Question

a business owner receives a $15,000 loan with 13% interest, charged at the end of each year.

match the expression that represents the total amount owed at the end of each year if nothing was paid on the balance.

column a\t\t\t\t\tcolumn b

1. ____ after 1 year\t\t\ta. $15,000(0.13)^t$
2. ____ after 2 years\t\t\tb. $15,000(1.13)^t$
3. ____ after 6 months\t\t\tc. $15,000(0.13)^2$
4. ____ after $t$ years\t\t\td. $15,000(0.13)^1$

\t\t\t\t\t\te. $15,000(1.13)^2$
\t\t\t\t\t\tf. $15,000(0.13)^{\frac{1}{2}}$
\t\t\t\t\t\tg. $15,000(1.13)^{\frac{1}{2}}$
\t\t\t\t\t\th. $15,000(1.13)^1$

Explanation:

Step1: Recall compound interest formula

The total amount owed with annual compound interest is $A = P(1+r)^t$, where $P$ is principal, $r$ is annual interest rate, $t$ is time in years.

Step2: Match 1 year amount

Substitute $P=15000$, $r=0.13$, $t=1$:
$A = 15000(1+0.13)^1 = 15000(1.13)^1$

Step3: Match 2 years amount

Substitute $P=15000$, $r=0.13$, $t=2$:
$A = 15000(1+0.13)^2 = 15000(1.13)^2$

Step4: Match 6 months amount

6 months = $\frac{1}{2}$ year. Substitute $t=\frac{1}{2}$:
$A = 15000(1+0.13)^\frac{1}{2} = 15000(1.13)^\frac{1}{2}$

Step5: Match $t$ years amount

Substitute general $t$:
$A = 15000(1+0.13)^t = 15000(1.13)^t$

Answer:

1. h. $15,000(1.13)^1$
2. e. $15,000(1.13)^2$
3. g. $15,000(1.13)^\frac{1}{2}$
4. b. $15,000(1.13)^t$