Use the compound interest formula Upper A equals Upper P left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript nt
to find the accumulated value of an investment of $ 15 comma 000 for 3 years at an interest rate of 4 % if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly.
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Part 1
a. What is the accumulated value if the money is compounded semiannually?
$
16,892.44 (Round to the nearest cent as needed.)
Part 2
b. What is the accumulated value if the money is compounded quarterly?
$
enter your response here (Round to the nearest cent as needed.)
Part 1:
To find the accumulated value if the money is compounded semiannually, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
where:
A = accumulated value
P = principal amount (initial investment) = $15,000
r = interest rate per period (annual rate) = 4% = 0.04
n = number of compounding periods per year
Since the money is compounded semiannually, n = 2 (2 compounding periods per year).
Plugging in the values:
A = 15000(1 + 0.04/2)^(2*3)
Calculating inside the parentheses:
A = 15000(1 + 0.02)^(6)
A = 15000(1.02)^(6)
Calculating the exponent:
A = 15000(1.124864)
Calculating the accumulated value:
A ≈ $16,892.44
Therefore, the accumulated value if the money is compounded semiannually is $16,892.44.
Part 2:
To find the accumulated value if the money is compounded quarterly, we can use the same formula, but with a different value for n.
Since the money is compounded quarterly, n = 4 (4 compounding periods per year).
Plugging in the values:
A = 15000(1 + 0.04/4)^(4*3)
Calculating inside the parentheses:
A = 15000(1 + 0.01)^(12)
A = 15000(1.01)^(12)
Calculating the exponent:
A = 15000(1.126825)
Calculating the accumulated value:
A ≈ $16,900.88
Therefore, the accumulated value if the money is compounded quarterly is $16,900.88.