# Eden finances a purchase of \$611.03 by making monthly payments of \$26.17 for 2.5 years.  What annual interest rate, compounded monthly, is she being charged? You are asked to find the annual interest rate if the monthly payment on an installment loan is \$26.17 payed over 30 months for a total of \$611.03.

From the lender's point of view this is an annuity. We can use:

`"PMT"="PV"i/(1-(1+i)^(-n))` where pmt is the monthly payment, pv is the present value (the loan amount), n is the number of compounding periods, and `i=r/m` where r is the annual interest rate and m the number of times compounded per year.

`26.17=611.03 i/(1-(1+i)^(-30))` where `i=r/12` and we are looking for r.

There is no simple way to solve for i. With only 30 compounding periods, you could find each monthly term and add them up to solve for i. You could also try a guess and check method.

Usually you would use technology (a spreadsheet or TVM solver in a graphing calculator.)

In a TI-8X you go under Apps->Finance->TVM solver.

N=30,PV=611.03,Pmt=-26.17,FV=0,p/Y=c/y=12 and then solve for i.

You will get `i~~20.4%`

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The annual interest rate is approximately 20.4%

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