How much more interest will Maria earn if she invests $1000 for one year at an x% annual interest rate, compounded semiannually, than if she invests $1000 for one year at an x% annual interest rate, compounded annually? A. 5x B. 10x C. x^2/200 D. x^2/400E. (10x + x^2/400)

To compare the interest Maria will earn with semiannual compounding versus annual compounding, we need to calculate the amount earned in both scenarios and then find the difference.

Let's break it down step by step:

For semiannual compounding, interest is compounded twice a year. The formula for compound interest is A = P(1 + r/n)^(nt), where: - A is the amount of money accumulated after n years, including interest. - P is the principal amount (the initial amount of money). - r is the annual interest rate (decimal). - n is the number of times that interest is compounded per year. - t is the time the money is invested for in years.

For Maria's semiannual compounding case (Compounded twice a year): A_semi = 1000 * (1 + x/2/100)^(2*1)

For annual compounding (Compounded once a year): A_annual = 1000 * (1 + x/100)^1

To find the extra interest earned from semiannual compounding: Interest_difference = A_semi - A_annual

Substituting the respective formulas, we have: Interest_difference = (1000 * (1 + x/200)^2) - (1000 * (1 + x/100))

We want to find a simplified version, so let's expand it and subtract it term by term: Interest_difference = (1000 + 1000 * (x/200) * 2 + 1000 * (x/200)^2) - (1000 + 1000 * (x/100)) Interest_difference = (1000 + 10x + (1000 * x^2)/(40000)) - (1000 + 10x) Interest_difference = (1000 * x^2)/(40000) Interest_difference = x^2/400

Therefore, the extra interest Maria will earn with semiannual compounding compared to annual compounding can be calculated as x^2/400.

So the correct answer is D. x^2/400.

Extra: Understanding compounding frequency is important for any student learning about interest and investments. Compounding refers to how often the interest is applied to the principal to calculate additional interest for the next period. This is powerful because it can turn small amounts of money into large amounts over time with reinvestment of earnings, which is also known as compound interest.

High compounding frequencies, like daily or monthly, typically allow money to grow faster than lower frequencies, like annual compounding, because interest is being calculated and added to the principal more frequently. With more frequent compounding, each interest payment starts earning interest on itself earlier. That's why investments with more frequent compounding (like Maria's semiannual example) will always earn more than an equivalent investment with less frequent compounding (like Maria's annual example) assuming the same interest rate and principal. This is the principle that makes Maria's semiannual investment accrue more interest than an annual one, resulting in the extra earnings we calculated. Understanding the effect of compounding frequencies is essential for anyone making investment decisions or studying finance.