Example Question - savings account interest
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Calculating Future Value and Annual Percentage Yield of Savings Account
The image contains a question which says: "A bank features a savings account that has an annual percentage rate of 3.8% with interest compounded monthly. Jace deposits $2,000 into the account. a) How much money will Jace have in the account in 1 year? Round answer to the nearest penny. b) What is the annual percentage yield (APY) for the savings account? Round to the nearest hundredth of a percent." Let's solve part a) first. We are given that the annual interest rate is 3.8% and it is compounded monthly. To find the total amount in the account after one year, we will use the formula for compound interest: A = P(1 + r/n)^(nt) Where: A = the future value of the investment/loan, including interest P = the principal investment amount ($2,000 in this case) r = the annual interest rate (decimal) (3.8% or 0.038) n = the number of times that interest is compounded per year (12 for monthly) t = the time the money is invested or borrowed for, in years (1 year in this case) Plugging in the values: A = 2000(1 + 0.038/12)^(12*1) A = 2000(1 + 0.0031667)^(12) A = 2000(1.0031667)^(12) Now use a calculator to evaluate the expression: A ≈ 2000(1.0031667)^12 A ≈ 2000 * 1.038726 A ≈ 2077.45 So, after one year, Jace will have approximately $2077.45 in the account, rounding to the nearest penny. For part b), the annual percentage yield (APY) takes into account the effect of compound interest over the year. It is calculated by the formula: APY = (1 + r/n)^(n) - 1 Again, substitute the values: APY = (1 + 0.038/12)^(12) - 1 APY = (1 + 0.0031667)^(12) - 1 Now evaluate the expression using the calculator: APY ≈ (1.0031667)^12 - 1 APY ≈ 1.038726 - 1 APY ≈ 0.038726 To express this as a percentage, multiply by 100: APY ≈ 0.038726 * 100 APY ≈ 3.8726% Rounding to the nearest hundredth of a percent, we get an APY of approximately 3.87%.
Calculating Compound Interest for a Savings Account
To solve this question, we can use the compound interest formula which 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 or borrowed for, in years. In this case, a grandmother deposits $5000 in an account that pays 9.5% compounded monthly, and we want to find the value of the account at the child's twenty-first birthday. Therefore, P = $5000, r = 9.5/100 = 0.095 (as a decimal), n = 12 (since interest is compounded monthly), and t = 21 years. Plugging in the values: A = 5000(1 + 0.095/12)^(12*21) A = 5000(1 + 0.00791667)^(252) A = 5000(1.00791667)^(252) Now we can calculate the value of A. A ≈ 5000(1.00791667)^252 Using a calculator to compute this value: A ≈ 5000 * (1.00791667)^252 A ≈ 5000 * 5.98472378 A ≈ 29923.619 So, the value of the account will be approximately $29,923.62 when rounded to the nearest dollar.
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