Question - Calculating Future Value and APY of a Savings Account
Solution:
The question in the image is asking to calculate the future value of a savings account with an annual percentage rate of 3.8% with interest compounded monthly after one year, given an initial deposit of $2,000. Additionally, it asks to calculate the annual percentage yield (APY) for the savings account.To solve the first part of the question, we use the future value formula for compound interest:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]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.Given:- $$ P = \$2,000 $$- $$ r = 3.8\% = 0.038 $$ (as a decimal)- $$ n = 12 $$ (since the interest is compounded monthly)- $$ t = 1 $$ year\[ A = 2000 \left(1 + \frac{0.038}{12}\right)^{12 \times 1} \]\[ A = 2000 \left(1 + 0.0031667\right)^{12} \]\[ A = 2000 \left(1.0031667\right)^{12} \]\[ A ≈ 2000 \times 1.03884 \]\[ A ≈ 2077.68 \]Jace will have approximately $2077.68 in the account after one year.For the second part of the question, we calculate the annual percentage yield (APY) which takes into account the effect of compound interest. APY can be calculated using the formula:\[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \]Using the same values for $$ r $$ and $$ n $$, we have:\[ APY = \left(1 + \frac{0.038}{12}\right)^{12} - 1 \]\[ APY = \left(1.0031667\right)^{12} - 1 \]\[ APY ≈ 1.03884 - 1 \]\[ APY ≈ 0.03884 \]\[ APY ≈ 3.88\% \]The annual percentage yield (APY) for the savings account is approximately 3.88%.
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