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Sagot :
Let's complete the table step-by-step by calculating Suzanne's and Derek's annual interest and balance over a 5-year period:
### Suzanne's Investment (Simple Interest)
Suzanne deposits \$7500 in an account earning simple interest at an annual rate of 8%.
1. Calculating Annual Interest:
- The annual interest for Suzanne is calculated as:
[tex]\[ \text{Interest} = \text{Principal} \times \text{Rate} = 7500 \times 0.08 = 600 \][/tex]
2. Calculating Balance for Each Year:
- Year 1:
[tex]\[ \text{Balance}_1 = \text{Principal} + \text{Interest} \times 1 = 7500 + 600 \times 1 = 8100 \][/tex]
- Year 2:
[tex]\[ \text{Balance}_2 = \text{Principal} + \text{Interest} \times 2 = 7500 + 600 \times 2 = 8700 \][/tex]
- Year 3:
[tex]\[ \text{Balance}_3 = \text{Principal} + \text{Interest} \times 3 = 7500 + 600 \times 3 = 9300 \][/tex]
- Year 4:
[tex]\[ \text{Balance}_4 = \text{Principal} + \text{Interest} \times 4 = 7500 + 600 \times 4 = 9900 \][/tex]
- Year 5:
[tex]\[ \text{Balance}_5 = \text{Principal} + \text{Interest} \times 5 = 7500 + 600 \times 5 = 10500 \][/tex]
### Derek's Investment (Compound Interest)
Derek deposits \$7500 in an account earning compound interest at an annual rate of 8%, compounded annually.
1. Calculating Annual Interest and Balance for Each Year:
- Year 0 (Initial Principal):
[tex]\[ \text{Balance}_0 = 7500 \][/tex]
- Year 1:
[tex]\[ \text{Interest}_1 = 7500 \times 0.08 = 600 \][/tex]
[tex]\[ \text{Balance}_1 = 7500 + 600 = 8100 \][/tex]
- Year 2:
[tex]\[ \text{Interest}_2 = 8100 \times 0.08 = 648 \][/tex]
[tex]\[ \text{Balance}_2 = 8100 + 648 = 8748 \][/tex]
- Year 3:
[tex]\[ \text{Interest}_3 = 8748 \times 0.08 = 699.84 \][/tex]
[tex]\[ \text{Balance}_3 = 8748 + 699.84 = 9447.84 \][/tex]
- Year 4:
[tex]\[ \text{Interest}_4 = 9447.84 \times 0.08 = 755.8272 \][/tex]
[tex]\[ \text{Balance}_4 = 9447.84 + 755.8272 = 10203.6672 \][/tex]
- Year 5:
[tex]\[ \text{Interest}_5 = 10203.6672 \times 0.08 = 816.293376 \][/tex]
[tex]\[ \text{Balance}_5 = 10203.6672 + 816.293376 = 11019.960576 \][/tex]
Here's the completed table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Year} & \begin{array}{c} \text{Suzanne's} \\ \text{Annual} \\ \text{Interest} \end{array} & \begin{array}{c} \text{Suzanne's} \\ \text{Balance} \end{array} & \begin{array}{c} \text{Derek's} \\ \text{Annual} \\ \text{Interest} \end{array} & \begin{array}{c} \text{Derek's} \\ \text{Balance} \end{array} \\ \hline 1 & 600.0 & 8100.0 & 600.0 & 8100.0 \\ 2 & 600.0 & 8700.0 & 648.0 & 8748.0 \\ 3 & 600.0 & 9300.0 & 699.84 & 9447.84 \\ 4 & 600.0 & 9900.0 & 755.8272 & 10203.6672 \\ 5 & 600.0 & 10500.0 & 816.293376 & 11019.960576 \\ \hline \end{array} \][/tex]
### Suzanne's Investment (Simple Interest)
Suzanne deposits \$7500 in an account earning simple interest at an annual rate of 8%.
1. Calculating Annual Interest:
- The annual interest for Suzanne is calculated as:
[tex]\[ \text{Interest} = \text{Principal} \times \text{Rate} = 7500 \times 0.08 = 600 \][/tex]
2. Calculating Balance for Each Year:
- Year 1:
[tex]\[ \text{Balance}_1 = \text{Principal} + \text{Interest} \times 1 = 7500 + 600 \times 1 = 8100 \][/tex]
- Year 2:
[tex]\[ \text{Balance}_2 = \text{Principal} + \text{Interest} \times 2 = 7500 + 600 \times 2 = 8700 \][/tex]
- Year 3:
[tex]\[ \text{Balance}_3 = \text{Principal} + \text{Interest} \times 3 = 7500 + 600 \times 3 = 9300 \][/tex]
- Year 4:
[tex]\[ \text{Balance}_4 = \text{Principal} + \text{Interest} \times 4 = 7500 + 600 \times 4 = 9900 \][/tex]
- Year 5:
[tex]\[ \text{Balance}_5 = \text{Principal} + \text{Interest} \times 5 = 7500 + 600 \times 5 = 10500 \][/tex]
### Derek's Investment (Compound Interest)
Derek deposits \$7500 in an account earning compound interest at an annual rate of 8%, compounded annually.
1. Calculating Annual Interest and Balance for Each Year:
- Year 0 (Initial Principal):
[tex]\[ \text{Balance}_0 = 7500 \][/tex]
- Year 1:
[tex]\[ \text{Interest}_1 = 7500 \times 0.08 = 600 \][/tex]
[tex]\[ \text{Balance}_1 = 7500 + 600 = 8100 \][/tex]
- Year 2:
[tex]\[ \text{Interest}_2 = 8100 \times 0.08 = 648 \][/tex]
[tex]\[ \text{Balance}_2 = 8100 + 648 = 8748 \][/tex]
- Year 3:
[tex]\[ \text{Interest}_3 = 8748 \times 0.08 = 699.84 \][/tex]
[tex]\[ \text{Balance}_3 = 8748 + 699.84 = 9447.84 \][/tex]
- Year 4:
[tex]\[ \text{Interest}_4 = 9447.84 \times 0.08 = 755.8272 \][/tex]
[tex]\[ \text{Balance}_4 = 9447.84 + 755.8272 = 10203.6672 \][/tex]
- Year 5:
[tex]\[ \text{Interest}_5 = 10203.6672 \times 0.08 = 816.293376 \][/tex]
[tex]\[ \text{Balance}_5 = 10203.6672 + 816.293376 = 11019.960576 \][/tex]
Here's the completed table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Year} & \begin{array}{c} \text{Suzanne's} \\ \text{Annual} \\ \text{Interest} \end{array} & \begin{array}{c} \text{Suzanne's} \\ \text{Balance} \end{array} & \begin{array}{c} \text{Derek's} \\ \text{Annual} \\ \text{Interest} \end{array} & \begin{array}{c} \text{Derek's} \\ \text{Balance} \end{array} \\ \hline 1 & 600.0 & 8100.0 & 600.0 & 8100.0 \\ 2 & 600.0 & 8700.0 & 648.0 & 8748.0 \\ 3 & 600.0 & 9300.0 & 699.84 & 9447.84 \\ 4 & 600.0 & 9900.0 & 755.8272 & 10203.6672 \\ 5 & 600.0 & 10500.0 & 816.293376 & 11019.960576 \\ \hline \end{array} \][/tex]
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