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\tcolumn b
____ after 1 year\ta. $15,000(0.13)^2
____ after 2 years\tb. $15,000(1.13)^{\frac{1}{2}}
____ after 6 months\tc. $15,000(0.13)^{\frac{1}{2}}
____ after $t$ years\td. $15,000(1.13)^t$
\t\te. $15,000(0.13)^1$
\t\tf. $15,000(0.13)^t$
\t\tg. $15,000(1.13)^2$

Explanation:

Step1: Identify compound interest formula

The total amount owed with annual compound interest is given by $A = P(1+r)^t$, where $P$ is principal, $r$ is annual interest rate, $t$ is time in years. Here, $P=15000$, $r=0.13$.

Step2: Calculate amount after 1 year

Substitute $t=1$ into the formula:
$A = 15000(1+0.13)^1 = 15000(1.13)^1$

Step3: Calculate amount after 2 years

Substitute $t=2$ into the formula:
$A = 15000(1+0.13)^2 = 15000(1.13)^2$

Step4: Calculate amount after 6 months

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

Step5: Generalize for t years

Substitute general $t$ into the formula:
$A = 15000(1.13)^t$

Answer:

• After 1 year: (matches no listed exact option, but the correct expression is $15000(1.13)^1$; note: options e uses 0.13 which is incorrect, this is likely a typo, but following the valid compound interest logic)
• After 2 years: g. $15,000(1.13)^2$
• After 6 months: b. $15,000(1.13)^\frac{1}{2}$
• After $t$ years: d. $15,000(1.13)^t$

Note: The options for "After 1 year" do not include the correct expression $15000(1.13)^1$; option e uses 0.13 instead of 1.13 which is incorrect, as 0.13 only calculates the interest, not the total amount owed.