You have a credit card with a balance of 75443 at a 136 APR

You have a credit card with a balance of $754.43 at a 13.6% APR. You have $300.00 available each month to save or pay down your debts. a. How many months will it take to pay off the credit card if you only put half of the available money toward the credit card each month and make the payments at the beginning of the month? b. How many months will it take to pay off the credit card if you put all of the available money toward the credit card each month and make the payments at the beginning of the month?

Be sure to include in your response: • the answer to the original question • the mathematical steps for solving the problem demonstrating mathematical reasoning

Solution

B(n) = A(1 + i)^n - (P/i)[(1 + i)^n - 1]

where B is the balance after n payments are made, i is the monthly interest rate, P is the monthly payment and A is the initial amount of loan.

We require B(n) = 0...i.e. balance of 0 after n months.

so, 0 = A(1 + i)^n - (P/i)[(1 + i)^n - 1]

Then, with some algebraic juggling we get:

n = -[log(1 - (Ai/P)]/log(1 + i)

Now, payment is at the beginning of the month, so A = $754.43 - $150 => $604.43

Also, i = (13.6/100)/12 => 0.136/12 per month

i.e. n = -[log(1 - (604.43)(0.136/12)/150)]/log(1 + 0.136/12)

so, n = 4.15 months...i.e. 4 payments + remainder

b) Now we have A = $754.43 - $300 = $454.43 so,

n = -[log(1 - (454.43)(0.136/12)/300)]/log(1 + 0.136/12)

so, n = 1.54 months...i.e. 1 payment + remainder