Question

Manually calculate the compound interest on an investment of $7,500 at 6% interest, compounded semiannually, for 1 year.

Answer 1

`\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$7500\\ r=rate\to 6\%\to (6)/(100)\dotfill &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semi-annually, thus twice} \end{array}\dotfill &2\\ t=years\dotfill &1 \end{cases} \\\\\\ A=7500\left(1+(0.06)/(2)\right)^(2\cdot 1)\implies A=7500(1.03)^2\implies A=7956.75`

Answer 2

The compound interest on an investment of $7,500 at 6% interest, compounded semiannually, for 1 year is $457.50.

Explanation:

When calculating compound interest, it is important to first understand the key components involved. Compound interest is the interest earned not only on the initial investment, but also on the accumulated interest from previous periods. In this case, the initial investment is $7,500 and the interest rate is 6%. However, the interest is compounded semiannually, meaning it is calculated twice a year.

To calculate the compound interest, we can use the formula A = P(1 + r/n)^nt, where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this scenario, the annual interest rate is 6%, which can be expressed as 0.06. Since the interest is compounded semiannually, n is equal to 2. Additionally, the investment is for 1 year, so t is equal to 1.

Substituting these values into the formula, we get A = $7,500(1 + 0.06/2)^2(1) = $7,500(1.03)^2 = $7,500(1.0609) = $7,957.18.

To find the compound interest, we subtract the initial investment of $7,500 from the final amount and get $7,957.18 - $7,500 = $457.18. This means that the compound interest earned on the investment is $457.18.

In conclusion, the compound interest on an investment of $7,500 at 6% interest, compounded semiannually, for 1 year is $457.50. By understanding the components and using the correct formula, we can accurately calculate compound interest for any investment.