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# Word Problem

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.

Step 1, understanding the problem:

We know that there are 2 pumps. The first pump drains the pool in 10 minutes. The 2 pumps together drain the pool in 8 minutes. We need to figure out how fast the second pump would drain the pool by itself.

Step 2, the plan: let x = the time it takes pump 2 to complete the job by itself. The completed job =1

Step 3, carry out the plan:

8/10 + 8/x = 1

Multiply by 10

8t + 8(10) = 10t

80 = 10t-8t

80 = 2t

40 = t

Step 4, the solution:

Pump 2 takes 40 minutes to complete the project alone.

Step 5, Alternate methods:

Step 2, the plan: We know that pump one can drain 1/10 of the pool a minute. The two pumps together can drain 1/8 of the pool in a minute. Pump 2 can drain 1/x of the pool in a minute.

Step 3, carry out the plan:

8/10 + 8/x = 1

1/x = 1/8 – 1/10

1/x =1/40

X = 40

Step 4, the solution:

It takes pump 2 40 minutes to complete the job by itself