At what rate of simple interest a certain sum will be doubled in 12 years ?

Options

A. `8 1/3%`
B. `6%` p.a
C. `6 2/3%` p.a
D. `7 1/2%` p.a

Answer

To find the rate of simple interest at which a certain sum will be doubled in 12 years, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let the principal amount be \( P \).
- The time period is given as \( 12 \) years.
- Let the rate of interest be \( R \) percent per annum.

2. **Determine the Amount:**
- When the principal is doubled, the amount \( A \) becomes \( 2P \).

3. **Use the Simple Interest Formula:**
- The formula for the amount in simple interest is:
\[ A = P + \text{Simple Interest} \]
- The simple interest (SI) can be calculated using the formula:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
- Here, \( T = 12 \) years.

4. **Set Up the Equation:**
- Substitute \( A \) and SI into the equation:
\[ 2P = P + \frac{P \times R \times 12}{100} \]

5. **Simplify the Equation:**
- Rearranging gives:
\[ 2P - P = \frac{P \times R \times 12}{100} \]
- This simplifies to:
\[ P = \frac{P \times R \times 12}{100} \]

6. **Cancel the Principal:**
- Since \( P \) is common on both sides, we can cancel it (assuming \( P \neq 0 \)):
\[ 1 = \frac{R \times 12}{100} \]

7. **Solve for Rate \( R \):**
- Rearranging gives:
\[ R = \frac{100}{12} \]
- Simplifying \( \frac{100}{12} \):
\[ R = 8.33\% \text{ or } 8 \frac{1}{3}\% \]

### Final Answer:
The rate of simple interest at which a certain sum will be doubled in 12 years is \( 8 \frac{1}{3}\% \) per annum.

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