The value [tex]\( 0.031 \)[/tex] in the equation [tex]\( A = 2,400 \left( 1 + \frac{0.031}{4} \right)^{el} \)[/tex] represents the annual interest rate. Let's break it down in detail:
1. Equation Explanation: The given equation is for calculating the future amount [tex]\( A \)[/tex] in a compound interest scenario. The formula for compound interest can be expressed as:
[tex]\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\][/tex]
where:
- [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 in years,
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
2. Interpretation of Variables:
- In the given equation [tex]\( A = 2,400 \left( 1 + \frac{0.031}{4} \right)^{el} \)[/tex],
- The principal amount [tex]\( P \)[/tex] is [tex]\( 2,400 \)[/tex] dollars.
- The term [tex]\(\frac{0.031}{4}\)[/tex] indicates that the annual interest rate [tex]\( r \)[/tex] is [tex]\( 0.031 \)[/tex], and it is compounded quarterly ([tex]\( n = 4 \)[/tex]).
- The exponent [tex]\( el \)[/tex] appears to replace [tex]\( nt \)[/tex], suggesting some substitution was made there and needs further context.
3. Identifying the Annual Interest Rate:
- The value [tex]\( 0.031 \)[/tex] specifically is found in the fraction [tex]\(\frac{0.031}{4}\)[/tex], which corresponds to the interest rate divided by the number of compounding periods per year ([tex]\( n = 4 \)[/tex]).
- This clearly establishes that [tex]\( 0.031 \)[/tex] is the annual interest rate.
4. Converting to Percentage:
- To express the annual interest rate as a percentage, multiply [tex]\( 0.031 \)[/tex] by 100.
[tex]\[
0.031 \times 100 = 3.1\%
\][/tex]
Therefore, the value [tex]\( 0.031 \)[/tex] represents the annual interest rate. When expressed as a percentage, the annual compounded interest rate is [tex]\( 3.1\% \)[/tex].
Thus, the correct interpretation from the choices provided is:
The value [tex]\( 0.031 \)[/tex] represents the interest rate, which means the annual compounded interest rate is [tex]\( 3.1\% \)[/tex].
