To solve the given problems, we will use the formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Let's solve the problems step-by-step:
### Case c: R18500 at 4.5% compounded yearly for eight years
1. Principal amount ([tex]\( P_c \)[/tex]) = R18500
2. Annual interest rate ([tex]\( r_c \)[/tex]) = 4.5% = 0.045 (decimal form)
3. Number of times interest is compounded per year ([tex]\( n_c \)[/tex]) = 1
4. Time the money is invested for ([tex]\( t_c \)[/tex]) = 8 years
Using the compound interest formula:
[tex]\[ A_c = 18500 \left(1 + \frac{0.045}{1}\right)^{1 \cdot 8} \][/tex]
The accumulated amount [tex]\( A_c \)[/tex] after 8 years is:
[tex]\[ A_c \approx R26308.861 \][/tex]
### Case d: R10000 at 11% compounded monthly for seven years
1. Principal amount ([tex]\( P_d \)[/tex]) = R10000
2. Annual interest rate ([tex]\( r_d \)[/tex]) = 11% = 0.11 (decimal form)
3. Number of times interest is compounded per year ([tex]\( n_d \)[/tex]) = 12
4. Time the money is invested for ([tex]\( t_d \)[/tex]) = 7 years
Using the compound interest formula:
[tex]\[ A_d = 10000 \left(1 + \frac{0.11}{12}\right)^{12 \cdot 7} \][/tex]
The accumulated amount [tex]\( A_d \)[/tex] after 7 years is:
[tex]\[ A_d \approx R21522.036 \][/tex]
### Case e: R150000 at 20% compounded daily for seven years
1. Principal amount ([tex]\( P_e \)[/tex]) = R150000
2. Annual interest rate ([tex]\( r_e \)[/tex]) = 20% = 0.20 (decimal form)
3. Number of times interest is compounded per year ([tex]\( n_e \)[/tex]) = 365
4. Time the money is invested for ([tex]\( t_e \)[/tex]) = 7 years
Using the compound interest formula:
[tex]\[ A_e = 150000 \left(1 + \frac{0.20}{365}\right)^{365 \cdot 7} \][/tex]
The accumulated amount [tex]\( A_e \)[/tex] after 7 years is:
[tex]\[ A_e \approx R608046.812 \][/tex]
### Summary
- The accumulated amount for R18500 at 4.5% compounded yearly for eight years is approximately R26308.861.
- The accumulated amount for R10000 at 11% compounded monthly for seven years is approximately R21522.036.
- The accumulated amount for R150000 at 20% compounded daily for seven years is approximately R608046.812.
