You ran a little short on your spring break vacation, so you put $1,000 on your other credit card. You can afford only the minimum payment of $60 per quarter. The interest rate on the credit card is 18 per year compounded quarterly. How many years will you need to pay off the $1,000?
Question
You ran a little short on your spring break vacation, so you put 60 per quarter. The interest rate on the credit card is 18 per year compounded quarterly. How many years will you need to pay off the $1,000?
Solution
To solve this problem, we need to use the formula for the payment of an ordinary annuity:
P = (r*PV) / (1 - (1 + r)^-n)
Where:
- P is the payment per period ($60)
- r is the interest rate per period (18% per year compounded quarterly means 4.5% per quarter, or 0.045 in decimal form)
- PV is the present value ($1000)
- n is the number of periods (which we're trying to find)
We can rearrange the formula to solve for n:
n = -log(1 - (r*PV/P)) / log(1 + r)
Substituting the given values:
n = -log(1 - (0.045*1000/60)) / log(1 + 0.045)
Calculating this gives approximately 24.6 quarters. Since there are 4 quarters in a year, this is equivalent to approximately 6.15 years.
Therefore, it will take you a little over 6 years to pay off the 60 per quarter.
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