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Pipes & Cistern Questions and Answers

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101.

A pump can fill a tank with water in 2 hours. Because of a leak, it took \( 2\tfrac{1}{3} \) hours to fill the tank. The leak can drain all the water of the tank in:

Answer: D

Work in rates: a pipe that fills a tank in \( n \) units of time fills \( \tfrac{1}{n} \) of it per unit, and rates add (an emptying pipe counting negative).

Without the leak the pump fills the tank at \( \tfrac{1}{2} \) per hour; with it, the effective rate is \( \tfrac{3}{7} \) per hour.

\[ \frac{1}{2} - \frac{1}{L} = \frac{1}{7/3} = \frac{3}{7} \]

\[ \frac{1}{L} = \frac{1}{2} - \frac{3}{7} = \frac{7 - 6}{14} = \frac{1}{14} \]

\[ L = 14 \text{ hours} \]

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102.

Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the tank will be filled in:

Answer: C

Work in rates: a pipe that fills a tank in \( n \) units of time fills \( \tfrac{1}{n} \) of it per unit, and rates add (an emptying pipe counting negative).

\[ \frac{1}{5} + \frac{1}{6} - \frac{1}{12} \]

Over the common denominator 60:

\[ \frac{12 + 10 - 5}{60} = \frac{17}{60} \text{ per hour} \]

\[ t = \frac{60}{17} = 3\tfrac{9}{17} \text{ hours} \]

The emptying pipe subtracts, so the tank still fills - but more slowly than the \( 2\tfrac{8}{11} \) hours A and B would manage alone.

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103.

Two pipes A and B can fill a cistern in 37 1/2 minutes and 45 minutes respectively. Both pipes are opened. The cistern will be filled in just half an hour, if the B is turned off after:

Answer: B

Let B be turned off after x minutes. Then,

Part filled by (A + B) in x min. + Part filled by A in (30 -x) min. = 1.

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104.

A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is:

Answer: C

Suppose, first pipe alone takes x hours to fill the tank .

Then, second and third pipes will take (x -5) and (x - 9) hours respectively to fill the tank.

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(2x - 5)(x - 9) = x(x - 5)

x2 - 18x + 45 = 0

(x - 15)(x - 3) = 0

x = 15. [neglecting x = 3]

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105.

Pipes A and B can fill a tank in 5 hours and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the tank will be filled in.

Answer: B

Work in rates: a pipe that fills a tank in \( n \) units of time fills \( \tfrac{1}{n} \) of it per unit, and rates add (an emptying pipe counting negative).

\[ \frac{1}{5} + \frac{1}{6} - \frac{1}{12} \]

Over the common denominator 60:

\[ \frac{12 + 10 - 5}{60} = \frac{17}{60} \text{ per hour} \]

\[ t = \frac{60}{17} = 3\tfrac{9}{17} \text{ hours} \]

The emptying pipe subtracts, so the tank still fills - but more slowly than the \( 2\tfrac{8}{11} \) hours A and B would manage alone.

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106.

Pipe A can fill a tank in 6 hours and pipe B can empty it in 8 hours. If both are opened together, how long will the tank take to fill ?

Answer: C

Correct answer: 24 hours

Treat the emptying pipe as a negative rate.

\[\frac{1}{6} - \frac{1}{8} = \frac{4 - 3}{24} = \frac{1}{24}\]

The tank fills at one twenty-fourth per hour, so it takes 24 hours. Had B been the faster pipe the net rate would be negative and the tank would never fill.

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107.

Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:

Answer: C

 

Work done by the waste pipe in 1 minute = 1 - 1 + 1
15 20 24

 

    = 1 - 11
15 120

 

    = -  1 .    [-ve sign means emptying]
40

 

 Volume of 1 part = 3 gallons.
40

Volume of whole = (3 x 40) gallons = 120 gallons.

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108.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Answer: C

Suppose pipe A alone takes x hours to fill the tank.

 

Then, pipes B and C will take x and x hours respectively to fill the tank.
2 4

 

1 + 2 + 4 = 1
x x x 5

 

7 = 1
x 5

 x = 35 hrs.

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109.

Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

Answer: C

Let the cistern be filled by pipe A alone in x hours.

Then, pipe B will fill it in (x + 6) hours.

 

1 + 1 = 1
x (x + 6) 4

 

x + 6 + x = 1
x(x + 6) 4

 x2 - 2x - 24 = 0

 (x -6)(x + 4) = 0

 x = 6.     [neglecting the negative value of x]

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110.

Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

Answer: A

 

Part filled by A in 1 min = 1 .
20

 

Part filled by B in 1 min = 1 .
30

 

Part filled by (A + B) in 1 min = 1 + 1 = 1 .
20 30 12

 Both pipes can fill the tank in 12 minutes.

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