A sum amount of Rs. 1067.20 at simple interest in 4 years and in `2 1/2` years it amount to Rs. 1012. What will be the rate of interest per annum?
साधारण ब्याज की किसी दर से कोई धनराशि `2 1//2` वर्ष में रू0 1012 तथा 4 वर्ष में रू0 1,067.20 होती है । ब्याज की वार्षिक दर होगी

Options

A. 0.025
B. 0.03
C. 0.04
D. 0.05

Explanation

To find the rate of interest per annum, we can follow these steps:

### Step 1: Identify the given amounts and time periods
- Amount after 2.5 years (A1) = Rs. 1012
- Amount after 4 years (A2) = Rs. 1067.20
- Time for A1 = 2.5 years
- Time for A2 = 4 years

### Step 2: Calculate the interest earned in 4 years
The interest earned over the 4 years can be calculated as:
\[ \text{Interest (I)} = A2 - P \]
where \( P \) is the principal amount. We will find \( P \) later.

### Step 3: Calculate the interest earned in 2.5 years
Similarly, the interest earned over the 2.5 years is:
\[ \text{Interest (I1)} = A1 - P \]

### Step 4: Determine the difference in interest over the time period
The difference in the amount earned between the two periods is:
\[ \text{Difference in Amount} = A2 - A1 = 1067.20 - 1012 = 55.20 \]
This difference represents the interest earned in the additional time of \( 4 - 2.5 = 1.5 \) years.

### Step 5: Calculate the interest per year
To find the interest earned per year, we divide the difference in interest by the time difference:
\[ \text{Interest per year} = \frac{55.20}{1.5} = 36.80 \]

### Step 6: Calculate the principal amount
Now, we can find the principal amount using the interest earned in 2.5 years:
\[ I1 = A1 - P \implies P = A1 - I1 \]
Substituting \( I1 \) with \( 36.80 \times 2.5 \):
\[ I1 = 36.80 \times 2.5 = 92 \]
So,
\[ P = 1012 - 92 = 920 \]

### Step 7: Calculate the total interest earned in 4 years
Now, we can calculate the total interest earned in 4 years:
\[ I = 36.80 \times 4 = 147.20 \]

### Step 8: Calculate the rate of interest
The rate of interest can be calculated using the formula:
\[ \text{Rate} = \left( \frac{I}{P \times T} \right) \times 100 \]
Where \( T \) is the time period in years. For 4 years:
\[ \text{Rate} = \left( \frac{147.20}{920 \times 4} \right) \times 100 \]
\[ \text{Rate} = \left( \frac{147.20}{3680} \right) \times 100 = 4\% \]

### Final Answer
The rate of interest per annum is **4%** or **0.04**.

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