Principal Amount Calculation with Varying Compound Interest

This question requires finding the initial principal amount invested over 3 years, given the final amount and different compound interest rates for each year.

Compound Interest Formula Application

For compound interest with varying rates per period, the formula is:

$ A = P \times \left(1 + \frac{R_1}{100}\right) \times \left(1 + \frac{R_2}{100}\right) \times \left(1 + \frac{R_3}{100}\right) $ Where:

• $A$ = Final Amount = ₹20,000
• $P$ = Principal Amount (to be calculated)
• $R_1$ = Rate for Year 1 = 6%
• $R_2$ = Rate for Year 2 = 12%
• $R_3$ = Rate for Year 3 = 18%

Calculation Steps

1. Substitute Values:

   Plug the given values into the formula: $ 20,000 = P \times \left(1 + \frac{6}{100}\right) \times \left(1 + \frac{12}{100}\right) \times \left(1 + \frac{18}{100}\right) $

2. Calculate Growth Factors:

   Calculate the value of each year's growth factor: $ \left(1 + 0.06\right) = 1.06 $ $ \left(1 + 0.12\right) = 1.12 $ $ \left(1 + 0.18\right) = 1.18 $

3. Combine Factors and Solve for P:

   Multiply the growth factors: $ 1.06 \times 1.12 \times 1.18 = 1.391096 $ The equation simplifies to: $ 20,000 = P \times 1.391096 $ Now, solve for P: $ P = \frac{20,000}{1.391096} $

4. Determine Principal Amount:

   Calculating P directly gives approximately ₹14,377.15. To confirm the correct option, we verify backwards:

   Let Principal = ₹14,276.58 Amount after Year 1 = $ 14,276.58 \times 1.06 = 15,133.17 $ (approx) Amount after Year 2 = $ 15,133.17 \times 1.12 = 16,949.15 $ (approx) Amount after Year 3 = $ 16,949.15 \times 1.18 = 19,999.99 \approx 20,000 $

   This calculation confirms the principal amount.

Principal Amount Found

The principal amount invested is ₹14,276.58.