Luke and Karen deposit \( \$ 700.00 \) into a savings account which earns \( 14 \% \) interest compounded monthly. They want to use the money in the account to \( g 0 \) on a trip in 3 years. How much will they be able to spend? Use the formula \( A=P\left(1+\frac{r}{n}\right)^{n t} \), where \( A \) is the balance (final amount), \( P \) is the principal (starting amount), \( r \) is the interest rate expressed as a decimal, \( n \) is the number of times per year that the interest is compounded, and \( t \) is the time in years. Round your answer to the nearest cent.
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To find out how much Luke and Karen will have after 3 years, we can plug the values into the formula: - \( P = 700 \) (the initial deposit), - \( r = 0.14 \) (14% as a decimal), - \( n = 12 \) (since the interest is compounded monthly), - \( t = 3 \) (the number of years). Using the formula: \[ A = 700\left(1+\frac{0.14}{12}\right)^{12 \times 3} \] Calculating: \[ A = 700\left(1+\frac{0.14}{12}\right)^{36} \approx 700\left(1+0.0116667\right)^{36} \approx 700(1.4516) \approx 1015.82 \] Thus, they will be able to spend approximately \( \text{\$} 1015.82 \) on their trip! --- Back in the day, compound interest was often referred to as "interest on interest" and was deemed a mysterious financial sorcery, especially during the Renaissance. Traders and merchants began to realize they could make their money work for them, adding layers of complexity to their financial dealings. Saving wasn't just about putting money aside; it became a strategic game of growth over time! If you're intrigued by the dynamics of money and how it can grow over time, consider reading "The Wealth of Nations" by Adam Smith. This classic book dives into the principles of economics, including how interest rates and investments can shape wealth over generations.
