Use the compound-interest formula to find the account balance [tex]$A$[/tex] where [tex]$P$[/tex] is principal, r is interest rate, [tex]$n$[/tex] is number of compounding periods per year, t is time, in years, and A is account balance.
The account balance is approximately [tex]$22,584.83$[/tex]. Simplify your answer. Do not round until the final answer. Then round to two decimal places as needed.
Answer (1)
Calculate the value inside the parentheses: 1 + 365 0.096 ≈ 1.00026301 .
Calculate the exponent: 365 × 5 = 1825 .
Raise the value inside the parentheses to the power of the exponent: ( 1.00026301 ) 1825 ≈ 1.61595 .
Multiply the result by the principal amount: 13976 × 1.61595 ≈ 22584.83 . The account balance is $22584.83 .
Explanation
Understanding the Problem We are asked to find the account balance A using the compound interest formula, given the principal P , interest rate r , number of compounding periods per year n , and time t . The formula is: A = P ( 1 + n r ) n t We have the following values: Principal, P = $13 , 976 Annual interest rate, r = 9.6% = 0.096 Compounded daily, so n = 365 Time, t = 5 years
Calculating Intermediate Values First, we need to calculate the value inside the parentheses: 1 + n r = 1 + 365 0.096 1 + 365 0.096 ≈ 1.00026301 Next, we calculate the exponent: n t = 365 × 5 = 1825
Calculating the Account Balance Now, we raise the value inside the parentheses to the power of the exponent: ( 1 + 365 0.096 ) 1825 ≈ ( 1.00026301 ) 1825 ≈ 1.61595 Finally, we multiply the result by the principal amount: A = 13976 × 1.61595 ≈ 22584.83
Final Answer Therefore, the account balance after 5 years is approximately $22 , 584.83 .
Examples
Compound interest is a powerful concept used in many real-world financial scenarios. For example, when you deposit money into a savings account, the bank pays you interest, which is often compounded. Understanding compound interest helps you estimate how much your investments will grow over time. Similarly, loans, mortgages, and credit cards also use compound interest, so understanding the formula can help you make informed decisions about borrowing money. By understanding the variables and how they interact, you can project the future value of your investments or the total cost of a loan.
