A certain sum of money at compound interest grows up to Rs 12960 in 2 years and up to Rs. 13176 in 3 years. Find the rate of interest per annum.

Answer

To find the rate of interest per annum, we can use the information given about the amounts after 2 years and 3 years.

### Step 1: Understand the given amounts
- Amount after 2 years (A2) = Rs 12,960
- Amount after 3 years (A3) = Rs 13,176

### Step 2: Calculate the amount of interest earned in the third year
The interest earned in the third year can be calculated by subtracting the amount after 2 years from the amount after 3 years.

\[ \text{Interest for the 3rd year} = A3 - A2 = 13,176 - 12,960 = 216 \]

### Step 3: Relate the interest to the principal and rate
The interest earned in the third year is calculated on the amount after 2 years (A2). Therefore, we can express this as:

\[ \text{Interest for the 3rd year} = A2 \times \left(\frac{R}{100}\right) \]

Where R is the rate of interest per annum.

### Step 4: Substitute the values
Substituting the values we have:

\[ 216 = 12,960 \times \left(\frac{R}{100}\right) \]

### Step 5: Solve for R
To find R, we rearrange the equation:

\[ R = \frac{216 \times 100}{12,960} \]

Calculating this gives:

\[ R = \frac{21600}{12,960} \approx 1.6667 \]

### Step 6: Convert to percentage
Now, converting this to a percentage:

\[ R \approx 1.67\% \]

### Conclusion
The rate of interest per annum is approximately **1.67%**.

---