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 (1)
Calculate the amount after 1 year using the compound interest formula: A = P ( 1 + n r ) n t , where P = 5500 , r = 0.036 , n = 4 , and t = 1 , resulting in A = $5700.69 .
Calculate the annual percentage yield (APY) using the formula: A P Y = ( 1 + n r ) n − 1 , where r = 0.036 and n = 4 , resulting in A P Y = 3.65% .
The amount of money Brigeth will have in the account in 1 year is $5700.69 .
The annual percentage yield (APY) for the savings account is 3.65% .
Explanation
Understanding Compound Interest Formula First, we need to calculate the amount of money Brigeth will have in the account in 1 year. The formula for compound interest is: A = P ( 1 + n r ) n t 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 (as a decimal).
n is the number of times that interest is compounded per year.
t is the number of years the money is invested or borrowed for.
Calculating the Amount After 1 Year In this case, we have:
P = $5 , 500
r = 3.6% = 0.036
n = 4 (compounded quarterly)
t = 1 year Plugging these values into the formula, we get: A = 5500 ( 1 + 4 0.036 ) 4 × 1 A = 5500 ( 1 + 0.009 ) 4 A = 5500 ( 1.009 ) 4 A = 5500 × 1.0364892256099966 A = 5700.690740854981 Rounding to the nearest penny, we get $5700.69 .
Calculating the Annual Percentage Yield (APY) Next, we need to calculate the annual percentage yield (APY). The formula for APY is: A P Y = ( 1 + n r ) n − 1 where:
r is the annual interest rate (as a decimal).
n is the number of times that interest is compounded per year. In this case, we have:
r = 0.036
n = 4 Plugging these values into the formula, we get: A P Y = ( 1 + 4 0.036 ) 4 − 1 A P Y = ( 1 + 0.009 ) 4 − 1 A P Y = ( 1.009 ) 4 − 1 A P Y = 1.0364892256099966 − 1 A P Y = 0.0364892256099966 To express this as a percentage, we multiply by 100: A P Y = 0.0364892256099966 × 100 = 3.64892256099966% 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) for the savings account is 3.65% .
Examples
Understanding compound interest and APY is crucial for making informed financial decisions. For instance, when comparing different savings accounts or investment options, knowing the APY helps you determine which option will yield the highest return over time. This knowledge is also valuable when planning for long-term financial goals, such as retirement or purchasing a home, as it allows you to accurately project the growth of your investments.
