A sum of Rs. 8,400 becomes Rs. 10,164 in 2 years at a certain rate percentage p.a., interest compounded yearly. What will be the simple interest on the same sum for `4 (3)/(5)` years at the same rate?

Options

A. Rs. 3,864
B. Rs. 4,454
C. Rs. 4,032
D. Rs. 4,368

Answer

To solve the problem step by step, we need to find the simple interest on a sum of Rs. 8,400 for a period of \(4 \frac{3}{5}\) years at the same rate of interest that compounded the principal to Rs. 10,164 in 2 years.

### Step 1: Determine the Rate of Interest
We know the formula for compound interest is:
\[ A = P(1 + r)^n \]
Where:
- \(A\) = Amount after time \(n\)
- \(P\) = Principal amount
- \(r\) = Rate of interest per annum (in decimal)
- \(n\) = Time in years

Given:
- \(A = 10,164\)
- \(P = 8,400\)
- \(n = 2\)

Substituting the values into the formula:
\[ 10,164 = 8,400(1 + r)^2 \]

### Step 2: Simplify the Equation
Divide both sides by 8,400 to isolate \((1 + r)^2\):
\[ \frac{10,164}{8,400} = (1 + r)^2 \]
Calculating the left side:
\[ \frac{10,164}{8,400} = 1.21 \]
Thus, we have:
\[ (1 + r)^2 = 1.21 \]

### Step 3: Take the Square Root
Taking the square root of both sides gives:
\[ 1 + r = \sqrt{1.21} \]
Calculating the square root:
\[ 1 + r = 1.1 \]
Therefore:
\[ r = 1.1 - 1 = 0.1 \text{ or } 10\% \]

### Step 4: Calculate Simple Interest
Now we need to calculate the simple interest for \(4 \frac{3}{5}\) years at the same rate. First, convert \(4 \frac{3}{5}\) years into an improper fraction:
\[ 4 \frac{3}{5} = \frac{20 + 3}{5} = \frac{23}{5} \text{ years} \]

Using the simple interest formula:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P = 8,400\)
- \(R = 10\)
- \(T = \frac{23}{5}\)

Substituting the values:
\[ SI = \frac{8,400 \times 10 \times \frac{23}{5}}{100} \]

### Step 5: Simplify the Calculation
Calculating step by step:
1. Calculate \(8,400 \times 10 = 84,000\)
2. Now calculate \(84,000 \times \frac{23}{5} = \frac{84,000 \times 23}{5} = \frac{1,932,000}{5} = 386,400\)
3. Finally, divide by 100:
\[ SI = \frac{386,400}{100} = 3,864 \]

### Final Answer
The simple interest on Rs. 8,400 for \(4 \frac{3}{5}\) years at the same rate is **Rs. 3,864**.