The amount obtained by investing a certain sum at `r%` p.a.for 3 years at simple interest is equal to the simple interest on the same sum at the same rate for 13 years. The value of r is

Options

A. 8
B. 10
C. 16
D. 12

Answer

To solve the problem, we need to set up the equations based on the information provided.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the amount obtained by investing a certain sum (let's denote it as \( P \)) at \( r\% \) per annum for 3 years is equal to the simple interest earned on the same sum at the same rate for 13 years.

2. **Formulating the Amount for 3 Years**:
The formula for the amount \( A \) after \( t \) years at simple interest is given by:
\[ A = P + \text{SI} \]
Where \( \text{SI} = \frac{P \cdot r \cdot t}{100} \).
For 3 years, we have:
\[ A = P + \frac{P \cdot r \cdot 3}{100} \]
Simplifying this, we get:
\[ A = P \left(1 + \frac{3r}{100}\right) \]

3. **Formulating the Simple Interest for 13 Years**:
The simple interest for 13 years is:
\[ \text{SI} = \frac{P \cdot r \cdot 13}{100} \]

4. **Setting the Equations Equal**:
According to the problem, the amount obtained after 3 years is equal to the simple interest earned after 13 years:
\[ P \left(1 + \frac{3r}{100}\right) = \frac{P \cdot r \cdot 13}{100} \]

5. **Eliminating \( P \)**:
Since \( P \) is common on both sides and is not zero, we can divide both sides by \( P \):
\[ 1 + \frac{3r}{100} = \frac{r \cdot 13}{100} \]

6. **Rearranging the Equation**:
Now, we will rearrange the equation to isolate \( r \):
\[ 1 = \frac{r \cdot 13}{100} - \frac{3r}{100} \]
\[ 1 = \frac{10r}{100} \]
\[ 1 = \frac{r}{10} \]

7. **Solving for \( r \)**:
Multiply both sides by 10:
\[ r = 10 \]

### Final Answer:
The value of \( r \) is \( 10\% \).