jmc

For the compounded interest, we use the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$, where $A$ is the end balance, $P$ is the principal, $r$ is the interest rate, $t$ is the number of years, and $n$ is the number of times compounded in a year.

First we find out how much he would owe in $5$ years, which is $$\$10,000{\left(1+\frac{0.1}{4}\right)}^{4\cdot 5}\approx \$16,386.16$$

He pays off half of it in $5$ years, which is \frac{\16,\!386.16}{2}=\$8193.08$ He has \8193.08$lefttobecompoundedoverthenext$5$years.Thisthenbecomes$\8193.08+\13,\!425.32=\$21,618.40$ in ten years if he chooses the compounding interest.

For the simple interest, he would have to pay $0.12\cdot 10000=1200$ dollars per year. This means he would have to pay a total of $10000+10\cdot 1200=22000$ dollars in ten years.

Therefore, he should choose the compounded interest and save \22000-\21618.40=\381.6 \approx \boxed{382 \text{ dollars}}$.