A sum of money becomes Rs. 6690 after three years and Rs. 10035 after six years on compound interest. The sum is-

Options

A. Rs.4400
B. Rs.4445
C. Rs.4460
D. Rs.4520

Answer

To find the initial sum of money (the principal) that becomes Rs. 6690 after 3 years and Rs. 10035 after 6 years on compound interest, we can follow these steps:

### Step 1: Understand the relationship between the amounts
We know that the amount after 3 years (A1) is Rs. 6690 and the amount after 6 years (A2) is Rs. 10035. The difference in time between these two amounts is 3 years.

### Step 2: Set up the formula for compound interest
The formula for compound interest is given by:
\[ A = P(1 + r)^n \]
Where:
- \( A \) is the amount after \( n \) years,
- \( P \) is the principal amount (initial sum),
- \( r \) is the rate of interest per year,
- \( n \) is the number of years.

### Step 3: Write equations for both amounts
From the information provided:
1. After 3 years:
\[ 6690 = P(1 + r)^3 \] (Equation 1)

2. After 6 years:
\[ 10035 = P(1 + r)^6 \] (Equation 2)

### Step 4: Divide the two equations
To eliminate \( P \), we can divide Equation 2 by Equation 1:
\[ \frac{10035}{6690} = \frac{P(1 + r)^6}{P(1 + r)^3} \]
This simplifies to:
\[ \frac{10035}{6690} = (1 + r)^3 \]

### Step 5: Calculate the left side
Calculating \( \frac{10035}{6690} \):
\[ \frac{10035}{6690} = 1.5 \]
So we have:
\[ (1 + r)^3 = 1.5 \]

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

### Step 7: Substitute back to find \( P \)
Now we can substitute \( r \) back into either Equation 1 or Equation 2 to find \( P \). Using Equation 1:
\[ 6690 = P(1.1447)^3 \]
Calculating \( (1.1447)^3 \):
\[ (1.1447)^3 \approx 1.5 \]
Thus,
\[ 6690 = P \times 1.5 \]
Solving for \( P \):
\[ P = \frac{6690}{1.5} = 4460 \]

### Conclusion
The initial sum of money (the principal) is Rs. 4460.