Two taps P and Q can fill a tank in 24 hours and 18 hours respec tively. If the two taps are opened at 11 a.m., then at what time (in p.m.) should the tap P be closed to completely fill the tank at exactly 2 a.m.?

Options

A. 5
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will determine how long each tap fills the tank and when tap P should be closed to ensure the tank is filled exactly by 2 a.m.

### Step-by-Step Solution:

1. **Determine the rates of the taps:**
- Tap P can fill the tank in 24 hours. Therefore, the rate of tap P is:
\[ \text{Rate of P} = \frac{1}{24} \text{ tank/hour} \]
- Tap Q can fill the tank in 18 hours. Therefore, the rate of tap Q is:
\[ \text{Rate of Q} = \frac{1}{18} \text{ tank/hour} \]

2. **Calculate the combined rate of both taps:**
- When both taps are open, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} = \frac{1}{24} + \frac{1}{18} \]
- To add these fractions, find a common denominator. The least common multiple of 24 and 18 is 72:
\[ \frac{1}{24} = \frac{3}{72}, \quad \frac{1}{18} = \frac{4}{72} \]
- Thus,
\[ \text{Combined Rate} = \frac{3}{72} + \frac{4}{72} = \frac{7}{72} \text{ tank/hour} \]

3. **Determine the total time from 11 a.m. to 2 a.m.:**
- From 11 a.m. to 2 a.m. is a total of 15 hours.

4. **Calculate the total amount of tank filled by both taps in 15 hours:**
- The total amount filled in 15 hours is:
\[ \text{Total filled} = \text{Combined Rate} \times \text{Time} = \frac{7}{72} \times 15 \]
- Calculating this gives:
\[ \text{Total filled} = \frac{7 \times 15}{72} = \frac{105}{72} = \frac{35}{24} \text{ tanks} \]

5. **Determine how much of the tank is filled by tap P alone:**
- Let \( x \) be the number of hours tap P is open. Then tap Q is open for 15 hours.
- The equation for the total tank filled is:
\[ \frac{x}{24} + \frac{15}{18} = 1 \]
- Simplifying the second term:
\[ \frac{15}{18} = \frac{5}{6} \]
- Thus, the equation becomes:
\[ \frac{x}{24} + \frac{5}{6} = 1 \]
- Rearranging gives:
\[ \frac{x}{24} = 1 - \frac{5}{6} = \frac{1}{6} \]
- Multiplying both sides by 24:
\[ x = 4 \text{ hours} \]

6. **Determine when tap P should be closed:**
- Since tap P is opened at 11 a.m. and should be closed after 4 hours, it should be closed at:
\[ 11 \text{ a.m.} + 4 \text{ hours} = 3 \text{ p.m.} \]

### Final Answer:
Tap P should be closed at **3 p.m.** to ensure the tank is completely filled by 2 a.m.