Calculating Compound Interest Compounded Half-Yearly

This problem requires us to calculate the compound interest on a principal amount when the interest is compounded half-yearly. Understanding how the compounding frequency affects the interest rate and time period is crucial.

Understanding the Key Terms

• Principal (P): The initial amount of money deposited or borrowed. Here, P = Rs. 16400.
• Annual Interest Rate (R): The rate at which interest is charged or earned per year. Here, R = 9% per annum.
• Time (T): The duration for which the money is invested or borrowed. Here, T = 1 year.
• Compounding Frequency: How many times the interest is calculated and added to the principal within a year. Here, it is compounded half-yearly, meaning twice a year (n=2).

Adjusting Rate and Time for Half-Yearly Compounding

When interest is compounded half-yearly, we need to adjust the annual rate and the total time period:

• Rate per compounding period (r): The annual rate is divided by the number of compounding periods per year. $$ r = \frac{\text{Annual Rate}}{\text{Number of compounding periods per year}} = \frac{9\%}{2} = 4.5\% \text{ per half-year} $$ In decimal form, $r = 4.5 / 100 = 0.045$.
• Total number of compounding periods (N): The total time in years is multiplied by the number of compounding periods per year. $$ N = \text{Time in years} \times \text{Number of compounding periods per year} = 1 \text{ year} \times 2 = 2 \text{ half-years} $$

Compound Interest Formula for Half-Yearly Compounding

The formula for the amount (A) after N periods when the rate per period is r is:

The compound interest (CI) is the difference between the amount and the principal:

Step-by-Step Calculation of Compound Interest

Let's plug in the values into the formula:

1. Calculate the amount (A): $$ \text{A} = 16400 (1 + 0.045)^2 $$ $$ \text{A} = 16400 (1.045)^2 $$ First, calculate $(1.045)^2$: $$ (1.045)^2 = 1.045 \times 1.045 = 1.092025 $$ Now, calculate the amount: $$ \text{A} = 16400 \times 1.092025 $$ $$ \text{A} = 17909.21 $$ The total amount after 1 year compounded half-yearly is Rs. 17909.21.
2. Calculate the Compound Interest (CI): $$ \text{CI} = \text{A} - \text{P} $$ $$ \text{CI} = 17909.21 - 16400 $$ $$ \text{CI} = 1509.21 $$ The compound interest is Rs. 1509.21.

Alternatively, you can calculate the interest for each half-year period:

• Interest for the 1st half-year: $$ \text{I}_1 = \text{Principal} \times \text{Rate per period} = 16400 \times 0.045 = 738 $$ Amount after 1st half-year = $16400 + 738 = 17138$.
• Interest for the 2nd half-year: $$ \text{I}_2 = \text{Amount after 1st half-year} \times \text{Rate per period} = 17138 \times 0.045 = 771.21 $$ Amount after 2nd half-year = $17138 + 771.21 = 17909.21$.
• Total Compound Interest (CI): $$ \text{CI} = \text{I}_1 + \text{I}_2 = 738 + 771.21 = 1509.21 $$

Both methods confirm that the compound interest on Rs. 16400 for 1 year at 9% per annum compounded half-yearly is Rs. 1509.21.

Conclusion

Based on the calculations, the compound interest is Rs. 1509.21.

Revision Table: Compound Interest Concepts

Additional Information: Compounding Frequency Impact

The frequency of compounding significantly impacts the total compound interest earned over a period. The more frequently interest is compounded, the higher the total interest will be, assuming the same annual rate.

• Annual Compounding: Interest calculated once a year. N = T, r = R/100.
• Half-Yearly Compounding: Interest calculated twice a year. N = 2T, r = R/200.
• Quarterly Compounding: Interest calculated four times a year. N = 4T, r = R/400.
• Monthly Compounding: Interest calculated twelve times a year. N = 12T, r = R/1200.

In this problem, compounding half-yearly means the interest earned in the first six months is added to the principal, and then the interest for the next six months is calculated on this new, larger principal. This process leads to higher interest compared to annual compounding over the same period.