The semi annual compound amounts of a sum of money in 1 year and 2 years are 400 and 441 respectively. Find the annual compound interest for 2 years.

To tackle the problem of finding the annual compound interest based on the semi-annual compound amounts, we need to first understand the relationships involved in compound interest and how the semi-annual compounding affects the total amounts over the specified periods.

Understanding Compound Interest

Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. When compounding is done semi-annually, the interest is added twice a year. This means that the interest rate is effectively halved, and the amount is compounded every six months.

Given Information

• Amount after 1 year (compounded semi-annually): 400
• Amount after 2 years (compounded semi-annually): 441

Finding the Interest Rate

Let's denote the principal amount as P and the semi-annual interest rate as r. Since there are two compounding periods in a year, we can set up the following equations:

• After 1 year: P(1 + r)^2 = 400
• After 2 years: P(1 + r)^4 = 441

Now we have two equations involving P and r. To find r, we can express P from the first equation:

P = 400 / (1 + r)^2

Substituting P into the Second Equation

Now, substitute this expression for P into the second equation:

(400 / (1 + r)^2)(1 + r)^4 = 441

This simplifies to:

400(1 + r)^2 = 441

Solving for r

Now we can solve for (1 + r)^2:

(1 + r)^2 = 441 / 400 = 1.1025

Taking the square root of both sides gives:

1 + r = √1.1025 ≈ 1.05

Thus, we can find r:

r ≈ 1.05 - 1 = 0.05

Calculating the Annual Interest Rate

Since we found that the semi-annual interest rate r is approximately 0.05 (or 5%), the annual compound interest rate, which is compounded twice, can be calculated as:

Annual Rate = (1 + r)^2 - 1

Substituting the value of r:

Annual Rate = (1 + 0.05)^2 - 1 = 1.1025 - 1 = 0.1025

This translates to an annual compound interest rate of approximately 10.25%.

Final Thoughts

So, the annual compound interest for 2 years, based on the semi-annual compounding amounts provided, is around 10.25%. This illustrates how understanding the compounding frequency and applying the right formulas can help us solve complex financial problems effectively.