Match List-I with List-II
| List-I | List-II | ||
| (A) | If an inlet pipe can fill a tank in \( x \) hours, but due to a leakage in the bottom it is filled in \( y \) hours, then time (in hours) taken to empty the tank due to leakage | (I) | \( \displaystyle\frac{yz+zx-xy}{xyz} \) |
| (B) | If two inlet pipes take \( x \) & \( y \) hours respectively to fill the tank and the third outlet pipe takes \( z \) hours to empty the tank. If all three of them are opened together, then the net part of the tank filled in 1 hour | (II) | \( \displaystyle\frac{1}{x} \) |
| (C) | If a pipe can fill the tank completely in \( x \) hours, then the part of the tank filled in 1 hour | (III) | \( \left(\displaystyle\frac{xy}{y-x}\right) \) |
| (D) | If two inlet pipes can fill a tank in \( x \) & \( y \) hours. If both pipes are opened at the same time, then net part of the tank filled in 1 hour | (IV) | \( \displaystyle\frac{x+y}{xy} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (I), (C) - (II), (D) - (IV)
The key quantity is the part of the tank filled in 1 hour: a pipe that fills the tank in x hours fills \( \frac{1}{x} \) of it per hour; an outlet or a leak counts as negative.
(A) If an inlet pipe can fill a tank in \( x \) hours, but due to a leakage in the bottom it is filled in \( y \) hours, then time (in hours) taken to empty the tank due to leakage
This matches (III) \( \left(\frac{xy}{y-x}\right) \). If the leak alone empties the tank in t hours:
\( \displaystyle \frac{1}{x}-\frac{1}{t}=\frac{1}{y} \)
\( \displaystyle \frac{1}{t}=\frac{1}{x}-\frac{1}{y} \)
\( \displaystyle =\frac{y-x}{xy} \)
\( \displaystyle t=\frac{xy}{y-x} \)
(B) If two inlet pipes take \( x \) & \( y \) hours respectively to fill the tank and the third outlet pipe takes \( z \) hours to empty the tank. If all three of them are opened together, then the net part of the tank filled in 1 hour
This matches (I) \( \frac{yz+zx-xy}{xyz} \):
\( \displaystyle \frac{1}{x}+\frac{1}{y}-\frac{1}{z} \)
\( \displaystyle =\frac{yz+zx-xy}{xyz} \)
(C) If a pipe can fill the tank completely in \( x \) hours, then the part of the tank filled in 1 hour
This matches (II) \( \frac{1}{x} \), the rate of the pipe.
(D) If two inlet pipes can fill a tank in \( x \) & \( y \) hours. If both pipes are opened at the same time, then net part of the tank filled in 1 hour
This matches (IV) \( \frac{x+y}{xy} \):
\( \displaystyle \frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy} \)
So the matching is (A)-(III), (B)-(I), (C)-(II), (D)-(IV).