Solved on Feb 07, 2024

Find the accumulated value of a $15,000 investment over 7 years at 6% interest rate, compounded a) semiannually, b) quarterly, c) monthly, d) continuously. Round answers to nearest cent.
a) $\$19,525.31$ b) $\$19,540.85$ c) $\$19,547.15$ d) $\$19,551.50$

STEP 1

Assumptions
1. The principal investment amount (P) is $15,000.
2. The annual interest rate (r) is 6%.
3. The time (t) for which the money is invested is 7 years.
4. For part a, the compounding frequency (n) is semiannually (2 times a year).
5. For part b, the compounding frequency (n) is quarterly (4 times a year).
6. For part c, the compounding frequency (n) is monthly (12 times a year).
7. For part d, the compounding is continuous.

STEP 2

First, we will calculate the accumulated value for part a, where the money is compounded semiannually. We will use the compound interest formula:
$A = P\left(1+\frac{r}{n}\right)^{nt}$

STEP 3

Convert the annual interest rate from a percentage to a decimal by dividing by 100.
$r = 6\% = \frac{6}{100} = 0.06$

STEP 4

Plug in the values for P, r, n, and t to calculate the accumulated value for semiannual compounding.
$A = \$15,000\left(1+\frac{0.06}{2}\right)^{2 \times 7}$

STEP 5

Calculate the accumulated value for semiannual compounding.
$A = \$15,000\left(1+\frac{0.06}{2}\right)^{14}$
$A = \$15,000\left(1+0.03\right)^{14}$
$A = \$15,000\left(1.03\right)^{14}$

STEP 6

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times 1.5107$
$A \approx \$22,660.50$

STEP 7

Round the answer to the nearest cent for part a.
Accumulated value if the money is compounded semiannually: $22,660.50

STEP 8

Next, we will calculate the accumulated value for part b, where the money is compounded quarterly. We will use the same compound interest formula with n changed to 4.
$A = P\left(1+\frac{r}{n}\right)^{nt}$

STEP 9

Plug in the values for P, r, n, and t to calculate the accumulated value for quarterly compounding.
$A = \$15,000\left(1+\frac{0.06}{4}\right)^{4 \times 7}$

STEP 10

Calculate the accumulated value for quarterly compounding.
$A = \$15,000\left(1+\frac{0.06}{4}\right)^{28}$
$A = \$15,000\left(1+0.015\right)^{28}$
$A = \$15,000\left(1.015\right)^{28}$

STEP 11

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times 1.5277$
$A \approx \$22,915.50$

STEP 12

Round the answer to the nearest cent for part b.
Accumulated value if the money is compounded quarterly: $22,915.50

STEP 13

Now, we will calculate the accumulated value for part c, where the money is compounded monthly. We will use the same compound interest formula with n changed to 12.
$A = P\left(1+\frac{r}{n}\right)^{nt}$

STEP 14

Plug in the values for P, r, n, and t to calculate the accumulated value for monthly compounding.
$A = \$15,000\left(1+\frac{0.06}{12}\right)^{12 \times 7}$

STEP 15

Calculate the accumulated value for monthly compounding.
$A = \$15,000\left(1+\frac{0.06}{12}\right)^{84}$
$A = \$15,000\left(1+0.005\right)^{84}$
$A = \$15,000\left(1.005\right)^{84}$

STEP 16

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times 1.5386$
$A \approx \$23,079$

STEP 17

Round the answer to the nearest cent for part c.
Accumulated value if the money is compounded monthly: $23,079

STEP 18

Finally, we will calculate the accumulated value for part d, where the money is compounded continuously. We will use the continuous compound interest formula:
$A = Pe^{rt}$

STEP 19

Plug in the values for P, r, and t to calculate the accumulated value for continuous compounding.
$A = \$15,000e^{0.06 \times 7}$

STEP 20

Calculate the accumulated value for continuous compounding.
$A = \$15,000e^{0.42}$

STEP 21

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times e^{0.42}$
$A \approx \$15,000 \times 1.5210$
$A \approx \$22,815$

STEP 22

Round the answer to the nearest cent for part d.
Accumulated value if the money is compounded continuously: $22,815

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STEP 1

STEP 2

First, we will calculate the accumulated value for part a, where the money is compounded semiannually. We will use the compound interest formula:A=P(1+rn)ntA = P\left(1+\frac{r}{n}\right)^{nt}A=P(1+nr​)nt

STEP 3

Convert the annual interest rate from a percentage to a decimal by dividing by 100.r=6%=6100=0.06r = 6\% = \frac{6}{100} = 0.06r=6%=1006​=0.06

STEP 4

Plug in the values for P, r, n, and t to calculate the accumulated value for semiannual compounding.A=$15,000(1+0.062)2×7A = \$15,000\left(1+\frac{0.06}{2}\right)^{2 \times 7}A=$15,000(1+20.06​)2×7

STEP 5

Calculate the accumulated value for semiannual compounding.A=$15,000(1+0.062)14A = \$15,000\left(1+\frac{0.06}{2}\right)^{14}A=$15,000(1+20.06​)14A=$15,000(1+0.03)14A = \$15,000\left(1+0.03\right)^{14}A=$15,000(1+0.03)14A=$15,000(1.03)14A = \$15,000\left(1.03\right)^{14}A=$15,000(1.03)14

STEP 6

Compute the value using a calculator or a computational tool.A≈$15,000×1.5107A \approx \$15,000 \times 1.5107A≈$15,000×1.5107A≈$22,660.50A \approx \$22,660.50A≈$22,660.50

STEP 7

Round the answer to the nearest cent for part a.Accumulated value if the money is compounded semiannually: $22,660.50

STEP 8

Next, we will calculate the accumulated value for part b, where the money is compounded quarterly. We will use the same compound interest formula with n changed to 4.A=P(1+rn)ntA = P\left(1+\frac{r}{n}\right)^{nt}A=P(1+nr​)nt

STEP 9

Plug in the values for P, r, n, and t to calculate the accumulated value for quarterly compounding.A=$15,000(1+0.064)4×7A = \$15,000\left(1+\frac{0.06}{4}\right)^{4 \times 7}A=$15,000(1+40.06​)4×7

STEP 10

Calculate the accumulated value for quarterly compounding.A=$15,000(1+0.064)28A = \$15,000\left(1+\frac{0.06}{4}\right)^{28}A=$15,000(1+40.06​)28A=$15,000(1+0.015)28A = \$15,000\left(1+0.015\right)^{28}A=$15,000(1+0.015)28A=$15,000(1.015)28A = \$15,000\left(1.015\right)^{28}A=$15,000(1.015)28

STEP 11

Compute the value using a calculator or a computational tool.A≈$15,000×1.5277A \approx \$15,000 \times 1.5277A≈$15,000×1.5277A≈$22,915.50A \approx \$22,915.50A≈$22,915.50

STEP 12

Round the answer to the nearest cent for part b.Accumulated value if the money is compounded quarterly: $22,915.50

STEP 13

Now, we will calculate the accumulated value for part c, where the money is compounded monthly. We will use the same compound interest formula with n changed to 12.A=P(1+rn)ntA = P\left(1+\frac{r}{n}\right)^{nt}A=P(1+nr​)nt

STEP 14

Plug in the values for P, r, n, and t to calculate the accumulated value for monthly compounding.A=$15,000(1+0.0612)12×7A = \$15,000\left(1+\frac{0.06}{12}\right)^{12 \times 7}A=$15,000(1+120.06​)12×7

STEP 15

Calculate the accumulated value for monthly compounding.A=$15,000(1+0.0612)84A = \$15,000\left(1+\frac{0.06}{12}\right)^{84}A=$15,000(1+120.06​)84A=$15,000(1+0.005)84A = \$15,000\left(1+0.005\right)^{84}A=$15,000(1+0.005)84A=$15,000(1.005)84A = \$15,000\left(1.005\right)^{84}A=$15,000(1.005)84

STEP 16

Compute the value using a calculator or a computational tool.A≈$15,000×1.5386A \approx \$15,000 \times 1.5386A≈$15,000×1.5386A≈$23,079A \approx \$23,079A≈$23,079

STEP 17

Round the answer to the nearest cent for part c.Accumulated value if the money is compounded monthly: $23,079

STEP 18

Finally, we will calculate the accumulated value for part d, where the money is compounded continuously. We will use the continuous compound interest formula:A=PertA = Pe^{rt}A=Pert

STEP 19

Plug in the values for P, r, and t to calculate the accumulated value for continuous compounding.A=$15,000e0.06×7A = \$15,000e^{0.06 \times 7}A=$15,000e0.06×7

STEP 20

Calculate the accumulated value for continuous compounding.A=$15,000e0.42A = \$15,000e^{0.42}A=$15,000e0.42

STEP 21

Compute the value using a calculator or a computational tool.A≈$15,000×e0.42A \approx \$15,000 \times e^{0.42}A≈$15,000×e0.42A≈$15,000×1.5210A \approx \$15,000 \times 1.5210A≈$15,000×1.5210A≈$22,815A \approx \$22,815A≈$22,815

STEP 22

Round the answer to the nearest cent for part d.Accumulated value if the money is compounded continuously: $22,815

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First, we will calculate the accumulated value for part a, where the money is compounded semiannually. We will use the compound interest formula:
$A = P\left(1+\frac{r}{n}\right)^{nt}$

Convert the annual interest rate from a percentage to a decimal by dividing by 100.
$r = 6\% = \frac{6}{100} = 0.06$

Plug in the values for P, r, n, and t to calculate the accumulated value for semiannual compounding.
$A = \$15,000\left(1+\frac{0.06}{2}\right)^{2 \times 7}$

Calculate the accumulated value for semiannual compounding.
$A = \$15,000\left(1+\frac{0.06}{2}\right)^{14}$
$A = \$15,000\left(1+0.03\right)^{14}$
$A = \$15,000\left(1.03\right)^{14}$

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times 1.5107$
$A \approx \$22,660.50$

Round the answer to the nearest cent for part a.
Accumulated value if the money is compounded semiannually: $22,660.50

Next, we will calculate the accumulated value for part b, where the money is compounded quarterly. We will use the same compound interest formula with n changed to 4.
$A = P\left(1+\frac{r}{n}\right)^{nt}$

Plug in the values for P, r, n, and t to calculate the accumulated value for quarterly compounding.
$A = \$15,000\left(1+\frac{0.06}{4}\right)^{4 \times 7}$

Calculate the accumulated value for quarterly compounding.
$A = \$15,000\left(1+\frac{0.06}{4}\right)^{28}$
$A = \$15,000\left(1+0.015\right)^{28}$
$A = \$15,000\left(1.015\right)^{28}$

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times 1.5277$
$A \approx \$22,915.50$

Round the answer to the nearest cent for part b.
Accumulated value if the money is compounded quarterly: $22,915.50

Now, we will calculate the accumulated value for part c, where the money is compounded monthly. We will use the same compound interest formula with n changed to 12.
$A = P\left(1+\frac{r}{n}\right)^{nt}$

Plug in the values for P, r, n, and t to calculate the accumulated value for monthly compounding.
$A = \$15,000\left(1+\frac{0.06}{12}\right)^{12 \times 7}$

Calculate the accumulated value for monthly compounding.
$A = \$15,000\left(1+\frac{0.06}{12}\right)^{84}$
$A = \$15,000\left(1+0.005\right)^{84}$
$A = \$15,000\left(1.005\right)^{84}$

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times 1.5386$
$A \approx \$23,079$

Round the answer to the nearest cent for part c.
Accumulated value if the money is compounded monthly: $23,079

Finally, we will calculate the accumulated value for part d, where the money is compounded continuously. We will use the continuous compound interest formula:
$A = Pe^{rt}$

Plug in the values for P, r, and t to calculate the accumulated value for continuous compounding.
$A = \$15,000e^{0.06 \times 7}$

Calculate the accumulated value for continuous compounding.
$A = \$15,000e^{0.42}$

Compute the value using a calculator or a computational tool.
$A \approx \$15,000 \times e^{0.42}$
$A \approx \$15,000 \times 1.5210$
$A \approx \$22,815$

Round the answer to the nearest cent for part d.
Accumulated value if the money is compounded continuously: $22,815