$Fixed installment loans?$

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$Fixed installment loans?$

You plan to purchase items for$500and will finance the cost with a fixed installment loan. The seller tells you that you will have to pay 20 dollars per month for 36 months.

a) What is the finance charge?

b)What is the interest rate? (Round to the nearest hundredth)

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a) The toal payment is
\[36 \times 20= \$ 720.\]
So the finance charge is
\[720-500=\$ 220.\]
(b) If $P_n$ is tjhe money owed to the bank after $n$ months. Then we have the formula
\[ (*) P_n=(1+r)^nP_0 - M \frac{(1+r)^n-1}{r},\]
where $M$ is the onthly payments, $r$ is the monthly interest rate, and $P_0$ is the initial loan amoiunt. Substituting the given values in formula (*) we get
\[0=P_{36}=(1+r)^{36}500 - 20 \frac{(1+r)^{36}-1}{r}.\]
So to find the interest rate we should solve
\[(1+r)^{36}500 - 20 \frac{(1+r)^{36}-1}{r}=0,\]

This could be done by graphing the function and finding its $x$-intercept. Doing so we get
\[r=0.02121. \]
So the yearly interest rate is
\[I=12*0.02121=0.25452=25.452 \%.\]
So the interest rate is about 25.46 %.

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This took me about 30 minutes to write. Please offer higher amounts if future, otherwise users may not have enough incentive to accept.
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You can use Desmos for graphing functions: https://www.desmos.com/calculator

$Fixed installment loans?$

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