SOLUTION: Using the Loan Formula P_0= D(1-(1+r/k)^-Nk)` dividing by `(R/k) Kyle is considering putting a 1000 laptop purchase on their credit card, which has an interest rate of 24% compo

Algebra ->  Finance -> SOLUTION: Using the Loan Formula P_0= D(1-(1+r/k)^-Nk)` dividing by `(R/k) Kyle is considering putting a 1000 laptop purchase on their credit card, which has an interest rate of 24% compo      Log On


   

Question 1190544: Using the Loan Formula P_0= D(1-(1+r/k)^-Nk)` dividing by `(R/k)
Kyle is considering putting a 1000 laptop purchase on their credit card, which has an interest rate of 24% compounded monthly, how long will it take them to pay off the purchase if they make payments of 30 a month?
I am just stuck on the second step, if anyone can help thanks!!
Answer by math_tutor2020(3835) About Me  (Show Source):
You can put this solution on YOUR website!

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%281-%281%2Br%2Fk%29%5E%28-N%29%29%2F%28r%2Fk%29

1000+=+30%281-%281%2B0.02%29%5E%28-N%29%29%2F%280.02%29

1000%2A0.02+=+30%281-%281.02%29%5E%28-N%29%29

20+=+30%281-%281.02%29%5E%28-N%29%29

20%2F30+=+1-%281.02%29%5E%28-N%29

2%2F3+=+1-%281.02%29%5E%28-N%29

1-%281.02%29%5E%28-N%29+=+2%2F3

-%281.02%29%5E%28-N%29+=+2%2F3-1

-%281.02%29%5E%28-N%29+=+-1%2F3

%281.02%29%5E%28-N%29+=+1%2F3

log%28%28%281.02%29%5E%28-N%29%29%29+=+log%28%281%2F3%29%29

-N%2Alog%28%281.02%29%29+=+log%28%281%2F3%29%29

N+=+-log%28%281%2F3%29%29%2Flog%28%281.02%29%29

N+=+55.4781076387803

That rounds up N = 56

Note what happens when N = 55
P+=+D%281-%281%2Br%2Fk%29%5E%28-N%29%29%2F%28r%2Fk%29

P+=+30%281-%281%2B0.02%29%5E%28-55%29%29%2F%280.02%29

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

If we tried N = 56, then,
P+=+D%281-%281%2Br%2Fk%29%5E%28-N%29%29%2F%28r%2Fk%29

P+=+30%281-%281%2B0.02%29%5E%28-56%29%29%2F%280.02%29

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.
https://www.calculator.net/credit-card-calculator.html
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