Understanding Compound Interest Compounded Half-Yearly

This question asks us to calculate the future amount of a sum of money invested at compound interest when the interest is compounded half-yearly. Compound interest means that the interest earned in each period is added to the principal for calculating the interest in the next period. When interest is compounded half-yearly, it means the interest is calculated and added to the principal twice a year.

Given Information

• Principal amount (P) = Rs. 2500
• Time period (T) = 1 year
• Annual interest rate (R) = 4%
• Compounding frequency = Half-yearly (twice a year)

Compound Interest Calculation Method

When interest is compounded \( n \) times a year, the formula for the amount (\( A \)) after \( T \) years is:

\( A = P \left(1 + \frac{R/100}{n}\right)^{n \times T} \)

In this case:

• Principal (\( P \)) = 2500
• Annual Rate (\( R \)) = 4%
• Time (\( T \)) = 1 year
• Number of times compounded per year (\( n \)) = 2 (since it's half-yearly)

The effective interest rate per period is \( \frac{R/100}{n} = \frac{4/100}{2} = \frac{0.04}{2} = 0.02 \). The total number of compounding periods is \( n \times T = 2 \times 1 = 2 \) periods.

Step-by-Step Calculation

Using the formula, we substitute the given values:

\( A = 2500 \left(1 + \frac{0.04}{2}\right)^{2 \times 1} \)

\( A = 2500 (1 + 0.02)^2 \)

\( A = 2500 (1.02)^2 \)

Now, we calculate \( (1.02)^2 \):

\( (1.02)^2 = 1.02 \times 1.02 = 1.0404 \)

Substitute this back into the equation for \( A \):

\( A = 2500 \times 1.0404 \)

\( A = 2601 \)

So, the amount after 1 year when compounded half-yearly is Rs. 2601.

Summary of Calculation

The sum of Rs 2500 will amount to Rs 2601 in 1 year at 4% interest rate, compounded half-yearly.

Revision Table: Key Concepts in Compound Interest

Additional Information: Compounding Frequency

The frequency of compounding significantly impacts the final amount. The more frequently interest is compounded, the higher the final amount will be, assuming the same annual interest rate. Common compounding frequencies include:

• Annually: \( n=1 \) (Interest calculated once a year)
• Half-yearly: \( n=2 \) (Interest calculated twice a year)
• Quarterly: \( n=4 \) (Interest calculated four times a year)
• Monthly: \( n=12 \) (Interest calculated twelve times a year)
• Daily: \( n=365 \) (Interest calculated 365 times a year)

In this specific problem, using half-yearly compounding means we divide the annual rate by 2 and multiply the number of years by 2 to find the rate per period and the total number of periods, respectively.