Arun invested a sum of money at a certain rate of simple interest for a period of four years. Had he invested the same sum for a period of six years. the total interest earned by him would have been fifty per cent more than the earlier interest amount. What was the rate of interest per cent per annum?

Options

A. 4
B. 8
C. 5
D. Cannot be determined

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI), which is given by:

\[ \text{SI} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount (the initial sum of money)
- \( R \) = Rate of interest per annum (in percentage)
- \( T \) = Time (in years)

### Step 1: Define the Simple Interest for both time periods

1. **For 4 years:**
\[ \text{SI}_1 = \frac{P \times R \times 4}{100} \]

2. **For 6 years:**
\[ \text{SI}_2 = \frac{P \times R \times 6}{100} \]

### Step 2: Establish the relationship between the two interests

According to the problem, the interest earned over 6 years is 50% more than the interest earned over 4 years. Therefore, we can express this relationship as:

\[ \text{SI}_2 = \text{SI}_1 + 0.5 \times \text{SI}_1 \]

This simplifies to:

\[ \text{SI}_2 = 1.5 \times \text{SI}_1 \]

### Step 3: Substitute the expressions for SI

Substituting the expressions for \(\text{SI}_1\) and \(\text{SI}_2\) into the equation gives us:

\[ \frac{P \times R \times 6}{100} = 1.5 \times \left(\frac{P \times R \times 4}{100}\right) \]

### Step 4: Simplify the equation

We can cancel \( \frac{P}{100} \) from both sides (assuming \( P \neq 0 \)):

\[ 6R = 1.5 \times 4R \]

This simplifies to:

\[ 6R = 6R \]

### Step 5: Analyze the result

The equation \( 6R = 6R \) is always true, which means that the rate of interest \( R \) cannot be determined from the information given. It implies that the relationship holds for any value of \( R \).

### Conclusion

Thus, we conclude that the rate of interest cannot be determined from the information provided.