Two taps can fill a tank in 20 minutes and 30 minutes respectively. There is an outlet tap at exactly half level of that rectangular tank which can pump out 100 litres of water per minute. If the outlet tap is open, then it takes 24 minutes to fill an empty tank. What is the volume of the tank?

Options

A. 1800 litres
B. 1500 litres
C. 1200 litres
D. 2400 litres

Answer

To solve the problem step by step, we will break down the information given and calculate the volume of the tank.

### Step 1: Determine the rates of the filling taps
- The first tap can fill the tank in 20 minutes. Therefore, its rate is:
\[ \text{Rate of Tap 1} = \frac{1}{20} \text{ tank per minute} \]
- The second tap can fill the tank in 30 minutes. Therefore, its rate is:
\[ \text{Rate of Tap 2} = \frac{1}{30} \text{ tank per minute} \]

### Step 2: Calculate the combined rate of both taps
To find the combined rate of both taps, we add their rates:
\[ \text{Combined Rate} = \frac{1}{20} + \frac{1}{30} \]
To add these fractions, we need a common denominator. The least common multiple of 20 and 30 is 60.
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Thus,
\[ \text{Combined Rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \text{ tank per minute} \]

### Step 3: Calculate the time taken to fill the entire tank without the outlet tap
The combined rate of both taps means they can fill the entire tank in:
\[ \text{Time to fill tank} = \frac{1}{\text{Combined Rate}} = 12 \text{ minutes} \]

### Step 4: Determine the effect of the outlet tap
The outlet tap pumps out water at a rate of 100 liters per minute. Since it is located at half the tank level, it will only affect the filling of the second half of the tank.

### Step 5: Calculate the time taken to fill the tank with the outlet tap open
According to the problem, when the outlet tap is open, it takes 24 minutes to fill the tank. This means:
- The first half of the tank is filled in 6 minutes (since it takes 12 minutes to fill the entire tank).
- The remaining half of the tank takes 24 minutes total, so it takes 24 - 6 = 18 minutes to fill the second half.

### Step 6: Calculate the effective rate with the outlet tap open
Let the volume of the tank be \( V \) liters. The rate of filling the second half of the tank with the outlet tap open can be expressed as:
\[ \text{Rate of filling} - \text{Rate of outlet} = \text{Effective Rate} \]
The effective rate for the second half of the tank is:
\[ \frac{V/2}{18} = \frac{V}{36} \text{ tank per minute} \]
The outlet tap's rate is:
\[ \frac{100}{V} \text{ tank per minute} \]
Thus, we have:
\[ \frac{1}{12} - \frac{100}{V} = \frac{1}{36} \]

### Step 7: Solve for the volume \( V \)
Rearranging the equation:
\[ \frac{1}{12} - \frac{1}{36} = \frac{100}{V} \]
Finding a common denominator for the left side:
\[ \frac{3}{36} - \frac{1}{36} = \frac{2}{36} = \frac{1}{18} \]
Thus:
\[ \frac{100}{V} = \frac{1}{18} \]
Cross-multiplying gives:
\[ 100 \cdot 18 = V \implies V = 1800 \text{ liters} \]

### Conclusion
The volume of the tank is **1800 liters**.