Dale takes out a loan of 8,000 dollars at a 15.2 percent interest rate compounded semi-annually. If he pays off the loan in 3 years, how much will he end up paying?

Step-by-step solution

1. 1

   Identify the four inputs of the compound interest formula.

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

   Here $P=8000$, the annual rate $r=0.152$, the compounding frequency $n=2$ (semi-annually means twice per year), and $t=3$ years. The word 'semi-annually' controls $n$ and nothing else.

2. 2

   Convert the annual rate into a period rate.

   $\frac{r}{n}=\frac{0.152}{2}=0.076$

   Each six-month period adds $7.6\%$. Using the full $15.2\%$ per period would roughly double the interest and land on the decoy answer near \18{,}698$.

3. 3

   Count the compounding periods.

   $nt=2\times3=6$

   Six periods, not three — the exponent counts periods, not years.

4. 4

   Evaluate the growth factor and the amount.

   $(1.076)^6=1.5519353585$

   $A=8000\times1.5519353585=12{,}415.4829$

   Rounded to the nearest cent, A=\12{,}415.48$, which is choice B.

5. 5

   Check by compounding period by period. Multiplying by $1.076$ six times: $8608.00$, $9262.21$, $9966.14$, $10723.56$, $11538.55$, $12415.48$. The final figure matches ✓, and the first intermediate value \8{,}608.00$ is exactly decoy D — that decoy is the balance after only one period.