At a certain rate per annum , the simple interest on a sum of money for one year is Rs.260 and the compound interest on the same sum for two years is Rs. 540.80 . The rate of interest per annum is

Options

A. 0.04
B. 0.06
C. 0.08
D. 0.1

Answer

To find the rate of interest per annum based on the given simple interest and compound interest, we can follow these steps:

### Step 1: Understand the given information
- Simple Interest (SI) for 1 year = Rs. 260
- Compound Interest (CI) for 2 years = Rs. 540.80

### Step 2: Calculate Simple Interest for 2 years
The formula for Simple Interest for 2 years is:
\[ \text{SI for 2 years} = \text{SI for 1 year} \times 2 \]
Substituting the values:
\[ \text{SI for 2 years} = 260 \times 2 = 520 \, \text{Rs.} \]

### Step 3: Find the difference between Compound Interest and Simple Interest
Now, we need to find the difference between the Compound Interest for 2 years and the Simple Interest for 2 years:
\[ \text{Difference} = \text{CI for 2 years} - \text{SI for 2 years} \]
Substituting the values:
\[ \text{Difference} = 540.80 - 520 = 20.80 \, \text{Rs.} \]

### Step 4: Relate the difference to the rate of interest
The difference between the Compound Interest and Simple Interest for 2 years is due to the interest earned on the interest from the first year. This difference can be expressed as:
\[ \text{Difference} = \text{SI for 1 year} \times \frac{r}{100} \]
Where \( r \) is the rate of interest per annum.

### Step 5: Set up the equation
We can set up the equation:
\[ 20.80 = 260 \times \frac{r}{100} \]

### Step 6: Solve for \( r \)
To find \( r \), we rearrange the equation:
\[ \frac{r}{100} = \frac{20.80}{260} \]
Now, calculate \( \frac{20.80}{260} \):
\[ \frac{20.80}{260} = 0.08 \]
Thus, we have:
\[ \frac{r}{100} = 0.08 \]

Multiplying both sides by 100 gives:
\[ r = 0.08 \times 100 = 8 \]

### Conclusion
The rate of interest per annum is **8%**.

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