A sum of money amounted to ₹ 720 in 2 years and ₹ 792 in 3 years when interest is compounded annually. The annual rate of interest, (in %) is:

Options

A. 0.05
B. 0.075
C. 0.07
D. 0.1

Answer

To solve the problem of finding the annual rate of interest when a sum of money amounts to ₹720 in 2 years and ₹792 in 3 years with compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Compound Interest Formula:**
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where:
- \(A\) is the amount after time \(t\),
- \(P\) is the principal amount (initial sum),
- \(R\) is the rate of interest per annum,
- \(t\) is the time in years.

2. **Set Up the Equations:**
From the problem, we have two scenarios:
- After 2 years, the amount is ₹720:
\[ 720 = P \left(1 + \frac{R}{100}\right)^2 \quad \text{(Equation 1)} \]
- After 3 years, the amount is ₹792:
\[ 792 = P \left(1 + \frac{R}{100}\right)^3 \quad \text{(Equation 2)} \]

3. **Divide Equation 2 by Equation 1:**
To eliminate \(P\), we divide Equation 2 by Equation 1:
\[ \frac{792}{720} = \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)^2} \]
This simplifies to:
\[ \frac{792}{720} = 1 + \frac{R}{100} \]

4. **Calculate the Left Side:**
Now calculate \(\frac{792}{720}\):
\[ \frac{792}{720} = 1.1 \]
Thus, we have:
\[ 1.1 = 1 + \frac{R}{100} \]

5. **Isolate \(\frac{R}{100}\):**
Subtract 1 from both sides:
\[ \frac{R}{100} = 1.1 - 1 = 0.1 \]

6. **Solve for \(R\):**
Multiply both sides by 100 to find \(R\):
\[ R = 0.1 \times 100 = 10 \]

7. **Conclusion:**
The annual rate of interest is:
\[ \boxed{10\%} \]