Part 2 of 4
HW Scores 00.82%, 44.5 of 40 points
Pointser 0 of 1
Use the compound interest formulas [tex]$A = P \left(1+\frac{r}{ n }\right)^{ nt }$[/tex] and [tex]$A = Pe ^{/ t }$[/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 monthly, d. compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
$29356.94
(Round your answer to the nearest cent. Do not include the $ symbol in your answer.)
b. What is the accumulated value if the money is compounded quarterly?
Answer (2)
Identify the principal (P), interest rate (r), number of times interest is compounded per year (n), and time in years (t).
Substitute the values into the compound interest formula: A = P ( 1 + n r ) n t .
Calculate the accumulated value (A) using the given values: A = 20000 ( 1 + 4 0.065 ) 4 × 6 .
Round the result to the nearest cent: 29447.16 .
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% when the money is compounded quarterly. We will use the compound interest formula to solve this problem.
Identifying the Formula and Variables The formula for compound interest is given by: A = P ( 1 + n r ) n t where:
A is the accumulated value
P is the principal amount
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
Assigning Values to Variables In this problem, we have:
P = 20000
r = 0.065 (6.5% as a decimal)
n = 4 (compounded quarterly)
t = 6
Calculating the Accumulated Value Now, we substitute these values into the formula: A = 20000 ( 1 + 4 0.065 ) 4 × 6 A = 20000 ( 1 + 0.01625 ) 24 A = 20000 ( 1.01625 ) 24 A = 20000 × 1.4723579526764852 A = 29447.159053529704 Rounding to the nearest cent, we get A = 29447.16 .
Final Answer Therefore, the accumulated value if the money is compounded quarterly is $29447.16.
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. Similarly, when taking out a loan, understanding the compounding frequency helps you assess the true cost of borrowing and compare different loan options effectively. This knowledge empowers you to make sound financial choices and achieve your long-term financial goals.
The accumulated value of an investment of $20,000 for 6 years at an interest rate of 6.5%, compounded quarterly, is approximately $28,647.29. This is computed using the compound interest formula with appropriate substitutions. The result is rounded to the nearest cent.
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