A sum of money amounts to Rs.4818 after 3yr and Rs.7227 after 6 yr on compound interest. The sum is

Options

A. Rs.3122
B. Rs.3212
C. Rs.2409
D. Rs.2490

Answer

To solve the problem step by step, we will use the concept of compound interest. The amounts after certain years are given, and we need to find the principal sum.

### Step 1: Understand the given information
We have two amounts:
- Amount after 3 years (A1) = Rs. 4818
- Amount after 6 years (A2) = Rs. 7227

### Step 2: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial sum)
- \( r \) = Rate of interest per annum
- \( t \) = Time in years

### Step 3: Set up the equations
From the information given, we can set up two equations:
1. For the amount after 3 years:
\[ 4818 = P \left(1 + \frac{r}{100}\right)^3 \] (Equation 1)

2. For the amount after 6 years:
\[ 7227 = P \left(1 + \frac{r}{100}\right)^6 \] (Equation 2)

### Step 4: Divide the two equations
To eliminate \( P \), we can divide Equation 2 by Equation 1:
\[ \frac{7227}{4818} = \frac{P \left(1 + \frac{r}{100}\right)^6}{P \left(1 + \frac{r}{100}\right)^3} \]
This simplifies to:
\[ \frac{7227}{4818} = \left(1 + \frac{r}{100}\right)^{6-3} \]
\[ \frac{7227}{4818} = \left(1 + \frac{r}{100}\right)^3 \]

### Step 5: Calculate the left side
Now we calculate \( \frac{7227}{4818} \):
\[ \frac{7227}{4818} = 1.5 \]
So we have:
\[ 1.5 = \left(1 + \frac{r}{100}\right)^3 \]

### Step 6: Take the cube root
To find \( 1 + \frac{r}{100} \), we take 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,
\[ \frac{r}{100} \approx 0.1447 \implies r \approx 14.47\% \]

### Step 7: Substitute back to find the principal
Now substitute \( 1 + \frac{r}{100} \) back into Equation 1 to find \( P \):
\[ 4818 = P \times (1.1447)^3 \]
Calculating \( (1.1447)^3 \):
\[ (1.1447)^3 \approx 1.5 \]
So,
\[ 4818 = P \times 1.5 \]
Now, solving for \( P \):
\[ P = \frac{4818}{1.5} \approx 3212 \]

### Final Answer
The principal sum is Rs. 3212.

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