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Cassandra is repaying an installment loan of \$3500 with 20 equal monthly payments of \$196 each. What is the annual percentage rate of the loan?

If there are N payments of \$M to pay off a principal of \$P, let

q = log2(1+1/N) = log(1+1/N)/log(2)

then a good approximation for the interest rate per payment is given by

r = ((1+M/P)^(1/q) - 1)^q - 1

In our case,
q = 0.070389
1/q = 14.206699
M/P = .0056

r = 0.011029

annual interest rate is 12r = 13.23%

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