A sum of money becomes Rs 35,680 after 3 years and Rs 53,520 after 6 years at a certain rate percentage p.a., interest compounded yearly. What is the compound interest on the same sum in the first case ? (Your answer should be nearest to an integer)

Options

A. Rs 11,983
B. Rs 11,893
C. Rs 10,842
D. Rs 11,938

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided in the question.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that a sum of money becomes Rs. 35,680 after 3 years and Rs. 53,520 after 6 years with compound interest. We need to find the compound interest for the first 3 years.

2. **Setting Up the Equations**:
We can use the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where \(A\) is the amount, \(P\) is the principal, \(R\) is the rate of interest, and \(t\) is the time in years.

For the first case (after 3 years):
\[ 35680 = P \left(1 + \frac{R}{100}\right)^3 \quad \text{(Equation 1)} \]

For the second case (after 6 years):
\[ 53520 = P \left(1 + \frac{R}{100}\right)^6 \quad \text{(Equation 2)} \]

3. **Expressing Equation 2 in terms of Equation 1**:
We can express the amount after 6 years in terms of the amount after 3 years:
\[ 53520 = 35680 \left(1 + \frac{R}{100}\right)^3 \]
This gives us:
\[ \frac{53520}{35680} = \left(1 + \frac{R}{100}\right)^3 \]

4. **Calculating the Ratio**:
Now, we calculate the left side:
\[ \frac{53520}{35680} = 1.5 \]
Therefore, we have:
\[ 1.5 = \left(1 + \frac{R}{100}\right)^3 \]

5. **Taking the Cube Root**:
To find \(1 + \frac{R}{100}\), we take the cube root of both sides:
\[ 1 + \frac{R}{100} = 1.5^{1/3} \]

6. **Calculating \(1.5^{1/3}\)**:
Using a calculator or estimation, we find:
\[ 1.5^{1/3} \approx 1.1447 \]
Thus:
\[ \frac{R}{100} \approx 0.1447 \quad \Rightarrow \quad R \approx 14.47 \]

7. **Finding the Principal (P)**:
Now, we can substitute \(R\) back into Equation 1 to find \(P\):
\[ 35680 = P \left(1 + \frac{14.47}{100}\right)^3 \]
\[ 35680 = P \cdot (1.1447)^3 \]
Calculating \((1.1447)^3 \approx 1.5\):
\[ 35680 = P \cdot 1.5 \]
Therefore:
\[ P = \frac{35680}{1.5} \approx 23786.67 \]

8. **Calculating the Compound Interest**:
The compound interest for the first 3 years is given by:
\[ \text{Compound Interest} = A - P \]
\[ \text{Compound Interest} = 35680 - 23786.67 \approx 11893.33 \]

9. **Rounding to the Nearest Integer**:
Finally, rounding \(11893.33\) gives us:
\[ \text{Compound Interest} \approx 11893 \]

### Final Answer:
The compound interest on the same sum in the first case is approximately **Rs. 11,893**.