Arun invested a sum of money at a certain rate of simple interest for a period of 4 yrs,had he invest for six year the total interst earned by him would have bean 50% 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, let's denote:

- Principal amount = P
- Rate of interest = R% per annum
- Time period for the first investment = 4 years
- Time period for the second investment = 6 years

### Step 1: Calculate the simple interest for 4 years
The formula for simple interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

For the first investment (4 years):
\[ \text{SI}_1 = \frac{P \times R \times 4}{100} \]

### Step 2: Calculate the simple interest for 6 years
For the second investment (6 years):
\[ \text{SI}_2 = \frac{P \times R \times 6}{100} \]

### Step 3: Set up the relationship between the two interests
According to the problem, the interest earned in 6 years is 50% more than the interest earned in 4 years:
\[ \text{SI}_2 = \text{SI}_1 + 0.5 \times \text{SI}_1 \]
This can be simplified to:
\[ \text{SI}_2 = 1.5 \times \text{SI}_1 \]

### Step 4: Substitute the expressions for SI
Substituting the expressions we derived for SI:
\[ \frac{P \times R \times 6}{100} = 1.5 \times \left( \frac{P \times R \times 4}{100} \right) \]

### Step 5: Simplify the equation
We can cancel \( \frac{P \times R}{100} \) from both sides (assuming P and R are not zero):
\[ 6 = 1.5 \times 4 \]
\[ 6 = 6 \]

This equation holds true, which means our setup is correct.

### Step 6: Find the rate of interest
Now, we need to express the relationship in terms of R. From the earlier relationship:
\[ \text{SI}_2 - \text{SI}_1 = 0.5 \times \text{SI}_1 \]
This implies:
\[ \text{SI}_2 = 2 \times \text{SI}_1 \]

Substituting the expressions for SI:
\[ \frac{P \times R \times 6}{100} = 2 \times \left( \frac{P \times R \times 4}{100} \right) \]

### Step 7: Solve for R
Cancelling \( \frac{P \times R}{100} \) from both sides gives:
\[ 6 = 2 \times 4 \]
\[ 6 = 8 \]

This indicates that our assumptions about the relationship were incorrect. We need to go back to the original equation.

### Step 8: Correct the relationship
We can express the interest earned in 6 years in terms of the interest earned in 4 years:
\[ \frac{P \times R \times 6}{100} = \frac{P \times R \times 4}{100} + \frac{1}{2} \left( \frac{P \times R \times 4}{100} \right) \]

This simplifies to:
\[ \frac{P \times R \times 6}{100} = \frac{P \times R \times 4}{100} + \frac{P \times R \times 2}{100} \]
\[ \frac{P \times R \times 6}{100} = \frac{P \times R \times 6}{100} \]

### Final Step: Conclusion
This shows that the rate of interest R can be any value as long as the principal P is constant. However, to find a specific rate, we need additional information about P or R.