Problem Solving

swerve wrote:Two water pumps, working simultaneously at their respective constant rates, took exactly four hours to fill a certain swimming pool. If the constant rate of one pump was 1.5 times the constant rate of the other, how many hours would it have taken the faster pump to fill the pool if it had worked alone at it's constant rate?

A. 5
B. 16/3
C. 11/2
D. 6
E. 20/3

The OA is E

Source: GMAT Prep

Let's pick some nice numbers that adhere to the given information.

...the constant rate of one pump was 1.5 times the constant rate of the other
So, the fast pump has a pumping rate that's 1.5 faster then the slow pump.
So, let's say the SLOW pump pumps at 2 gallons per hour
This means the FASTER pump pumps at 3 gallons per hour

Note: we don't know the volume of the pool yet.

Two water pumps, working together at their respective constant rates, took exactly 4 hours to fill a certain swimming pool
Their COMBINED RATE = 2 + 3 = 5 gallons per hour
If it took 4 hours for both pumps to fill the pool, then the volume of the pool = (4)(5) = 20 GALLONS

How many hours would it have taken the faster pump to fill the pool if it had worked alone at its constant rate?
The pool holds 20 GALLONS and the FASTER pump pumps at 3 gallons per hour
Time = output/rate
= 20/3
Answer: E

Hi All,

This is an example of a Work Formula question. Any time you have two entities (people, machines, water pumps, etc.) working on a job together, you can use the following formula:

(AB)/(A+B) = Total time to do the job together

Here, we're told that the total time = 4 hours and that one machine's rate is 1.5 times the other machine's rate...

If B = 1.5A then we have...

(A)(1.5A)/(A + 1.5A) = 4

1.5(A^2)/2.5A = 4

(3/2)(A^2) = 10A

A^2 = 20A/3

A = 20/3 hours to fill the pool alone

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

swerve wrote:Two water pumps, working simultaneously at their respective constant rates, took exactly four hours to fill a certain swimming pool. If the constant rate of one pump was 1.5 times the constant rate of the other, how many hours would it have taken the faster pump to fill the pool if it had worked alone at it's constant rate?

A. 5
B. 16/3
C. 11/2
D. 6
E. 20/3

The OA is E

Source: GMAT Prep

We are given that the rate of 1 pump is 1.5 times faster than the rate of the other pump. Since 1 pool is being filled and rate = work/time, the rate of the faster pump is 1/x, in which x = the time it takes for the faster pump to fill the pool, and the rate of the slower pump = 1/(1.5x) = 1/(3x/2) = 2/(3x).

Since, when the pumps work together, they take 4 hours to fill 1 pool, we can create the following equation:

work of faster pump + work of slower pump = 1

(1/x)4 + (2/3x)4 = 1

4/x + 8/(3x) = 1

Multiplying the entire equation by 3x, we have:

12 + 8 = 3x

20 = 3x

20/3 = x

Alternate Solution:

If the rate of the faster pump is 1.5 times that of the slower pump, then, it will take slower pump 1.5 times the amount of time to fill the entire pool compared to the faster pump. Let t denote the time (in hours) it takes for the faster pump to fill the pool alone. Then, it will take 1.5t hours for the slower pump to fill the pool alone.

Notice that in one hour, 1/t of the pool is filled by the faster pump and 1/(1.5t) of the pool is filled by the slower pump. We are also given that together they fill the entire pool in 4 hours, therefore 1/4 of the pool is filled in one hour when both pumps are running. We can create the following equation:

1/t + 1/(1.5t) = 1/4

Let's multiply each side of this equation by 12t:

12 + 8 = 3t

20 = 3t

t = 20/3

Answer: E