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Sagot :
Certainly! Let's analyze each investment option and determine which one yields the best return over 5 years.
### Option a: 13% Interest Per Annum Compounded Annually
In this option, the interest is compounded annually. The formula for compound interest is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n periods, including interest.
- [tex]\( P \)[/tex] (principal) is the initial amount of money ([tex]$10,000). - \( r \) is the annual interest rate (13% or 0.13). - \( n \) is the number of times interest is compounded per year (1 for annually). - \( t \) is the number of years the money is invested (5 years). Plugging in the values: \[ A_a = 10000 \left(1 + 0.13\right)^5 \] \[ A_a \approx 18424.35 \] ### Option b: 12% Interest Per Annum Compounded Monthly In this option, the interest is compounded monthly. The formula for compound interest remains the same: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( P \) (principal) is the initial amount ($[/tex]10,000).
- [tex]\( r \)[/tex] is the annual interest rate (12% or 0.12).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year (12 for monthly).
- [tex]\( t \)[/tex] is the number of years the money is invested (5 years).
Plugging in the values:
[tex]\[ A_b = 10000 \left(1 + \frac{0.12}{12}\right)^{12 \cdot 5} \][/tex]
[tex]\[ A_b \approx 18166.97 \][/tex]
### Option c: 14% Simple Interest for 5 Years
In this option, simple interest is used. The formula for simple interest is given by:
[tex]\[ A = P (1 + rt) \][/tex]
where:
- [tex]\( P \)[/tex] (principal) is the initial amount ([tex]$10,000). - \( r \) is the annual interest rate (14% or 0.14). - \( t \) is the number of years the money is invested (5 years). Plugging in the values: \[ A_c = 10000 (1 + 0.14 \cdot 5) \] \[ A_c = 10000 (1 + 0.70) \] \[ A_c = 10000 \cdot 1.70 \] \[ A_c = 17000.00 \] ### Conclusion The final amounts for each option after 5 years are: - Option a: \( \$[/tex]18424.35 \)
- Option b: [tex]\( \$18166.97 \)[/tex]
- Option c: [tex]\( \$17000.00 \)[/tex]
Comparing these amounts, you can see that Option a yields the highest return. Therefore, the best choice would be to invest the money at 13% interest per annum compounded annually for 5 years.
### Option a: 13% Interest Per Annum Compounded Annually
In this option, the interest is compounded annually. The formula for compound interest is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n periods, including interest.
- [tex]\( P \)[/tex] (principal) is the initial amount of money ([tex]$10,000). - \( r \) is the annual interest rate (13% or 0.13). - \( n \) is the number of times interest is compounded per year (1 for annually). - \( t \) is the number of years the money is invested (5 years). Plugging in the values: \[ A_a = 10000 \left(1 + 0.13\right)^5 \] \[ A_a \approx 18424.35 \] ### Option b: 12% Interest Per Annum Compounded Monthly In this option, the interest is compounded monthly. The formula for compound interest remains the same: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( P \) (principal) is the initial amount ($[/tex]10,000).
- [tex]\( r \)[/tex] is the annual interest rate (12% or 0.12).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year (12 for monthly).
- [tex]\( t \)[/tex] is the number of years the money is invested (5 years).
Plugging in the values:
[tex]\[ A_b = 10000 \left(1 + \frac{0.12}{12}\right)^{12 \cdot 5} \][/tex]
[tex]\[ A_b \approx 18166.97 \][/tex]
### Option c: 14% Simple Interest for 5 Years
In this option, simple interest is used. The formula for simple interest is given by:
[tex]\[ A = P (1 + rt) \][/tex]
where:
- [tex]\( P \)[/tex] (principal) is the initial amount ([tex]$10,000). - \( r \) is the annual interest rate (14% or 0.14). - \( t \) is the number of years the money is invested (5 years). Plugging in the values: \[ A_c = 10000 (1 + 0.14 \cdot 5) \] \[ A_c = 10000 (1 + 0.70) \] \[ A_c = 10000 \cdot 1.70 \] \[ A_c = 17000.00 \] ### Conclusion The final amounts for each option after 5 years are: - Option a: \( \$[/tex]18424.35 \)
- Option b: [tex]\( \$18166.97 \)[/tex]
- Option c: [tex]\( \$17000.00 \)[/tex]
Comparing these amounts, you can see that Option a yields the highest return. Therefore, the best choice would be to invest the money at 13% interest per annum compounded annually for 5 years.
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