Use the compound-interest formula to find the account balance [tex]$A$[/tex], where [tex]$P$[/tex] is principal, [tex]$r$[/tex] is interest rate, [tex]$n$[/tex] is the number of compounding periods per year, [tex]$t$[/tex] is time in years, and [tex]$A$[/tex] is the account balance.
Answer (2)
Substitute the given values into the compound interest formula: A = P ( 1 + n r ) n t .
Calculate the value inside the parenthesis: ( 1 + 365 0.046 ) ≈ 1.000126027 .
Calculate the exponent: ( 365 ) ( 4 ) = 1460 .
Calculate the account balance and round to two decimal places: A ≈ 19090.19 .
Explanation
Understanding the Problem We are given the principal amount P = $15 , 882 , the annual interest rate r = 4.6% = 0.046 , the number of compounding periods per year n = 365 (since it's compounded daily), and the time in years t = 4 . We want to find the account balance A using the compound interest formula.
Stating the Formula The compound interest formula is: A = P ( 1 + n r ) n t
Substituting the Values Now, we substitute the given values into the formula: A = 15882 ( 1 + 365 0.046 ) ( 365 ) ( 4 )
Calculating Inside Parenthesis First, let's calculate the value inside the parenthesis: 1 + 365 0.046 ≈ 1 + 0.000126027 ≈ 1.000126027
Calculating the Exponent Next, let's calculate the exponent: ( 365 ) ( 4 ) = 1460
Calculating the Power Now, we calculate the value of ( 1 + 365 0.046 ) 1460 :
( 1 + 365 0.046 ) 1460 ≈ ( 1.000126027 ) 1460 ≈ 1.201994
Calculating the Account Balance Finally, we multiply the result by the principal amount: A = 15882 × 1.201994 ≈ 19090.193978
Rounding the Answer Rounding the final answer to two decimal places, we get: A ≈ $19090.19
Examples
Compound interest is a powerful concept used in many real-life financial situations. For example, when you deposit money into a savings account, the bank pays you interest, which is often compounded. This means the interest earned is added to your principal, and the next interest calculation includes this new amount. Understanding compound interest can help you make informed decisions about investments, loans, and other financial products. It's also used in calculating the future value of investments, like retirement accounts, and the cost of loans, such as mortgages.
To find the account balance using the compound interest formula, you substitute the principal, interest rate, compounding frequency, and time into the formula. After performing the calculations, the account balance comes out to approximately $19,090.19. This approach helps understand how investments grow with compounded interest over time.
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