Use the compound interest formulas [tex]$A = P \left(1+\frac{ r }{ n }\right)^{ nt }$[/tex] and [tex]$A = P e^{r t }$[/tex] to solve the problem given. Round answers to the nearest cent.
Find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 6.5% if the money is
a. compounded semiannually,
b. compounded quarterly;
c. compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
$ 29356.94
b. What is the accumulated value if the money is compounded quarterly?
$ 29447.16
c. What is the accumulated value if the money is compounded monthly?
$
Answer (2)
Identify the principal amount P = $20 , 000 , the annual interest rate r = 0.065 , the number of years t = 6 , and the number of times compounded per year n = 12 .
Substitute these values into the compound interest formula: A = P ( 1 + n r ) n t .
Calculate the accumulated value: A = 20000 ( 1 + 12 0.065 ) 12 × 6 = 20000 ( 1 + 0.00541667 ) 72 = 20000 ( 1.00541667 ) 72 ≈ $29508.54 .
The accumulated value of the investment is 29508.54 .
Explanation
Understanding the Problem We are asked to find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 6.5% compounded monthly. We will use the compound interest formula: A = P ( 1 + n r ) n t where:
A is the accumulated value
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the number of years
Identifying the Values Identify the given values:
Principal amount, P = $20 , 000
Annual interest rate, r = 6.5% = 0.065
Number of years, t = 6
Number of times compounded per year, n = 12 (monthly)
Substituting the Values Substitute the values into the formula: A = 20000 ( 1 + 12 0.065 ) 12 × 6
Calculating the Accumulated Value Calculate the accumulated value: A = 20000 ( 1 + 12 0.065 ) 72 A = 20000 ( 1 + 0.00541667 ) 72 A = 20000 ( 1.00541667 ) 72 A = 20000 × 1.475427 A = 29508.54
Final Answer The accumulated value of the investment, rounded to the nearest cent, is $29508.54.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how different compounding frequencies affect your investment's growth can significantly impact your savings over time. Comparing monthly compounding to annual compounding helps you appreciate the power of earning interest on interest more frequently, leading to a larger nest egg when you retire. This concept is also vital when evaluating loan options, as it helps you understand the true cost of borrowing money.
The accumulated value of an investment of $20,000 at an interest rate of 6.5% compounded monthly is approximately $29,718.94. This calculation uses the compound interest formula with suitable substitutions for principal, rate, compounding frequency, and time. It demonstrates the impact of compounding on investment growth over time.
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