A certain sum of money amounts to Rs. 2420 in 2 years and Rs. 2662 in 3 years at some rate of compound interest, compounded annually. The rate of interest per annum is:

Answer

To find the rate of compound interest per annum based on the given amounts for 2 years and 3 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Amount after 2 years (A1) = Rs. 2420
- Amount after 3 years (A2) = Rs. 2662
- Time for A1 (T1) = 2 years
- Time for A2 (T2) = 3 years

2. **Set Up the Compound Interest Formula:**
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where:
- \(A\) = Amount
- \(P\) = Principal
- \(R\) = Rate of interest
- \(T\) = Time in years

3. **Write the Equations for Both Amounts:**
- For the first amount (after 2 years):
\[ A1 = P \left(1 + \frac{R}{100}\right)^{2} \]
Substituting the values:
\[ 2420 = P \left(1 + \frac{R}{100}\right)^{2} \quad \text{(Equation 1)} \]

- For the second amount (after 3 years):
\[ A2 = P \left(1 + \frac{R}{100}\right)^{3} \]
Substituting the values:
\[ 2662 = P \left(1 + \frac{R}{100}\right)^{3} \quad \text{(Equation 2)} \]

4. **Divide Equation 2 by Equation 1:**
\[ \frac{A2}{A1} = \frac{P \left(1 + \frac{R}{100}\right)^{3}}{P \left(1 + \frac{R}{100}\right)^{2}} \]
This simplifies to:
\[ \frac{2662}{2420} = \left(1 + \frac{R}{100}\right)^{3 - 2} \]
Therefore:
\[ \frac{2662}{2420} = 1 + \frac{R}{100} \]

5. **Calculate the Left Side:**
\[ \frac{2662 - 2420}{2420} = \frac{242}{2420} \]
Thus:
\[ 1 + \frac{R}{100} = \frac{2662}{2420} \]

6. **Isolate R:**
\[ \frac{R}{100} = \frac{2662 - 2420}{2420} \]
Simplifying:
\[ \frac{R}{100} = \frac{242}{2420} \]

7. **Calculate R:**
\[ R = \frac{242 \times 100}{2420} \]
Simplifying gives:
\[ R = 10 \]

### Final Answer:
The rate of interest per annum is **10%**.

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