Three water pumps, P, Q, and R, operate at constant individual rates. The ratio of the rat… — GRE General Test

Three water pumps, $P$ , $Q$ , and $R$ , operate at constant individual rates. The ratio of the rate of pump $P$ to the rate of pump $Q$ is $2 : 3$ . When all three pumps operate simultaneously, their combined rate is $3$ times the rate of pump $P$ alone. If pump $R$ working alone can drain a full reservoir in $24$ hours, how many hours would pump $Q$ working alone take to drain the same full reservoir?

$8$ hoursAnswer
B
$12$ hours
C
$16$ hours
D
$32$ hours
E
$56$ hours

Answer

8 hours

The correct answer is 8 hours. By setting the rate of pump Q as

\frac{3}{2}

times the rate of pump P, the combined rate of all three pumps is

r_P + \frac{3}{2} r_P + r_R = \frac{5}{2} r_P + r_R

. Setting this equal to

3 r_P

shows that pump R's rate is

\frac{1}{2} r_P

. Since pump R takes 24 hours (

r_R = \frac{1}{24}

), pump P's rate is

\frac{1}{12}

(taking 12 hours), and pump Q's rate is

\frac{3}{2} \times \frac{1}{12} = \frac{1}{8}

(taking 8 hours).

Step-by-Step Solution

Express the rates of pumps P and Q in terms of a common variable.

Let

r_P

r_Q

, and

r_R

be the rates of pumps

P

Q

, and

R

in reservoirs per hour. Given

r_P : r_Q = 2 : 3

, we have

r_Q = \frac{3}{2} r_P

Relating pump rates using the given ratio simplifies the system of equations to one variable.

Set up the combined rate equation and solve for

r_R

in terms of

r_P

r_P + r_Q + r_R = 3 r_P \implies r_P + \frac{3}{2} r_P + r_R = 3 r_P \implies \frac{5}{2} r_P + r_R = 3 r_P \implies r_R = \frac{1}{2} r_P

The total rate is the sum of individual rates, allowing us to express pump R's rate in terms of pump P's rate.

Calculate

r_P

and

r_Q

using the given rate for pump R.

Since pump

R

takes

24

hours alone,

r_R = \frac{1}{24}

. Therefore,

\frac{1}{2} r_P = \frac{1}{24} \implies r_P = \frac{1}{12}

. Then

r_Q = \frac{3}{2} \times \frac{1}{12} = \frac{1}{8}

Knowing pump R's explicit numerical rate allows finding the numerical rates for pumps P and Q.

Determine the time required for pump Q alone to drain the reservoir.

\text{Time for } Q = \frac{1}{r_Q} = \frac{1}{1/8} = 8 \text{ hours}.

The time required to complete one full job is the reciprocal of the rate.

Key Concept

Combined Work Rates and Ratio Relationships

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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