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The compound interest formula is given below, where [tex]\( R \)[/tex] is the future value of the investment, [tex]\( r \)[/tex] is the annual interest rate (as a decimal), [tex]\( n \)[/tex] is the number of times interest is compounded each year, [tex]\( t \)[/tex] is the number of years the principal is invested, and [tex]\( P \)[/tex] is the principal, which represents the original amount of money invested.

Mary invested \[tex]$1000 at 5% annual interest in an account that compounds interest 4 times per year. If she kept her money in the account for 5 years, how much will her future value be?

\[ P\left(1+\frac{r}{n}\right)^{nt} = R \]

A. \$[/tex]1,057.33
B. \[tex]$2,413.16
C. \$[/tex]5,254.73
D. \[tex]$1,282.04
E. \$[/tex]1,525.47

I don't know.



Answer :

GinnyAnswer
To find the future value of Mary's investment using the compound interest formula, we follow these steps:

1. Identify the given values:
- Principal amount ([tex]\( P \)[/tex]) = \[tex]$1000 - Annual interest rate (\( r \)) = 0.05 (since 5% as a decimal is 0.05) - Number of times interest is compounded per year (\( n \)) = 4 - Number of years (\( t \)) = 5 2. Write the compound interest formula: \[ R = P \left(1 + \frac{r}{n}\right)^{nt} \] 3. Substitute the given values into the formula: \[ R = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 5} \] 4. Calculate the value inside the parentheses: \[ 1 + \frac{0.05}{4} = 1 + 0.0125 = 1.0125 \] 5. Calculate the exponent: \[ nt = 4 \cdot 5 = 20 \] 6. Raise the value inside the parentheses to the power of 20: \[ (1.0125)^{20} \] 7. Multiply this result by the principal amount (\( P \)): \[ R = 1000 \times (1.0125)^{20} \approx 1000 \times 1.2820372317085844 \] 8. Calculate the final amount: \[ R \approx 1282.0372317085844 \] So, the future value of Mary's investment after 5 years will be approximately \$[/tex]1,282.04.

Based on the calculations, the correct answer is:
[tex]\[ \$ 1,282.04 \][/tex]

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