A sum becomes 1.69 times of itself in 2 years when invested at compound interest. What is the per annum rate of interest (in percentage)?

Options

A. 38
B. `34.5`
C. `13`
D. 30

Answer

To find the per annum rate of interest when a sum becomes 1.69 times itself in 2 years under compound interest, we can use the formula for compound interest.

### Step-by-Step Solution:

1. **Understanding the Compound Interest Formula**:
The formula for the amount \( A \) after \( n \) years with principal \( P \), rate of interest \( r \) (in decimal), and time \( n \) is given by:
\[ A = P(1 + r)^n \]

2. **Setting Up the Equation**:
In this case, the amount \( A \) is 1.69 times the principal \( P \), and the time \( n \) is 2 years. Therefore, we can write:
\[ 1.69P = P(1 + r)^2 \]

3. **Dividing Both Sides by \( P \)**:
Since \( P \) is not zero, we can divide both sides by \( P \):
\[ 1.69 = (1 + r)^2 \]

4. **Taking the Square Root**:
To solve for \( 1 + r \), we take the square root of both sides:
\[ 1 + r = \sqrt{1.69} \]

5. **Calculating the Square Root**:
We know that \( \sqrt{1.69} = 1.3 \), so:
\[ 1 + r = 1.3 \]

6. **Finding the Rate \( r \)**:
Now, we can find \( r \) by subtracting 1 from both sides:
\[ r = 1.3 - 1 = 0.3 \]

7. **Converting to Percentage**:
To convert \( r \) to a percentage, we multiply by 100:
\[ \text{Rate of interest} = 0.3 \times 100 = 30\% \]

### Final Answer:
The per annum rate of interest is **30%**.