Pumps X, Y, and Z, operating simultaneously at their respective constant rates, can fill a… — GMAT

Pumps X, Y, and Z, operating simultaneously at their respective constant rates, can fill an empty reservoir in $4$ hours. Operating alone at its constant rate, Pump X can fill $\frac{1}{5}$ of the reservoir in $2$ hours. The rate at which Pump Y fills the reservoir is $50\%$ greater than the rate at which Pump Z fills the reservoir. If Pump X and Pump Z work together for $3$ hours, and then Pump Y is turned on so that all three pumps work together until the reservoir is full, how many additional hours will it take to finish filling the remainder of the reservoir?

A
$1 \frac{18}{25}$ hours
B
$1 \frac{23}{25}$ hours
$2 \frac{2}{25}$ hoursAnswer
D
$2 \frac{1}{2}$ hours
E
$2 \frac{4}{5}$ hours

Answer

2 \frac{2}{25}

hours

The rate of Pump X is

\frac{1}{10}

reservoir/hour, and the combined rate of all three pumps is

\frac{1}{4}

reservoir/hour. Thus, the combined rate of Pumps Y and Z is

\frac{1}{4} - \frac{1}{10} = \frac{3}{20}

reservoir/hour. Given

r_y = 1.5 r_z

, setting

\frac{5}{2} r_z = \frac{3}{20}

yields

r_z = \frac{3}{50}

and

r_y = \frac{9}{100}

. The combined rate of Pumps X and Z is

\frac{1}{10} + \frac{3}{50} = \frac{4}{25}

. In

3

hours, Pumps X and Z complete

3 \times \frac{4}{25} = \frac{12}{25}

of the reservoir, leaving

\frac{13}{25}

un-filled. When all three pumps operate, their combined rate is

\frac{1}{4}

. The additional time required is

\frac{13/25}{1/4} = \frac{52}{25} = 2 \frac{2}{25}

hours.

Step-by-Step Solution

Determine the individual rate of Pump X and the total combined rate of all three pumps.

Pump X's rate

r_x = \frac{1/5}{2} = \frac{1}{10}

of the reservoir per hour. The total combined rate of Pumps X, Y, and Z is

r_x + r_y + r_z = \frac{1}{4}

of the reservoir per hour.

Work rate is equal to work completed divided by time elapsed.

Determine the individual work rates of Pump Y and Pump Z.

The combined rate of Y and Z is

r_y + r_z = \frac{1}{4} - \frac{1}{10} = \frac{3}{20}

. Since

r_y = 1.5 r_z = \frac{3}{2} r_z

, we have

\frac{5}{2} r_z = \frac{3}{20}

, which gives

r_z = \frac{3}{50}

and

r_y = \frac{9}{100}

Use the given relationship between the rates of Pump Y and Pump Z to solve for their individual values.

Calculate the work completed during the first 3 hours by Pumps X and Z working together, and find the remaining work.

The combined rate of X and Z is

r_x + r_z = \frac{1}{10} + \frac{3}{50} = \frac{8}{50} = \frac{4}{25}

. In 3 hours, they complete

3 \times \frac{4}{25} = \frac{12}{25}

of the reservoir. The remaining work is

1 - \frac{12}{25} = \frac{13}{25}

Work done equals combined rate multiplied by time worked.

Calculate the additional time required for all three pumps working together to finish the remaining work.

Time

t = \frac{\text{Remaining Work}}{\text{Combined Rate of X, Y, Z}} = \frac{13/25}{1/4} = \frac{52}{25} = 2 \frac{2}{25}

hours.

Time needed is remaining work divided by the total combined rate of all active pumps.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 30s

Question

Answer

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