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If [tex]$\$[/tex]300[tex]$ is invested at a rate of $[/tex]5\%[tex]$ per year and is compounded quarterly, how much will the investment be worth in 20 years?

Use the compound interest formula: $[/tex]A = P\left(1 + \frac{r}{n}\right)^{nt}[tex]$.

A. $[/tex]\[tex]$810.45$[/tex]
B. [tex]$\$[/tex]515.28[tex]$
C. $[/tex]\[tex]$384.61$[/tex]
D. [tex]$\$[/tex]109.67$


Sagot :

To determine how much a \[tex]$300 investment will be worth in 20 years with an annual interest rate of 5% compounded quarterly, we can use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( P \) is the principal amount (initial investment), - \( r \) is the annual interest rate (as a decimal), - \( n \) is the number of times the interest is compounded per year, - \( t \) is the number of years the money is invested for, - \( A \) is the amount of money accumulated after n years, including interest. For this problem: - \( P = 300 \) - \( r = 0.05 \) (since 5% as a decimal is 0.05) - \( n = 4 \) (because the interest is compounded quarterly) - \( t = 20 \) Now let's plug these values into the formula: \[ A = 300 \left(1 + \frac{0.05}{4}\right)^{4 \times 20} \] Simplify the fraction inside the parentheses: \[ \frac{0.05}{4} = 0.0125 \] So the formula now becomes: \[ A = 300 \left(1 + 0.0125\right)^{80} \] Add inside the parentheses: \[ 1 + 0.0125 = 1.0125 \] So we have: \[ A = 300 \left(1.0125\right)^{80} \] Evaluating the expression: \[ (1.0125)^{80} \approx 2.701484941 \] Finally, multiply by the principal amount: \[ A = 300 \times 2.701484941 \approx 810.45 \] So, the investment will be worth approximately \$[/tex]810.45 in 20 years. Therefore, the correct answer is:

[tex]\[ \$810.45 \][/tex]
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