Pumps A and B work independently at their respective constant rates to empty a water stora… — GRE General Test

Pumps $A$ and $B$ work independently at their respective constant rates to empty a water storage reservoir. Working together, Pump $A$ and Pump $B$ can empty the reservoir in $12$ hours. If Pump $A$ 's rate of pumping is $1.5$ times the rate of Pump $B$ , how many hours would it take Pump $B$ working alone at its constant rate to empty the entire reservoir?

Answer: 30 hours

Answer

It would take Pump B working alone 30 hours to empty the entire reservoir.

Since Pump A's rate is

1.5

times Pump B's rate, their combined rate is

1.5 r_B + r_B = 2.5 r_B

reservoirs per hour. Working together for

12

hours empties

12 \times 2.5 r_B = 30 r_B

reservoirs. Setting

30 r_B = 1

reservoir yields

r_B = \frac{1}{30}

reservoir per hour. Therefore, Pump B working alone requires

30

hours to empty the reservoir.

Step-by-Step Solution

Express Pump A's rate in terms of Pump B's rate.

Let

r_B

be the rate of Pump B in reservoirs per hour. Then the rate of Pump A is

r_A = 1.5 r_B

Expressing both rates in terms of a single variable simplifies setting up the combined rate equation.

Calculate the combined pumping rate.

Combined rate =

r_A + r_B = 1.5 r_B + r_B = 2.5 r_B

When two machines work simultaneously, their individual rates add together.

Set up the work equation using the given combined time of 12 hours.

\text{Total Work} = \text{Combined Rate} \times 12 = 2.5 r_B \times 12 = 30 r_B = 1 \text{ reservoir}

The total work performed equals the combined rate multiplied by the total time taken.

Solve for the time required for Pump B to complete the job alone.

\text{Time}_B = \frac{1}{r_B} = 30 \text{ hours}

The time needed for Pump B alone to empty

1

reservoir at rate

r_B

is the reciprocal of

r_B

Key Concept

Combined Work Rates and Rate Ratios

Alternative Method

Choose a convenient hypothetical capacity for the reservoir. Let Pump B's rate be

2

units per hour. Then Pump A's rate is

1.5 \times 2 = 3

units per hour. Working together, they empty

2 + 3 = 5

units per hour. In

12

hours, the reservoir holds

5 \times 12 = 60

units. Pump B alone at

2

units per hour takes

60 / 2 = 30

hours.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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