Suppose you make a purchase for $1000 using your credit card. Also, let%u2019s a

Question

Suppose you make a purchase for $1000 using your credit card. Also, let%u2019s assume that your credit card charges 24% annual interest rate and that it is compounded monthly. We will also assume that the minimum payment on the credit card is 5% of the total. How long will it take you pay off your purchase, assuming that you make only the minimum payment on the credit card.

Previous balance

End of Month

Previous balance

New interest (Previous balance*monthly interest) New total (Preview balance + new interest) Min. Payment (5% of new total) New balance (New total-min payment) 1 $1000 2 3 4 ...

Explanation / Answer

Monthly interest = 2%

Month 1: interest = 2%*1000 = 20

New total = 1000+20 = 1020

Min payment = 5%*1020 = 51

New balance = 1020-51 = 969

For month 2, previous balance = 969. Interest = 2%*969=19.38 and so on as explained above.... The below table gives the details for 4 months.

If you observe the pattern, let's say the previous balance was P. Then new interest = P*2%=0.02P and new total = P+0.02P = 1.02P.

Min payment = 5%*1.02P = 0.051P

So new balance = new total-min payment = 1.02P-0.051P = 0.969P

This can be seen in the above table also. Initial principal was 1000. New balance after month 1 was 1000*0.969 = $969. New balance after month 2 was 969*0.969 = 938.96. New balance after month 3 was 938.96*0.969 = 909.85. New balance after month 4 was 909.85*0.969 = 881.65.

So we have to multiply previous balance by 0.969 to get new balance. This is a geometric series with the first value being 969 and the ratio being 0.969. The nth value in this series is first value * ratio ^(n-1) = 969*0.969^(n-1), or in other words, the new balance at the end of month n is 969*0.969^(n-1).

For this to become 0.01, 969*0.969^(n-1) = 0.01. Solving this, we get n=365.59. So the new balance falls below 1 cent in the 366th month.

Min payment in month 2= 0.051P = 0.051 * new balance at end of month 1

Min payment in month 3= 0.051P = 0.051 * new balance at end of month 2

Min payment in month 4= 0.051P = 0.051 * new balance at end of month 3..and so on..

or in other words, Min payment in month n= 0.051P = 0.051 * new balance at end of month (n-1)

But we know formula for new balance at month n = 969*0.969^(n-1). So formula for new balance at month n-1 = 969*0.969^(n-2).

So Min payment in month n= 0.051P = 0.051 * 969*0.969^(n-2) = 49.419*0.969^(n-2)

Again, this is a geometric series with the first value being 51 and ratio being 0.969

Summation to nth term is first term * (1-ratio ^ n) / (1- ratio)

Summation till 366th term is 51 * (1-0.969^366) / (1-0.969) = $1645.15

Hope you liked this detailed explanation ! Let me know in case of any queries.

End of month Previous balance New interest (Previous balance*monthly interest) New total (Previous balance + new interest) Min payment (5% of new total) New balance (new total-min payment) 1 1000.00 20.00 1020.00 51.00 969.00 2 969.00 19.38 988.38 49.42 938.96 3 938.96 18.78 957.74 47.89 909.85 4 909.85 18.20 928.05 46.40 881.65

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