If $\$700$ is invested at $9\%$ compounded $\newline$ (A) annually, $\newline$ (B) quarterly, $\newline$ (C) monthly, $\newline$ what is the amount after $6$ years? How much interest is earned? $\newline$ (A) If it is compounded annually, what is the amount? $\newline$ $\$1173.97$ (Round to the nearest cent.) $\newline$ How much interest is earned? $\newline$ $\$473.97$ (Round to the nearest cent.) $\newline$ (B) If it is compounded quarterly, what is the amount? $\newline$ $\$$ \square (Round to the nearest cent.)

Full solution

Q. If

\$700

is invested at

9\%

compounded

\newline

(A) annually,

\newline

(B) quarterly,

\newline

\newline

what is the amount after

6

years? How much interest is earned?

\newline

(A) If it is compounded annually, what is the amount?

\newline

\$1173.97

(Round to the nearest cent.)

\newline

How much interest is earned?

\newline

\$473.97

(Round to the nearest cent.)

\newline

(B) If it is compounded quarterly, what is the amount?

\newline

\$

\square (Round to the nearest cent.)

Identify values: Identify the values of $P$ , $r$ , $n$ , and $t$ for the quarterly compounding scenario. $\newline$ Principal amount ( $P$ ) = $\$700$ $\newline$ Interest rate ( $r$ ) = $0.09$ $\newline$ Compounding periods per year ( $n$ ) = $4$ (since interest is compounded quarterly) $\newline$ Time in years ( $t$ ) = $r$ $1$ years
Use compound interest formula: Use the compound interest formula $A = P(1 + (r/n))^{(nt)}$ to calculate the amount after $6$ years for quarterly compounding. $\newline$ Substitute $P = 700$ , $r = 0.09$ , $n = 4$ , and $t = 6$ into the formula. $\newline$ $A = 700(1 + (0.09/4))^{(4 \times 6)}$
Simplify expression and calculate: Simplify the expression inside the parentheses and calculate the exponent. $\newline$ $A = 700(1 + 0.0225)^{24}$ $\newline$ $A = 700(1.0225)^{24}$
Calculate value using calculator: Calculate the value of $(1.0225)^{24}$ using a calculator. $\newline$ $(1.0225)^{24} \approx 2.0398873$
Multiply principal amount: Multiply the principal amount by the calculated value to find the final amount. $\newline$ $A = 700 \times 2.0398873$ $\newline$ $A \approx 1427.92111$
Round final amount: Round the final amount to the nearest cent. $A \approx \$1427.92$
Calculate interest earned: Calculate the interest earned by subtracting the principal from the final amount. $\newline$ Interest earned = $A - P$ $\newline$ Interest earned = $1427.92 - 700$ $\newline$ Interest earned = $\$727.92$