Welcome back to Professor Baker’s Math Class! Today, we are tackling a topic that affects almost everyone's daily life: Personal Finance. Specifically, in Section 4-4, we are looking at Credit Cards and the mathematics of paying off consumer debt.

Understanding the fine print of a credit card statement isn't just good math—it is a vital life skill. Let's break down the key formulas from our class notes to see exactly how banks calculate what you owe.

1. Calculating the Finance Charge

One of the most important concepts to master is determining the amount subject to finance charges. According to our text (Quantitative Literacy: Thinking Between the Lines), the calculation generally follows this order:

  • Start with your Previous Balance.
  • Subtract any Payments you made.
  • Add any new Purchases.

Once you have that amount, you apply the monthly interest rate. Remember, the APR (Annual Percentage Rate) must be divided by 12 to get the monthly rate ($r$).

$$ \text{Finance Charge} = \left( \text{Previous Balance} - \text{Payments} + \text{Purchases} \right) \times \frac{\text{APR}}{12} $$

Example: If you have a balance subject to charges of $700 and an APR of 22.8%, your monthly finance charge isn't just a random fee—it is specifically calculated as:

$$ \$700 \times \frac{0.228}{12} = \$700 \times 0.019 = \$13.30 $$

2. The Trap of the Minimum Payment

The most eye-opening part of this lesson is the Minimum Payment Balance Formula. This formula helps us predict what your balance will be after $t$ months if you stop making new charges and only pay the minimum required percentage ($m$).

The formula creates an exponential pattern, showing that your balance decreases very slowly over time:

$$ \text{Balance after } t \text{ payments} = \text{Initial Balance} \times [(1+r)(1-m)]^t $$

Where:

  • $r$ is the monthly interest rate (decimal).
  • $m$ is the minimum payment percentage (decimal).
  • $t$ is the number of monthly payments.

This explains why paying only the minimum can keep you in debt for years! As we saw in the class slides, even after 5 years (60 months) of payments on a large balance, the principal goes down much slower than you might expect.

3. How to Get Out of Debt

So, how do we solve for $t$ (time) to see when the balance will be paid off? We have two main methods:

  1. Logarithms: We can solve the exponential equation using logs to find the exact month the balance drops below a certain amount (like $100).
  2. Trial and Error: Using a calculator to plug in different values for $t$ until you find the answer.

Alternatively, treating your credit card debt like an installment loan (Section 4.2) allows you to calculate a fixed monthly payment that will clear the debt in a specific timeframe (e.g., 2 years). This is usually a much better financial strategy than sticking to minimum payments.

Make sure to review the attached PDF notes for the step-by-step examples on how to apply these formulas. Good luck with the assignment!