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Sagot :
Sure, let’s tackle the problem of calculating the total value of an investment under different compounding periods.
### Given:
- Principal ([tex]\( P \)[/tex]): K25,000,000
- Annual Interest Rate ([tex]\( r \)[/tex]): 12.5%
- Time ([tex]\( t \)[/tex]): 5 years
### 1) Compounded Monthly
When interest is compounded monthly, the formula to calculate the future value ([tex]\( A \)[/tex]) is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the time the money is invested for in years.
For monthly compounding:
- [tex]\( n = 12 \)[/tex] (since interest is compounded 12 times a year)
Plugging in the given values into the equation:
[tex]\[ A = 25,000,000 \left(1 + \frac{0.125}{12}\right)^{12 \times 5} \][/tex]
From the results provided:
[tex]\[ A \approx 46,555,402.13 \][/tex]
So, when compounded monthly, the total value of the investment after 5 years is approximately K46,555,402.13.
### 2) Compounded Daily
When interest is compounded daily, the formula to calculate the future value ([tex]\( A \)[/tex]) is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the time the money is invested for in years.
For daily compounding:
- [tex]\( n = 365 \)[/tex] (since interest is compounded 365 times a year)
Plugging in the given values into the equation:
[tex]\[ A = 25,000,000 \left(1 + \frac{0.125}{365}\right)^{365 \times 5} \][/tex]
From the results provided:
[tex]\[ A \approx 46,701,151.83 \][/tex]
So, when compounded daily, the total value of the investment after 5 years is approximately K46,701,151.83.
### Conclusion
- When the investment of K25,000,000 is compounded monthly at an annual interest rate of 12.5% for 5 years, the total value becomes approximately K46,555,402.13.
- When the same investment is compounded daily, it becomes approximately K46,701,151.83.
### Given:
- Principal ([tex]\( P \)[/tex]): K25,000,000
- Annual Interest Rate ([tex]\( r \)[/tex]): 12.5%
- Time ([tex]\( t \)[/tex]): 5 years
### 1) Compounded Monthly
When interest is compounded monthly, the formula to calculate the future value ([tex]\( A \)[/tex]) is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the time the money is invested for in years.
For monthly compounding:
- [tex]\( n = 12 \)[/tex] (since interest is compounded 12 times a year)
Plugging in the given values into the equation:
[tex]\[ A = 25,000,000 \left(1 + \frac{0.125}{12}\right)^{12 \times 5} \][/tex]
From the results provided:
[tex]\[ A \approx 46,555,402.13 \][/tex]
So, when compounded monthly, the total value of the investment after 5 years is approximately K46,555,402.13.
### 2) Compounded Daily
When interest is compounded daily, the formula to calculate the future value ([tex]\( A \)[/tex]) is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the time the money is invested for in years.
For daily compounding:
- [tex]\( n = 365 \)[/tex] (since interest is compounded 365 times a year)
Plugging in the given values into the equation:
[tex]\[ A = 25,000,000 \left(1 + \frac{0.125}{365}\right)^{365 \times 5} \][/tex]
From the results provided:
[tex]\[ A \approx 46,701,151.83 \][/tex]
So, when compounded daily, the total value of the investment after 5 years is approximately K46,701,151.83.
### Conclusion
- When the investment of K25,000,000 is compounded monthly at an annual interest rate of 12.5% for 5 years, the total value becomes approximately K46,555,402.13.
- When the same investment is compounded daily, it becomes approximately K46,701,151.83.
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