In a public bathroom there are n taps 1, 2, 3... n. Tap 1 and tap 2 take equal time to fill the tank while tap 3 takes half the time taken by tap 2 and tap 4 takes half the time taken by tap 3. Similarly each next number of tap takes half the time taken by previous number of tap i.e., `K^(th)` tap takes half the time taken by `(K-1)^(th)` tap.
If the 10th tap takes 2 hours to fill the tank alone then what is the ratio of efficiency of 8th tap and 12th tap, respectively?

Options

A. a. `4:1`
B. b. `5:3`
C. c. `16:1`
D. d. `1:16`

Answer

To solve the problem, we need to determine the time taken by the 8th and 12th taps to fill the tank, and then find the ratio of their efficiencies.

### Step-by-step Solution:

1. **Understand the Time Taken by Each Tap:**
- We know that the 10th tap takes 2 hours to fill the tank.
- Each tap takes half the time of the previous tap. Therefore, we can calculate the time taken by the taps leading up to the 10th tap.

2. **Calculate the Time for the 9th Tap:**
- Since the 10th tap takes 2 hours, the 9th tap takes half of that:
\[ \text{Time taken by 9th tap} = \frac{2}{2} = 1 \text{ hour} \]

3. **Calculate the Time for the 8th Tap:**
- The 8th tap takes half the time of the 9th tap:
\[ \text{Time taken by 8th tap} = \frac{1}{2} = 0.5 \text{ hours} = 8 \text{ hours} \]

4. **Calculate the Time for the 11th Tap:**
- The 11th tap takes half the time of the 10th tap:
\[ \text{Time taken by 11th tap} = \frac{2}{2} = 1 \text{ hour} \]

5. **Calculate the Time for the 12th Tap:**
- The 12th tap takes half the time of the 11th tap:
\[ \text{Time taken by 12th tap} = \frac{1}{2} = 0.5 \text{ hours} \]

6. **Calculate the Efficiency of Each Tap:**
- Efficiency is inversely proportional to the time taken. Therefore, we can express the efficiencies as:
- Efficiency of the 8th tap:
\[ \text{Efficiency of 8th tap} = \frac{1}{\text{Time taken by 8th tap}} = \frac{1}{8} \]
- Efficiency of the 12th tap:
\[ \text{Efficiency of 12th tap} = \frac{1}{\text{Time taken by 12th tap}} = \frac{1}{0.5} = 2 \]

7. **Calculate the Ratio of Efficiencies:**
- Now, we find the ratio of the efficiency of the 8th tap to the efficiency of the 12th tap:
\[ \text{Ratio of efficiency (8th:12th)} = \frac{\frac{1}{8}}{2} = \frac{1}{16} \]

### Final Answer:
The ratio of the efficiency of the 8th tap to the 12th tap is \( \frac{1}{16} \).