A certain sum amounts to Rs. 14641 in 4 years @ 10% p.a compounded annually. What is the value of principal ?

Options

A. Rs 6000
B. Rs 12000
C. Rs. 10000
D. data insufficient

Answer

To find the principal amount (P) that amounts to Rs. 14,641 in 4 years at a rate of 10% per annum compounded annually, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{100}\right)^n \]

Where:
- \( A \) = Amount after time \( n \) (in this case, Rs. 14,641)
- \( P \) = Principal amount (the value we need to find)
- \( r \) = Rate of interest (10% in this case)
- \( n \) = Number of years (4 years in this case)

### Step 1: Substitute the known values into the formula

We know:
- \( A = 14641 \)
- \( r = 10 \)
- \( n = 4 \)

Substituting these values into the formula gives:

\[ 14641 = P \left(1 + \frac{10}{100}\right)^4 \]

### Step 2: Simplify the equation

Calculating \( 1 + \frac{10}{100} \):

\[ 1 + \frac{10}{100} = 1 + 0.1 = 1.1 \]

Now substitute this back into the equation:

\[ 14641 = P (1.1)^4 \]

### Step 3: Calculate \( (1.1)^4 \)

Now we need to calculate \( (1.1)^4 \):

\[ (1.1)^4 = 1.1 \times 1.1 \times 1.1 \times 1.1 \]

Calculating step-by-step:

1. \( 1.1 \times 1.1 = 1.21 \)
2. \( 1.21 \times 1.1 = 1.331 \)
3. \( 1.331 \times 1.1 = 1.4641 \)

So, \( (1.1)^4 = 1.4641 \).

### Step 4: Substitute back into the equation

Now we substitute \( (1.1)^4 \) back into our equation:

\[ 14641 = P \times 1.4641 \]

### Step 5: Solve for \( P \)

To find \( P \), divide both sides by \( 1.4641 \):

\[ P = \frac{14641}{1.4641} \]

Calculating this gives:

\[ P \approx 10000 \]

### Conclusion

The principal amount is Rs. 10,000.