The equation [tex][tex]$A=2,400\left(1+\frac{0.031}{4}\right)^{21}$[/tex][/tex] represents the amount of money earned on a compound interest savings account. What does the value 0.031 represent?

A. The value 0.031 represents the interest rate, which means the annual compounded interest rate is [tex][tex]$3.1 \%$[/tex][/tex].
B. The value 0.031 represents the interest rate, which means the annual compounded interest rate is [tex][tex]$0.31 \%$[/tex][/tex].
C. The value 0.031 represents the investment period, which means the investment is invested for 0.031 years.
D. The value 0.031 represents the investment period, which means the investment is invested for 3.1 years.



Answer :

Let's break down the given equation: [tex]\(A=2,400\left(1+\frac{0.031}{4}\right)^{21}\)[/tex].

This equation appears to be in the standard 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 [tex]\(n\)[/tex] periods,
- [tex]\(P\)[/tex] is the principal amount (initial investment),
- [tex]\(r\)[/tex] is the annual interest rate,
- [tex]\(n\)[/tex] is the number of times interest is compounded per year,
- [tex]\(t\)[/tex] is the time the money is invested for in years.

From the equation:
- [tex]\(P\)[/tex] is 2400, representing the initial investment.
- The expression [tex]\(\left(1 + \frac{0.031}{4}\right)\)[/tex] implies that the annual interest rate is [tex]\(0.031\)[/tex], and it is compounded quarterly (4 times a year).
- The exponent [tex]\(21\)[/tex] represents the total number of compounding periods.

Now, focusing on the value [tex]\(0.031\)[/tex]:

The fraction [tex]\(\frac{0.031}{4}\)[/tex] indicates that [tex]\(0.031\)[/tex] is divided by 4, which fits the structure of the compound interest formula. In this context, [tex]\(0.031\)[/tex] is the annual interest rate before being divided by the number of compounding periods per year. Therefore, the value [tex]\(0.031\)[/tex] represents the annual interest rate.

Given this analysis, the annual interest rate [tex]\(0.031\)[/tex] translates to [tex]\(3.1\%\)[/tex].

Thus, the correct interpretation is:
The value 0.031 represents the interest rate, which means the annual compounded interest rate is [tex]\(3.1\%\)[/tex].