Understanding Compound Interest Calculation

This question asks us to calculate the total amount payable after a certain period when interest is compounded half-yearly. We are given the principal amount, the annual interest rate, and the time duration.

Key Information Provided:

• Principal Amount (P) = Rs. 10000
• Annual Interest Rate (R) = 40% per annum
• Time Period (T) = 1.5 years
• Compounding Frequency = Half-yearly

Compound Interest Formula

The formula for calculating the amount (A) when interest is compounded is:

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (as a decimal)
• n = the number of times that interest is compounded per year
• t = the number of years the money is invested or borrowed for

Adapting the Formula for Half-Yearly Compounding

Since the interest is compounded half-yearly, we need to adjust the rate and the number of periods:

• The rate per compounding period (let's call it r') is half of the annual rate: $$ r' = \frac{R}{2} = \frac{40\%}{2} = 20\% $$ In decimal form, this is $0.20$.
• The total number of compounding periods (let's call it N) is twice the number of years: $$ N = T \times 2 = 1.5 \text{ years} \times 2 = 3 $$ So, there are 3 compounding periods.

Now we can use a simplified version of the compound interest formula for these adjusted values:

Step-by-Step Calculation

1. Substitute the values:

   P = 10000

   r' = 0.20

   N = 3

   $$ A = 10000 (1 + 0.20)^3 $$
2. Calculate the term inside the parenthesis: $$ A = 10000 (1.20)^3 $$
3. Calculate the exponent:

   $(1.20)^3 = 1.20 \times 1.20 \times 1.20 = 1.44 \times 1.20 = 1.728$

4. Calculate the final amount: $$ A = 10000 \times 1.728 $$ $$ A = 17280 $$

Conclusion

The total amount to be paid after 1.5 years, with the interest compounded half-yearly at an annual rate of 40%, is Rs. 17280.