A sum amounts to Rs 8,028 in 3 years and to Rs 12,042 in 6 years at a certain rate percent per annum, when the interest is compounded yearly. The sum is :

Options

A. Rs 5,352
B. Rs 5,235
C. Rs 5,325
D. Rs 5,253

Answer

To find the principal sum (P) that amounts to Rs 8,028 in 3 years and Rs 12,042 in 6 years under compound interest, we can follow these steps:

### Step 1: Understand the relationship between amounts
We know that the amount after 3 years (A1) is Rs 8,028 and after 6 years (A2) is Rs 12,042. The formula for compound interest is:
\[ A = P(1 + r)^n \]
where A is the amount, P is the principal, r is the rate of interest, and n is the number of years.

### Step 2: Set up the equations
From the information given:
1. For 3 years:
\[ A_1 = P(1 + r)^3 = 8028 \]
2. For 6 years:
\[ A_2 = P(1 + r)^6 = 12042 \]

### Step 3: Divide the two equations
To eliminate P, we can divide the second equation by the first:
\[ \frac{A_2}{A_1} = \frac{P(1 + r)^6}{P(1 + r)^3} \]
This simplifies to:
\[ \frac{12042}{8028} = (1 + r)^{6 - 3} \]
\[ \frac{12042}{8028} = (1 + r)^3 \]

### Step 4: Calculate the left side
Now calculate the left side:
\[ \frac{12042}{8028} = 1.5 \]
So we have:
\[ (1 + r)^3 = 1.5 \]

### Step 5: Solve for (1 + r)
Now, take the cube root of both sides:
\[ 1 + r = (1.5)^{1/3} \]
Calculating this gives:
\[ 1 + r \approx 1.1447 \]
Thus:
\[ r \approx 0.1447 \text{ or } 14.47\% \]

### Step 6: Substitute back to find P
Now, we can substitute r back into one of the original equations to find P. Using the first equation:
\[ 8028 = P(1 + 0.1447)^3 \]
Calculating \( (1 + 0.1447)^3 \):
\[ (1.1447)^3 \approx 1.5 \]
Thus:
\[ 8028 = P \cdot 1.5 \]
Now solve for P:
\[ P = \frac{8028}{1.5} \approx 5352 \]

### Final Answer
The principal sum (P) is approximately Rs 5,352.

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