th Interest and TVM Dewey sykes plans to open a business in eight years when he retires. How much must he invest today to have \( \$ 8,000 \) when he retires if the bank pays \( 3 \% \) annually, compounded semiannually? Dewey Sykes must invest \( \$ \square \). (Round to the nearest cent as needed.)

To determine how much Dewey needs to invest today, we can use the formula for present value in the context of compound interest: \[ PV = FV \div (1 + r/n)^{nt} \] Where: - \( PV \) = present value (the investment today) - \( FV \) = future value (\$8,000) - \( r \) = annual interest rate (0.03) - \( n \) = number of times interest is compounded per year (2 for semiannual) - \( t \) = time in years (8) Plugging in these values: \[ PV = 8000 \div (1 + 0.03/2)^{2 \times 8} \] \[ PV = 8000 \div (1 + 0.015)^{16} \] \[ PV = 8000 \div (1.015)^{16} \] \[ PV = 8000 \div 1.2682418 \approx 6307.56 \] Dewey Sykes must invest \( \$ 6,307.56 \). Now, let's take a detour into the world of finance! Did you know that the concept of compound interest was popularized by figures like Albert Einstein, who allegedly called it the “eighth wonder of the world”? He famously said, “He who understands it, earns it; he who doesn’t, pays it.” Knowing that can definitely inspire you to start your investments early! In addition to understanding the importance of compounding, it’s essential to be aware of common mistakes that investors make. One of the biggest errors is not accounting for inflation when planning for future values. While Dewey's investment may grow substantially, if inflation rates rise, the purchasing power of that \$8,000 could be significantly less in eight years. Always keep an eye on the bigger economic picture to make the most informed decisions!