He actually borrowed P=21349-3000=18349 (present value)
Assume the monthly interest is i.
then future value due to loan:
F1=P(1+i)^n=18349(1+i)^(5*12)=18349(1+i)^60
future value from monthly payment of A=352
F2=A((1+i)^n-1)/i=352((1+i)^60-1)/i
Since F1=F2 for the same loan, we have
18349(1+i)^60=352((1+i)^60-1)/i
Simplify notation by defining R=1+i, then
18349(R^60)-352(R^60-1)/(R-1)=0
Simplify further by multiplication by (R-1)
f(R)=18349*R^60*(R-1)-352(R^60-1)=0
Solve for R by trial and error, or by iteration to get R=1.004732
The APR is therefore
12*(1.004732-1)=0.056784, or 5.678% approx.
