Answer :
To determine which investment option will earn you the most money, we will evaluate each option and its final value after 8 years. The investment options and their details are as follows:
1. Option A: [tex]\(3.99\%\)[/tex] interest compounded monthly.
2. Option B: [tex]\(4\%\)[/tex] interest compounded quarterly.
3. Option C: [tex]\(4.175\%\)[/tex] interest compounded annually.
4. Option D: [tex]\(4.2\%\)[/tex] simple interest.
### Option A: [tex]\(3.99\%\)[/tex] Compounded Monthly
For compound interest, the formula to calculate the future value [tex]\(A\)[/tex] is given by:
[tex]\[ A = P \left(1 + \frac{R}{N}\right)^{N \cdot T} \][/tex]
where:
- [tex]\(P\)[/tex] is the principal amount ([tex]$12,000$[/tex]),
- [tex]\(R\)[/tex] is the annual interest rate (0.0399),
- [tex]\(N\)[/tex] is the number of times the interest is compounded per year (12 for monthly),
- [tex]\(T\)[/tex] is the number of years (8).
By substituting the given values:
[tex]\[ A = 12000 \left(1 + \frac{0.0399}{12}\right)^{12 \cdot 8} = 16,503.58 \][/tex]
So, the amount after 8 years at [tex]\(3.99\%\)[/tex] compounded monthly is [tex]\( \$16,503.58\)[/tex].
### Option B: [tex]\(4\%\)[/tex] Compounded Quarterly
Using the same compound interest formula, we substitute the following for quarterly compounding:
- [tex]\(P = 12000\)[/tex],
- [tex]\(R = 0.04\)[/tex],
- [tex]\(N = 4\)[/tex] (for quarterly),
- [tex]\(T = 8\)[/tex].
[tex]\[ A = 12000 \left(1 + \frac{0.04}{4}\right)^{4 \cdot 8} = 16,499.29 \][/tex]
So, the amount after 8 years at [tex]\(4\%\)[/tex] compounded quarterly is [tex]\( \$16,499.29\)[/tex].
### Option C: [tex]\(4.175\%\)[/tex] Compounded Annually
For annual compounding, use:
- [tex]\(P = 12000\)[/tex],
- [tex]\(R = 0.04175\)[/tex],
- [tex]\(N = 1\)[/tex] (for annually),
- [tex]\(T = 8\)[/tex].
[tex]\[ A = 12000 \left(1 + \frac{0.04175}{1}\right)^{1 \cdot 8} = 16,645.21 \][/tex]
So, the amount after 8 years at [tex]\(4.175\%\)[/tex] compounded annually is [tex]\( \$16,645.21\)[/tex].
### Option D: [tex]\(4.2\%\)[/tex] Simple Interest
For simple interest, the formula is:
[tex]\[ A = P(1 + r \cdot t) \][/tex]
where:
- [tex]\(P\)[/tex] is the principal amount ([tex]$12,000$[/tex]),
- [tex]\(r\)[/tex] is the annual interest rate (0.042),
- [tex]\(t\)[/tex] is the number of years (8).
[tex]\[ A = 12000 \left(1 + 0.042 \cdot 8\right) = 16,032.00 \][/tex]
So, the amount after 8 years at [tex]\(4.2\%\)[/tex] simple interest is [tex]\( \$16,032.00\)[/tex].
### Conclusion
By comparing the amounts from each option after 8 years:
- Option A: [tex]\( \$16,503.58\)[/tex]
- Option B: [tex]\( \$16,499.29\)[/tex]
- Option C: [tex]\( \$16,645.21\)[/tex]
- Option D: [tex]\( \$16,032.00\)[/tex]
The investment option that will earn you the most money is Option C: [tex]\(4.175\%\)[/tex] compounded annually, which results in [tex]\( \$16,645.21\)[/tex].
1. Option A: [tex]\(3.99\%\)[/tex] interest compounded monthly.
2. Option B: [tex]\(4\%\)[/tex] interest compounded quarterly.
3. Option C: [tex]\(4.175\%\)[/tex] interest compounded annually.
4. Option D: [tex]\(4.2\%\)[/tex] simple interest.
### Option A: [tex]\(3.99\%\)[/tex] Compounded Monthly
For compound interest, the formula to calculate the future value [tex]\(A\)[/tex] is given by:
[tex]\[ A = P \left(1 + \frac{R}{N}\right)^{N \cdot T} \][/tex]
where:
- [tex]\(P\)[/tex] is the principal amount ([tex]$12,000$[/tex]),
- [tex]\(R\)[/tex] is the annual interest rate (0.0399),
- [tex]\(N\)[/tex] is the number of times the interest is compounded per year (12 for monthly),
- [tex]\(T\)[/tex] is the number of years (8).
By substituting the given values:
[tex]\[ A = 12000 \left(1 + \frac{0.0399}{12}\right)^{12 \cdot 8} = 16,503.58 \][/tex]
So, the amount after 8 years at [tex]\(3.99\%\)[/tex] compounded monthly is [tex]\( \$16,503.58\)[/tex].
### Option B: [tex]\(4\%\)[/tex] Compounded Quarterly
Using the same compound interest formula, we substitute the following for quarterly compounding:
- [tex]\(P = 12000\)[/tex],
- [tex]\(R = 0.04\)[/tex],
- [tex]\(N = 4\)[/tex] (for quarterly),
- [tex]\(T = 8\)[/tex].
[tex]\[ A = 12000 \left(1 + \frac{0.04}{4}\right)^{4 \cdot 8} = 16,499.29 \][/tex]
So, the amount after 8 years at [tex]\(4\%\)[/tex] compounded quarterly is [tex]\( \$16,499.29\)[/tex].
### Option C: [tex]\(4.175\%\)[/tex] Compounded Annually
For annual compounding, use:
- [tex]\(P = 12000\)[/tex],
- [tex]\(R = 0.04175\)[/tex],
- [tex]\(N = 1\)[/tex] (for annually),
- [tex]\(T = 8\)[/tex].
[tex]\[ A = 12000 \left(1 + \frac{0.04175}{1}\right)^{1 \cdot 8} = 16,645.21 \][/tex]
So, the amount after 8 years at [tex]\(4.175\%\)[/tex] compounded annually is [tex]\( \$16,645.21\)[/tex].
### Option D: [tex]\(4.2\%\)[/tex] Simple Interest
For simple interest, the formula is:
[tex]\[ A = P(1 + r \cdot t) \][/tex]
where:
- [tex]\(P\)[/tex] is the principal amount ([tex]$12,000$[/tex]),
- [tex]\(r\)[/tex] is the annual interest rate (0.042),
- [tex]\(t\)[/tex] is the number of years (8).
[tex]\[ A = 12000 \left(1 + 0.042 \cdot 8\right) = 16,032.00 \][/tex]
So, the amount after 8 years at [tex]\(4.2\%\)[/tex] simple interest is [tex]\( \$16,032.00\)[/tex].
### Conclusion
By comparing the amounts from each option after 8 years:
- Option A: [tex]\( \$16,503.58\)[/tex]
- Option B: [tex]\( \$16,499.29\)[/tex]
- Option C: [tex]\( \$16,645.21\)[/tex]
- Option D: [tex]\( \$16,032.00\)[/tex]
The investment option that will earn you the most money is Option C: [tex]\(4.175\%\)[/tex] compounded annually, which results in [tex]\( \$16,645.21\)[/tex].
