Question

71. Suppose that you currently have one credit card with a balance of $10,000 at an annual rate of 24.00% interest. You have stopped adding any additional charges to this card and are determined to pay off the balance. You have worked out the formulabn = b0r n − R(1 + r+r2 +....+ r n−1), where b0 is the initial balance, bn is the balance after you have maden payments, r= 1 + i, wherei is the monthly interest rate, and R is the amount you are planning to pay each month.
c. How long does it take to pay off this debt if you can afford to pay a constant $250 per month? Give the
answer in years and months.

Answer 1

Answer:

It will take 71,35 months, wich can be rounded to 72, or 6 years

Step-by-step explanation:

The debt will be paid when $b_{n} = 0$, the formula can be written as:

$b_{n} = b_{0} r^{n} - R (\frac{1-r^{n} }{1-r} ) = 0$

Solving for n:

$\frac{R}{1-r} = \frac{b_{0}r^{n}}{1-r^{n} } = \frac{b_{0}r^{n}}{r^{n}(\frac{1}{r^{n} } -1) }$

$\frac{1}{r^{n}} = \frac{b_{0}(1-r)}{R}+1= 0,28$

$r^{n}= \frac{1}{0,28} = 3,5714$

Solving the exponential equation:

$r^{n} = 3,5714 => n = log_{r} (3,5714) = \frac{log(3,5714)}{log(r)} =71,35$

It will take 71,35 months, wich can be rounded to 72, or 6 years