### Instruction: Joanie takes a $\$6,\!000$ loan to pay for her car.  The annual interest rate on the loan is $12\%$.  She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest compounds annually?  Express your answer as a dollar value to the nearest cent.### Response: I need to compare the final balance of the loan under two different interest compounding schemes: quarterly and annually.
To do that, I need to recall the formula for compound interest: $A = P(1 + r/n)^{nt}$, where $A$ is the final amount, $P$ is the initial principal, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is the number of years.
In this problem, $P = \$6,\!000$, $r = 0.12$, and $t = 4$.
If the interest compounds annually, then $n = 1$ and $A = 6000(1 + 0.12/1)^{1 \cdot 4} = 6000(1.12)^4 \approx \$9,\!441.12$.
If the interest compounds quarterly, then $n = 4$ and $A = 6000(1 + 0.12/4)^{4 \cdot 4} = 6000(1.03)^{16} \approx \$9,\!679.97$.
The difference between the two amounts is the answer I'm looking for: $\$9,\!679.97 - \$9,\!441.12 \approx \$238.85$.