Math

Question 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\$19,525.31 b) $19,540.85\$19,540.85 c) $19,547.15\$19,547.15 d) $19,551.50\$19,551.50

Studdy Solution

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(1+rn)ntA = 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%=6100=0.06r = 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(1+0.062)2×7A = \$15,000\left(1+\frac{0.06}{2}\right)^{2 \times 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+0.03)14A = \$15,000\left(1+0.03\right)^{14}
A=$15,000(1.03)14A = \$15,000\left(1.03\right)^{14}

STEP 6

Compute the value using a calculator or a computational tool.
A$15,000×1.5107A \approx \$15,000 \times 1.5107
A$22,660.50A \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(1+rn)ntA = 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(1+0.064)4×7A = \$15,000\left(1+\frac{0.06}{4}\right)^{4 \times 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+0.015)28A = \$15,000\left(1+0.015\right)^{28}
A=$15,000(1.015)28A = \$15,000\left(1.015\right)^{28}

STEP 11

Compute the value using a calculator or a computational tool.
A$15,000×1.5277A \approx \$15,000 \times 1.5277
A$22,915.50A \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(1+rn)ntA = 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(1+0.0612)12×7A = \$15,000\left(1+\frac{0.06}{12}\right)^{12 \times 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+0.005)84A = \$15,000\left(1+0.005\right)^{84}
A=$15,000(1.005)84A = \$15,000\left(1.005\right)^{84}

STEP 16

Compute the value using a calculator or a computational tool.
A$15,000×1.5386A \approx \$15,000 \times 1.5386
A$23,079A \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=PertA = Pe^{rt}

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}

STEP 20

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

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$15,000×1.5210A \approx \$15,000 \times 1.5210
A$22,815A \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

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord