Let's solve these step by step.
(a) Simple Interest
Simple interest can be calculated using the formula:
[tex]\[ \text{Simple Interest (SI)} = P \cdot r \cdot t \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Given:
- [tex]\( P = $600 \)[/tex]
- [tex]\( r = 9\% = 0.09 \)[/tex] (as a decimal)
- [tex]\( t = 3 \)[/tex] years
The future value [tex]\( FV_{\text{simple}} \)[/tex] with simple interest is obtained by adding the simple interest to the principal amount:
[tex]\[ FV_{\text{simple}} = P + \text{SI} = P + (P \cdot r \cdot t) \][/tex]
[tex]\[ FV_{\text{simple}} = $600 + ($600 \cdot 0.09 \cdot 3) \][/tex]
[tex]\[ FV_{\text{simple}} = $600 + $162 \][/tex]
[tex]\[ FV_{\text{simple}} = $762 \][/tex]
The future value with simple interest is indeed [tex]$762, as given in your question.
(b) Compound Interest
Compound interest is calculated using the formula:
\[ A = P \cdot (1 + r)^n \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money),
- \( r \) is the annual interest rate (in decimal form),
- \( n \) is the number of times that interest is compounded per year.
Given that the interest is compounded annually, \( n \) will also represent the number of years in this context, which makes the formula become:
\[ A = P \cdot (1 + r)^t \]
Given:
- \( P = $[/tex]600 \)
- [tex]\( r = 9\% = 0.09 \)[/tex] (as a decimal)
- [tex]\( t = 3 \)[/tex] years
Let's calculate:
[tex]\[ A = $600 \cdot (1 + 0.09)^3 \][/tex]
[tex]\[ A = $600 \cdot (1.09)^3 \][/tex]
[tex]\[ A = $600 \cdot 1.09 \cdot 1.09 \cdot 1.09 \][/tex]
[tex]\[ A = $600 \cdot 1.295029 \][/tex]
[tex]\[ A \approx $777.02 \][/tex]
Rounding off to the nearest cent, the future value of the account that pays interest compounded annually is approximately $777.02.
