At what rate of simple interest a certain sum will be doubled in 14 years

Options

A. (a) `7 1/7` %
B. (b) `7 1/9` %
C. (c) `8 1/5` %
D. (d) `9 1/3` %

Answer

To solve the problem of determining the rate of simple interest at which a certain sum will be doubled in 14 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the rate of simple interest (R) that will double a principal amount (P) in 14 years (T).

2. **Define the Variables**:
- Let the principal amount be \( P \).
- The final amount after 14 years will be \( F = 2P \).
- The time period is \( T = 14 \) years.

3. **Use the Formula for Simple Interest**:
The formula for the final amount \( F \) in terms of principal \( P \), rate \( R \), and time \( T \) is given by:
\[ F = P + \text{SI} \]
where SI (Simple Interest) is calculated as:
\[ \text{SI} = \frac{P \cdot R \cdot T}{100} \]

4. **Set Up the Equation**:
Since \( F = 2P \), we can write:
\[ 2P = P + \frac{P \cdot R \cdot T}{100} \]

5. **Simplify the Equation**:
Subtract \( P \) from both sides:
\[ 2P - P = \frac{P \cdot R \cdot T}{100} \]
This simplifies to:
\[ P = \frac{P \cdot R \cdot T}{100} \]

6. **Cancel \( P \)**:
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ 1 = \frac{R \cdot T}{100} \]

7. **Substitute the Time**:
Substitute \( T = 14 \):
\[ 1 = \frac{R \cdot 14}{100} \]

8. **Solve for R**:
Rearranging gives:
\[ R = \frac{100}{14} \]
Simplifying this:
\[ R = 7.14\% \]

### Final Answer:
The rate of simple interest at which a certain sum will be doubled in 14 years is approximately **7.14%**.