Steven opened a savings account and deposited $\$100.00$ as principal. The account earns $9\%$ interest, compounded annually. What is the balance after $5$ years? $\newline$ Use the formula $A = P(1 + \frac{r}{n})^{nt}$ , where $A$ is the balance (final amount), $P$ is the principal (starting amount), $r$ is the interest rate expressed as a decimal, $n$ is the number of times per year that the interest is compounded, and $t$ is the time in years. $\newline$ Round your answer to the nearest cent. $\newline$ $\$$ ____

Full solution

Q. Steven opened a savings account and deposited

\$100.00

as principal. The account earns

9\%

interest, compounded annually. What is the balance after

5

years?

\newline

Use the formula

A = P(1 + \frac{r}{n})^{nt}

, where

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

____

Identify Variables: Identify the values for the variables in the compound interest formula $A = P(1 + \frac{r}{n})^{nt}$ . Here, $P = \$100$ (the principal), $r = 9\%$ or $0.09$ (the interest rate as a decimal), $n = 1$ (since the interest is compounded annually), and $t = 5$ years (the time period).
Substitute Values: Substitute the values into the compound interest formula. $A = 100(1 + (0.09)/(1))^{(1\times5)}$
Simplify Expression: Simplify the expression inside the parentheses and then calculate the exponent. $\newline$ $A = 100(1 + 0.09)^5$ $\newline$ $A = 100(1.09)^5$
Calculate Exponent: Calculate the value of $(1.09)^5$ . $(1.09)^5 \approx 1.53862$ (rounded to five decimal places for intermediate calculation)
Find Final Balance: Multiply the principal by the result from Step $4$ to find the final balance. $\newline$ $A = 100 \times 1.53862$ $\newline$ $A \approx 153.862$
Round Final Balance: Round the final balance to the nearest cent. $A \approx \$(153.86)$