A sum of money invested at simple interest becomes 13/10 of itself in 2 years and 6 months. What is the rate (in percentage) of interest per annum?

Options

A. 10
B. 15
C. 12
D. 18

Answer

To solve the problem step by step, we will use the formula for simple interest.

### Step 1: Understand the given information
We know that the amount becomes \( \frac{13}{10} \) of the principal after 2 years and 6 months. Let's denote the principal amount as \( P \).

### Step 2: Convert time into years
The time given is 2 years and 6 months. We can convert this into years:
\[ \text{Time} = 2 \text{ years} + 6 \text{ months} = 2 + \frac{6}{12} = 2.5 \text{ years} \]

### Step 3: Write the formula for the amount
The formula for the amount \( A \) in simple interest is:
\[ A = P + \left( P \times \frac{R \times T}{100} \right) \]
Where:
- \( A \) is the amount
- \( P \) is the principal
- \( R \) is the rate of interest per annum
- \( T \) is the time in years

### Step 4: Substitute the known values into the formula
From the problem, we know:
\[ A = \frac{13}{10} P \]
Substituting into the formula gives:
\[ \frac{13}{10} P = P + \left( P \times \frac{R \times 2.5}{100} \right) \]

### Step 5: Simplify the equation
First, we can subtract \( P \) from both sides:
\[ \frac{13}{10} P - P = \left( P \times \frac{R \times 2.5}{100} \right) \]
This simplifies to:
\[ \frac{3}{10} P = \left( P \times \frac{R \times 2.5}{100} \right) \]

### Step 6: Cancel \( P \) from both sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{3}{10} = \frac{R \times 2.5}{100} \]

### Step 7: Solve for \( R \)
Now, we can cross-multiply to solve for \( R \):
\[ 3 \times 100 = 10 \times R \times 2.5 \]
This simplifies to:
\[ 300 = 25R \]
Now, divide both sides by 25:
\[ R = \frac{300}{25} = 12 \]

### Step 8: Conclusion
Thus, the rate of interest per annum is:
\[ \boxed{12\%} \]