Rs. 282464 is divided between Ram and Shyam so that the amount that Ram receives in 3 years is the same as that Shyam receives in 6 years. If the interest is compounded annually at the rate of 4% per annum then shyam's share is

Options

A. 125000
B. 1352000
C. 152000
D. 108200

Answer

To solve the problem step by step, we need to find Shyam's share of the total amount of Rs. 282,464 given that the amount Ram receives in 3 years is equal to the amount Shyam receives in 6 years, with an interest rate of 4% compounded annually.

### Step 1: Define the Variables
Let:
- \( P_1 \) = Principal amount received by Ram
- \( P_2 \) = Principal amount received by Shyam

According to the problem, we have:
\[ P_1 + P_2 = 282464 \]

### Step 2: Write the Amount Formulas
The amount received by Ram after 3 years is given by the formula for compound interest:
\[ A_1 = P_1 \left(1 + \frac{R}{100}\right)^t \]
Where:
- \( R = 4\% \)
- \( t = 3 \)

Thus,
\[ A_1 = P_1 \left(1 + \frac{4}{100}\right)^3 = P_1 \left(1.04\right)^3 \]

The amount received by Shyam after 6 years is:
\[ A_2 = P_2 \left(1 + \frac{R}{100}\right)^t \]
Where:
- \( R = 4\% \)
- \( t = 6 \)

Thus,
\[ A_2 = P_2 \left(1 + \frac{4}{100}\right)^6 = P_2 \left(1.04\right)^6 \]

### Step 3: Set the Amounts Equal
Since \( A_1 = A_2 \):
\[ P_1 \left(1.04\right)^3 = P_2 \left(1.04\right)^6 \]

### Step 4: Simplify the Equation
We can simplify the equation:
\[ \frac{P_1}{P_2} = \frac{(1.04)^6}{(1.04)^3} = (1.04)^{6-3} = (1.04)^3 \]

Calculating \( (1.04)^3 \):
\[ (1.04)^3 = 1.124864 \]

So,
\[ \frac{P_1}{P_2} = 1.124864 \]

### Step 5: Express \( P_1 \) in Terms of \( P_2 \)
Let \( P_2 = x \):
\[ P_1 = 1.124864x \]

### Step 6: Substitute Back into the Total
Substituting \( P_1 \) in the total amount equation:
\[ 1.124864x + x = 282464 \]
\[ 2.124864x = 282464 \]

### Step 7: Solve for \( x \)
\[ x = \frac{282464}{2.124864} \]
Calculating \( x \):
\[ x \approx 132812 \]

### Step 8: Find Shyam's Share
Shyam's share \( P_2 \) is approximately Rs. 132812.

### Final Answer
Shyam's share is Rs. 132812.

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