A sum amounts to Rs 18,600 after 3 years and to Rs 27,900 after 6 years, at a certain rate percent P.a., when the interest is compounded annually. The sum is:

Options

A. Rs 11,800
B. Rs 12,400
C. Rs 14,400
D. Rs 14,600

Answer

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

### Step 1: Understand the Problem
We have two amounts:
- After 3 years, the amount is Rs 18,600.
- After 6 years, the amount is Rs 27,900.

We need to find the principal sum (P) that was invested.

### Step 2: Set Up the Compound Interest Formula
The formula for the amount (A) after time (T) with principal (P) and rate of interest (R) compounded annually is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

### Step 3: Write the Equations
From the information given:
1. For 3 years:
\[ 18600 = P \left(1 + \frac{R}{100}\right)^3 \] (Equation 1)

2. For 6 years:
\[ 27900 = P \left(1 + \frac{R}{100}\right)^6 \] (Equation 2)

### Step 4: Divide the Two Equations
To eliminate P, we can divide Equation 2 by Equation 1:
\[ \frac{27900}{18600} = \frac{P \left(1 + \frac{R}{100}\right)^6}{P \left(1 + \frac{R}{100}\right)^3} \]
This simplifies to:
\[ \frac{27900}{18600} = \left(1 + \frac{R}{100}\right)^{6-3} \]
\[ \frac{27900}{18600} = \left(1 + \frac{R}{100}\right)^3 \]

### Step 5: Calculate the Left Side
Now, calculate the left side:
\[ \frac{27900}{18600} = 1.5 \]
So we have:
\[ 1.5 = \left(1 + \frac{R}{100}\right)^3 \]

### Step 6: Solve for \(1 + \frac{R}{100}\)
Taking the cube root of both sides:
\[ 1 + \frac{R}{100} = \sqrt[3]{1.5} \]
Calculating the cube root:
\[ 1 + \frac{R}{100} \approx 1.1447 \]
Thus,
\[ \frac{R}{100} \approx 0.1447 \]
So,
\[ R \approx 14.47\% \]

### Step 7: Substitute Back to Find P
Now, substitute \(R\) back into either Equation 1 or Equation 2 to find P. We will use Equation 1:
\[ 18600 = P \left(1 + \frac{14.47}{100}\right)^3 \]
Calculating:
\[ 18600 = P \cdot (1.1447)^3 \]
Calculating \( (1.1447)^3 \approx 1.5 \):
\[ 18600 = P \cdot 1.5 \]
So,
\[ P = \frac{18600}{1.5} = 12400 \]

### Final Answer
The principal sum (P) is Rs 12,400.

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