Let's analyze the given exponential expression step by step:
[tex]\[ 5,000\left(1+\frac{0.04}{12}\right)^{12 t} \][/tex]
This expression follows the general form of the compound interest formula:
[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 (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed for.
In Rick's expression:
- [tex]\( P = 5,000 \)[/tex] dollars (the principal amount).
- [tex]\( r = 0.04 \)[/tex] (the annual interest rate, 4% in decimal form).
- [tex]\( n = 12 \)[/tex] (the number of times interest is compounded per year, as the interest is compounded monthly).
- [tex]\( t \)[/tex] is the number of years (in this case, 3 years, but it is generally represented as [tex]\( t \)[/tex]).
The value that represents the number of times per year that interest is compounded is the value of [tex]\( n \)[/tex], which is found as the denominator in the fraction within the expression [tex]\( \left(1+\frac{0.04}{12}\right) \)[/tex].
Therefore, the value in the expression that represents the number of times per year that interest is compounded is 12. This indicates that the interest is compounded monthly.
