To solve the problem, we need to find the rate of interest per annum given the amounts at two different times.
1. **Identify the given values**:
- Amount after 2.5 years (A1) = Rs. 1,012
- Amount after 4 years (A2) = Rs. 1,067.20
- Time for A1 (T1) = 2.5 years
- Time for A2 (T2) = 4 years
2. **Calculate the difference in amounts**:
- The difference in amounts (A2 - A1) gives us the interest earned between 2.5 years and 4 years.
\[
\text{Interest} = A2 - A1 = 1,067.20 - 1,012 = Rs. 55.20
\]
3. **Calculate the time difference**:
- The time difference between the two amounts is:
\[
T2 - T1 = 4 - 2.5 = 1.5 \text{ years}
\]
4. **Calculate the rate of interest**:
- The interest earned in 1.5 years is Rs. 55.20. To find the annual interest, we can use the formula for simple interest:
\[
\text{Simple Interest} = \frac{P \times R \times T}{100}
\]
Rearranging the formula to find R (rate of interest):
\[
R = \frac{\text{Interest} \times 100}{P \times T}
\]
However, we need to find the principal (P) first.
5. **Find the principal using the first amount**:
- Using the first amount (A1) after 2.5 years:
\[
A1 = P + \text{Interest for 2.5 years}
\]
Let the interest for 2.5 years be \( I_1 \):
\[
I_1 = \frac{P \times R \times 2.5}{100}
\]
Thus,
\[
1,012 = P + \frac{P \times R \times 2.5}{100}
\]
6. **Using the second amount**:
- Similarly, for the second amount (A2):
\[
A2 = P + \text{Interest for 4 years}
\]
Let the interest for 4 years be \( I_2 \):
\[
I_2 = \frac{P \times R \times 4}{100}
\]
Thus,
\[
1,067.20 = P + \frac{P \times R \times 4}{100}
\]
7. **Set up equations**:
- Now we have two equations:
\[
1. \quad 1,012 = P + \frac{P \times R \times 2.5}{100} \quad (1)
\]
\[
2. \quad 1,067.20 = P + \frac{P \times R \times 4}{100} \quad (2)
\]
8. **Subtract equation (1) from equation (2)**:
\[
(1,067.20 - 1,012) = \left(\frac{P \times R \times 4}{100} - \frac{P \times R \times 2.5}{100}\right)
\]
\[
55.20 = \frac{P \times R \times (4 - 2.5)}{100}
\]
\[
55.20 = \frac{P \times R \times 1.5}{100}
\]
\[
P \times R = \frac{55.20 \times 100}{1.5} = 3680
\]
9. **Substituting back to find R**:
- Substitute \( P \) in terms of \( R \) into either equation to find the rate. For simplicity, we can assume a value for \( P \) or solve for \( R \) directly.
- We can use the first equation to express \( P \):
\[
P = 1,012 - \frac{P \times R \times 2.5}{100}
\]
Substitute \( P \times R = 3680 \):
\[
P = 1,012 - \frac{3680 \times 2.5}{100}
\]
\[
P = 1,012 - 92
\]
\[
P = 920
\]
10. **Find the rate**:
- Now substitute \( P \) back into \( P \times R = 3680 \):
\[
920 \times R = 3680
\]
\[
R = \frac{3680}{920} = 4
\]
Thus, the rate of interest per annum is \( 4\% \).
### Final Answer:
The rate of interest per annum is **4%**.
