A bank features a savings account that has an annual percentage rate of $3.6\%$ with interest compounded quarterly. Brigeth deposits $\$5,500$ into the account.
How much money will Brigeth have in the account in 1 year?
Answer = $\$\square$ Round answer to the nearest penny.
What is the annual percentage yield (APY) for the savings account?
$APY = \square\%$ Round to the nearest hundredth of a percent.
Answer (2)
Calculate the quarterly interest rate: 4 0.036 = 0.009 .
Calculate the amount after 1 year using the compound interest formula: A = 5500 ( 1 + 0.009 ) 4 ≈ 5700.69 .
Calculate the APY using the formula: A P Y = ( 1 + 0.009 ) 4 − 1 ≈ 0.0364889 ≈ 3.65% .
The amount of money Brigeth will have in the account in 1 year is $5700.69 , and the annual percentage yield (APY) is 3.65% .
Explanation
Understanding the Problem We are given that Brigeth deposits $5 , 500 into a savings account with an annual percentage rate of 3.6% compounded quarterly. We need to find the amount of money Brigeth will have in the account in 1 year and the annual percentage yield (APY).
Calculating the Quarterly Interest Rate First, we need to find the quarterly interest rate. Since the annual interest rate is 3.6% , the quarterly interest rate is 4 3.6% = 4 0.036 = 0.009 .
Calculating the Amount After 1 Year Next, we calculate the amount of money Brigeth will have in the account in 1 year. We use the compound interest formula: A = P ( 1 + r ) n , where P is the principal amount, r is the interest rate per compounding period, and n is the number of compounding periods. In this case, P = 5500 , r = 0.009 , and n = 4 (since the interest is compounded quarterly). Therefore, A = 5500 ( 1 + 0.009 ) 4 = 5500 ( 1.009 ) 4 ≈ 5500 ( 1.0364889 ) ≈ 5700.689 . Rounding to the nearest penny, we get $5700.69 .
Calculating the APY Now, we calculate the annual percentage yield (APY). The formula for APY is A P Y = ( 1 + r ) n − 1 , where r is the interest rate per compounding period and n is the number of compounding periods per year. In this case, r = 0.009 and n = 4 . Therefore, A P Y = ( 1 + 0.009 ) 4 − 1 = ( 1.009 ) 4 − 1 ≈ 1.0364889 − 1 = 0.0364889 . To express this as a percentage, we multiply by 100: 0.0364889 × 100 = 3.64889% . Rounding to the nearest hundredth of a percent, we get 3.65% .
Final Answer Therefore, the amount of money Brigeth will have in the account in 1 year is $5700.69 , and the annual percentage yield (APY) is 3.65% .
Examples
Understanding compound interest and APY is crucial for making informed financial decisions. For example, when choosing between different savings accounts or investment options, comparing their APYs allows you to determine which one will provide the highest return on your investment over time. This knowledge helps you maximize your savings and achieve your financial goals more effectively. Compound interest is also applicable when calculating loan payments, mortgages, and other financial products.
Brigeth will have approximately $5700.69 in her savings account after 1 year, and the annual percentage yield (APY) for the account is about 3.65%.
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