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 number of compounding periods per year, [tex]$t$[/tex] is time, in years, and [tex]$A$[/tex] is account balance.
| [tex]$P$[/tex] | [tex]$r$[/tex] | compounded | [tex]$t$[/tex] |
|---|---|---|---|
| [tex]13,976$[/tex] | [tex]9.6 \%[/tex] | Daily | 5 |
The account balance is approximately $ $\boxed{\phantom{0000}}$ (Simplify your answer. Do not round until the final answer. Then round to two decimal places as needed.)
Answer (2)
Substitute the given values into the compound interest formula: A = P ( 1 + n r ) n t .
Plug in P = 13976 , r = 0.096 , n = 365 , and t = 5 into the formula: A = 13976 ( 1 + 365 0.096 ) 365 ⋅ 5 .
Simplify the expression and calculate the final amount: A ≈ 13976 × 1.61595 .
The account balance after 5 years is approximately: 22584.83 .
Explanation
Understanding the Problem We are given the principal amount P = $13 , 976 , the annual interest rate r = 9.6% = 0.096 , the number of compounding periods per year n = 365 (daily), and the time in years t = 5 . We need to find the account balance A using the compound interest formula.
Stating the Formula The compound interest formula is given by: A = P ( 1 + n r ) n t We will substitute the given values into this formula.
Calculations Substituting the given values, we have: A = 13976 ( 1 + 365 0.096 ) 365 ⋅ 5 Now, we simplify the expression inside the parentheses: 1 + 365 0.096 ≈ 1 + 0.0002630137 = 1.0002630137 Next, we calculate the exponent: 365 ⋅ 5 = 1825 So, we have: A = 13976 ( 1.0002630137 ) 1825 Now, we calculate the value of ( 1.0002630137 ) 1825 :
( 1.0002630137 ) 1825 ≈ 1.61595 Finally, we calculate the account balance A :
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 tool for growing wealth over time. For example, if you invest $10,000 in a retirement account with an average annual return of 7% compounded monthly, after 30 years, your investment could grow to approximately $81,162.74. This demonstrates how consistent investing and the effect of compounding can significantly increase your savings. Understanding compound interest helps in making informed financial decisions, whether it's for retirement planning, saving for a down payment on a house, or simply growing your savings.
To find the account balance using compound interest, we substitute the given values into the formula. After calculating, we find that the account balance after 5 years is approximately $22,584.83. This illustrates the power of compound interest over time.
;
