ 2**

**t = 40**

Thus, it takes the pump 2 40 minutes to drain the pool by itself.