A sum of Rs 16,000 invested at compound interest, amounts to Rs 21,160 in two years. What is the rate per cent per annum?

Options

A. 16.5
B. 17.5
C. 15
D. 10

Answer

To find the rate of compound interest per annum, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{100}\right)^t \]

Where:
- \(A\) is the amount after time \(t\),
- \(P\) is the principal amount (initial investment),
- \(r\) is the rate of interest per annum,
- \(t\) is the time in years.

### Step-by-Step Solution:

1. **Identify the given values**:
- Principal \(P = 16,000\)
- Amount \(A = 21,160\)
- Time \(t = 2\) years

2. **Substitute the values into the compound interest formula**:
\[ 21,160 = 16,000 \left(1 + \frac{r}{100}\right)^2 \]

3. **Divide both sides by 16,000** to isolate the compound interest term:
\[ \frac{21,160}{16,000} = \left(1 + \frac{r}{100}\right)^2 \]

4. **Calculate the left side**:
\[ \frac{21,160}{16,000} = 1.3225 \]
So, we have:
\[ 1.3225 = \left(1 + \frac{r}{100}\right)^2 \]

5. **Take the square root of both sides** to solve for \(1 + \frac{r}{100}\):
\[ \sqrt{1.3225} = 1 + \frac{r}{100} \]
\[ 1.15 = 1 + \frac{r}{100} \]

6. **Subtract 1 from both sides**:
\[ 1.15 - 1 = \frac{r}{100} \]
\[ 0.15 = \frac{r}{100} \]

7. **Multiply both sides by 100** to find \(r\):
\[ r = 0.15 \times 100 = 15 \]

8. **Conclusion**:
The rate of interest per annum is \(15\%\).