A bank features a savings account that has an annual percentage rate of $4.1 \%$ with interest compounded semiannually. Jayden deposits \$2,500 into the account.
How much money will Jayden 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 future value using the compound interest formula: A = P ( 1 + n r ) n t , where P = 2500 , r = 0.041 , and n = 2 . The future value A is 2603.55. − C a l c u l a t e t h e ann u a lp erce n t a g ey i e l d ( A P Y ) u s in g t h e f or m u l a : APY = (1 + \frac{r}{n})^n - 1 , w h ere r = 0.041 an d n = 2$. The APY is 0.0414 . - Convert the APY to a percentage: 0.0414 × 100 = 4.14% . - The amount of money Jayden will have in the account in 1 year is $\boxed{ 2603.55} , and the APY is 4.14% .
Explanation
Understanding the Problem We are given a savings account with an annual percentage rate (APR) of 4.1% and interest compounded semiannually. Jayden deposits $2,500 into the account, and we want to find the amount of money in the account after 1 year and the annual percentage yield (APY).
Compound Interest Formula First, we need to calculate the amount of money Jayden will have in the account in 1 year. The formula for compound interest is: A = P ( 1 + r ) n where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (the initial deposit)
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
Identifying the Values In this case:
$P = 2 , 500
The annual interest rate is 4.1%, so r = 0.041
The interest is compounded semiannually, so the number of times interest is compounded per year is 2. Since the time period is 1 year, n = 2 .
Calculating the Future Value Now, we can plug these values into the formula: A = 2500 ( 1 + 2 0.041 ) 2 A = 2500 ( 1 + 0.0205 ) 2 A = 2500 ( 1.0205 ) 2 A = 2500 ( 1.04142025 ) A = 2603.550625 Rounding to the nearest penny, we get $2603.55.
APY Formula 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 stated annual interest rate
n is the number of compounding periods per year
Calculating APY In this case:
r = 0.041
n = 2 Plugging these values into the formula, we get: A P Y = ( 1 + 2 0.041 ) 2 − 1 A P Y = ( 1 + 0.0205 ) 2 − 1 A P Y = ( 1.0205 ) 2 − 1 A P Y = 1.04142025 − 1 A P Y = 0.04142025 To express APY as a percentage, we multiply by 100: A P Y = 0.04142025 × 100 = 4.142025 Rounding to the nearest hundredth of a percent, we get 4.14%.
Final Answer Therefore, the amount of money Jayden will have in the account in 1 year is $2603.55, and the annual percentage yield (APY) for the savings account is 4.14%.
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 buying a home, as it allows you to estimate the growth of your investments more accurately. By understanding these concepts, you can make smarter choices about where to save and invest your money.
Jayden will have approximately $2603.55 in his savings account after 1 year, and the Annual Percentage Yield (APY) for the account is approximately 4.14%.
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