To calculate the interest that Mark will pay on his credit card balance, we need to consider the APR (Annual Percentage Rate), the compounding period, and the duration for which the balance is outstanding.
First, let's calculate the interest for one month. The formula to calculate the interest for a given period with daily compounding is:
Interest = Principal * (1 + (r/n))^(n*t) - Principal
Where:
- Principal is the initial balance,
- r is the annual interest rate (expressed as a decimal),
- n is the number of compounding periods per year,
- t is the time in years.
In this case, Principal = $4,000, r = 13% (0.13 as a decimal), n = 365 (daily compounding), and t = 1/12 (one month is 1/12 of a year).
Calculating the interest for one month:
Interest = $4,000 * (1 + (0.13/365))^(365*(1/12)) - $4,000
Simplifying the equation:
Interest = $4,000 * (1.0003561643835616) - $4,000
Interest ≈ $4,001.42 - $4,000
Interest ≈ $1.42
Therefore, if Mark doesn't pay off his balance in full after one month, he will pay approximately $1.42 in interest.
Next, let's calculate the interest for six months. Using the same formula and plugging in t = 6/12 (six months is half a year):
Interest = $4,000 * (1 + (0.13/365))^(365*(6/12)) - $4,000
Simplifying the equation:
Interest = $4,000 * (1.0003561643835616)^(182.5) - $4,000
Interest ≈ $4,000 * (1.071798546166585) - $4,000
Interest ≈ $4,287.19 - $4,000
Interest ≈ $287.19
Therefore, if Mark doesn't make any payments for six months, he will pay approximately $287.19 in interest.