IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.
Sagot :
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].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
