A sum of money at simple interest amounts to Rs 1,012 in `2(1)/(2)` years and to Rs 1,067.20 in 4 years. The rate of interset per annum is :

Options

A. 2.5 %
B. 3%
C. 4%
D. 5%

Answer

To solve the problem step by step, we need to find the rate of interest per annum given the amounts at different times.

### Step 1: Identify the amounts and times given in the problem.
- Amount after 2.5 years = Rs 1,012
- Amount after 4 years = Rs 1,067.20

### Step 2: Calculate the simple interest for the time period from 2.5 years to 4 years.
- Time difference = 4 years - 2.5 years = 1.5 years
- Amount at 4 years = Rs 1,067.20
- Amount at 2.5 years = Rs 1,012

Using the formula for simple interest (SI):
\[ \text{SI} = \text{Amount at 4 years} - \text{Amount at 2.5 years} \]
\[ \text{SI} = 1,067.20 - 1,012 = 55.20 \]

### Step 3: Calculate the simple interest per year.
- Simple interest for 1.5 years = Rs 55.20
- To find the interest for 1 year, we divide by 1.5:
\[ \text{SI for 1 year} = \frac{55.20}{1.5} = 36.80 \]

### Step 4: Calculate the principal amount using the amount at 2.5 years.
- Amount at 2.5 years = Principal + Simple Interest for 2.5 years
- Simple Interest for 2.5 years = SI for 1 year × 2.5
\[ \text{SI for 2.5 years} = 36.80 \times 2.5 = 92 \]
- Therefore, the Principal (P) can be calculated as:
\[ P = \text{Amount at 2.5 years} - \text{SI for 2.5 years} \]
\[ P = 1,012 - 92 = 920 \]

### Step 5: Use the formula for simple interest to find the rate of interest.
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest per annum
- T = Time in years

We know:
- SI = 92 (for 2.5 years)
- P = 920
- T = 2.5 years

Substituting the values into the formula:
\[ 92 = \frac{920 \times R \times 2.5}{100} \]

### Step 6: Solve for R.
Rearranging the equation gives:
\[ R = \frac{92 \times 100}{920 \times 2.5} \]
Calculating the right-hand side:
\[ R = \frac{9200}{2300} = 4 \]

### Conclusion
The rate of interest per annum is **4%**.