James invests \$20,000 in an account that offers a compound interest rate of 8.3% per year. Which of the following is the correct equation for how much James will have in year 6?

A. [tex]A(6) = 20,000 \cdot (1 + 0.83)^{6-1}[/tex]

B. [tex]A(6) = 20,000 \cdot (1 + 0.083)^6[/tex]

C. [tex]A(6) = 20,000 \cdot (1 + 0.083)^{6+1}[/tex]

D. [tex]A(6) = 20,000 \cdot (1 + 0.083)^{6-1}[/tex]



Answer :

To determine the amount James will have in his account after 6 years, we need to use the compound interest formula. The compound interest formula is:

[tex]\[ A(t) = P \cdot (1 + r)^t \][/tex]

where:
- [tex]\( A(t) \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested for in years.

Given the information:
- The principal amount [tex]\( P \)[/tex] is \[tex]$20,000. - The annual interest rate \( r \) is 8.3%, which can be written as a decimal \[0.083\]. - The duration \( t \) is 6 years. Substituting these values into the formula, we get: \[ A(6) = 20,000 \cdot (1 + 0.083)^6 \] This matches option B from the given choices. Therefore: \[ \text{Correct equation: } A(6) = 20,000 \cdot (1 + 0.083)^6 \] Using this equation and solving it, James will have approximately \$[/tex]32,270.13 in his account after 6 years.