A sum amounts to `Rs. 2420` in 2 years when invested at compound interest. If principal is `Rs. 2000`, then what is the rate of interest (in %)?

Options

A. A) `15`
B. B) `10`
C. C) `8`
D. D) `12`

Answer

To find the rate of interest when a sum amounts to Rs. 2420 in 2 years with a principal of Rs. 2000, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 2000
- Amount (A) = Rs. 2420
- Time (T) = 2 years

### Step 2: Calculate the ratio of Amount to Principal
We can find the ratio of the amount to the principal:
\[ \text{Ratio} = \frac{A}{P} = \frac{2420}{2000} \]

### Step 3: Simplify the ratio
Simplifying the ratio:
\[ \frac{2420}{2000} = \frac{242}{200} = \frac{121}{100} \]

### Step 4: Find the ratio for 1 year
Since the amount is compounded annually, we need to find the equivalent ratio for 1 year. We take the square root of the ratio we found:
\[ \text{Ratio for 1 year} = \sqrt{\frac{121}{100}} = \frac{\sqrt{121}}{\sqrt{100}} = \frac{11}{10} \]

### Step 5: Calculate the interest earned in 1 year
From the ratio, we can see that Rs. 10 becomes Rs. 11 in 1 year. This means the interest earned in 1 year is:
\[ \text{Interest} = 11 - 10 = 1 \text{ rupee} \]

### Step 6: Calculate the rate of interest
Now, we can calculate the rate of interest using the formula:
\[ \text{Rate of Interest} = \left(\frac{\text{Interest}}{\text{Principal}}\right) \times 100 \]
Substituting the values:
\[ \text{Rate of Interest} = \left(\frac{1}{10}\right) \times 100 = 10\% \]

### Conclusion
The rate of interest is **10%**.

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