An electric bettle has two coils. When one coil is switched on it takes 15 minutes and the other takes 30 minutes to boil certain mass of water. The ratio of times taken by them, when connected in series and in parallel to boil the same mass of water is :

Options

A. `1:3`
B. `5:2`
C. `7:3`
D. `9:2`

Answer

To solve the problem, we need to determine the time taken by two coils of an electric kettle to boil the same mass of water when connected in series and in parallel. We will then find the ratio of these times.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Time taken by Coil 1 (T1) = 15 minutes
- Time taken by Coil 2 (T2) = 30 minutes

2. **Relate Resistance to Time:**
- The power (P) of each coil can be expressed as:
\[ P = \frac{V^2}{R} \]
- The heat provided to boil the water is the same for both coils, so we can write:
\[ \frac{V^2}{R_1} \cdot T_1 = \frac{V^2}{R_2} \cdot T_2 \]
- Canceling \(V^2\) from both sides gives:
\[ \frac{T_2}{T_1} = \frac{R_2}{R_1} \]
- Substituting the values of \(T_1\) and \(T_2\):
\[ \frac{30}{15} = \frac{R_2}{R_1} \implies R_2 = 2R_1 \]

3. **Calculate Time for Series Connection:**
- When connected in series, the total resistance \(R_s\) is:
\[ R_s = R_1 + R_2 = R_1 + 2R_1 = 3R_1 \]
- The time taken (T1') in series is given by:
\[ \frac{V^2}{R_s} \cdot T_1' = \frac{V^2}{R_1} \cdot T_1 \]
- Substituting \(R_s\):
\[ \frac{V^2}{3R_1} \cdot T_1' = \frac{V^2}{R_1} \cdot 15 \]
- Canceling \(V^2\) and \(R_1\) gives:
\[ \frac{T_1'}{3} = 15 \implies T_1' = 45 \text{ minutes} \]

4. **Calculate Time for Parallel Connection:**
- The equivalent resistance \(R_p\) when connected in parallel is:
\[ R_p = \frac{R_1 \cdot R_2}{R_1 + R_2} = \frac{R_1 \cdot 2R_1}{R_1 + 2R_1} = \frac{2R_1^2}{3R_1} = \frac{2R_1}{3} \]
- The time taken (T2') in parallel is given by:
\[ \frac{V^2}{R_p} \cdot T_2' = \frac{V^2}{R_1} \cdot T_1 \]
- Substituting \(R_p\):
\[ \frac{V^2}{\frac{2R_1}{3}} \cdot T_2' = \frac{V^2}{R_1} \cdot 15 \]
- Canceling \(V^2\) and \(R_1\) gives:
\[ \frac{3T_2'}{2} = 15 \implies T_2' = 10 \text{ minutes} \]

5. **Calculate the Ratio of Times:**
- The ratio of times taken in series to parallel is:
\[ \text{Ratio} = \frac{T_1'}{T_2'} = \frac{45}{10} = \frac{9}{2} \]

### Final Answer:
The ratio of times taken by the coils when connected in series and in parallel is \( \frac{9}{2} \).