A certain sum at compound interest amounts to ? 12960 in 2 yr and ? 13176 in 3 yr. Find the rate per cent per annum.

Options

A. `1( 1)/( 3) %`
B. `2 ( 1)/( 3) %`
C. `1(2)/( 3)%`
D. `2 ( 2)/( 3)%`

Answer

To solve the problem, we need to find the rate of compound interest per annum given that a certain sum amounts to ₹12960 in 2 years and ₹13176 in 3 years.

### Step-by-Step Solution:

1. **Identify the amounts and time periods:**
- Amount after 2 years (A1) = ₹12960
- Amount after 3 years (A2) = ₹13176

2. **Understand the relationship between the amounts:**
- The amount after 3 years can be expressed in terms of the amount after 2 years:
\[ A2 = A1 \times (1 + \frac{r}{100}) \]
- Here, \( r \) is the rate of interest per annum.

3. **Set up the equation using the amounts:**
- Substitute the known amounts into the equation:
\[ 13176 = 12960 \times (1 + \frac{r}{100}) \]

4. **Solve for \( r \):**
- Divide both sides by 12960:
\[ \frac{13176}{12960} = 1 + \frac{r}{100} \]
- Calculate the left side:
\[ \frac{13176}{12960} = 1.016 \]
- Therefore:
\[ 1.016 = 1 + \frac{r}{100} \]

5. **Isolate \( r \):**
- Subtract 1 from both sides:
\[ 0.016 = \frac{r}{100} \]
- Multiply both sides by 100 to find \( r \):
\[ r = 0.016 \times 100 = 1.6 \]

6. **Conclusion:**
- The rate of interest per annum is \( r = 1.6\% \).

### Final Answer:
The rate of interest per annum is **1.6%**.

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