Question 1190544
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N = number of months remaining  
P = loan balance when N months remain
D = 30 = monthly payment
r = 0.24 = decimal form of the 24% interest rate
k = 12 = number of payments per year (aka monthly)


r/k = 0.24/12 = 0.02


If we plugged in P = 1000 and solved for N, we would find the month number when the balance is $1000
In other words, it would tell us how long the loan is.


{{{P = D(1-(1+r/k)^(-N))/(r/k)}}}


{{{1000 = 30(1-(1+0.02)^(-N))/(0.02)}}}


{{{1000*0.02 = 30(1-(1.02)^(-N))}}}


{{{20 = 30(1-(1.02)^(-N))}}}


{{{20/30 = 1-(1.02)^(-N)}}}


{{{2/3 = 1-(1.02)^(-N)}}}


{{{1-(1.02)^(-N) = 2/3}}}


{{{-(1.02)^(-N) = 2/3-1}}}


{{{-(1.02)^(-N) = -1/3}}}


{{{(1.02)^(-N) = 1/3}}}


{{{log(((1.02)^(-N))) = log((1/3))}}}


{{{-N*log((1.02)) = log((1/3))}}}


{{{N = -log((1/3))/log((1.02))}}}


{{{N = 55.4781076387803}}}


That rounds up N = 56


Note what happens when N = 55
{{{P = D(1-(1+r/k)^(-N))/(r/k)}}}


{{{P = 30(1-(1+0.02)^(-55))/(0.02)}}}


{{{P = 995.24362567013}}}
which is a bit short of $1,000


If we tried N = 56, then,
{{{P = D(1-(1+r/k)^(-N))/(r/k)}}}


{{{P = 30(1-(1+0.02)^(-56))/(0.02)}}}


{{{P = 1005.14080948052}}}
which is a bit over the target. So this confirms that N is somewhere between 55 and 56
We round N up to the nearest whole number so that we guarantee to make the final monthly payment.



This is a handy calculator to check your work.
<a href = "https://www.calculator.net/credit-card-calculator.html">https://www.calculator.net/credit-card-calculator.html</a>
Type $1000 for the balance, 24% for the rate, and $30 for the min payment. Click the "minimum payment" radio button and click "calculate".
The calculator should say "It will take 4 years and 8 months to payoff the balance"
This is equivalent to 56 months because 12*4+8 = 48+8 = 56.


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Answer:  56 months
This is equivalent to 4 years, 8 months
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