Suppose you owe $16,000 on a credit card that carries an APR of 24%. Because the balance is so high, you choose to stop charging and pay off the card. You can afford to make only the minimum monthly payment, which is 5% of the balance. Then your balance after t months is given by
B = 16,000(1.02 ✕ 0.95)t
 dollars.
How many payments will you have to make to get the balance below $400? (Enter the smallest such number of payments as an integer.)

We are given the formula for the balance after $ t $ months:

\[
B = 16,000(1.02 \times 0.95)^t
\]

First, simplify the expression inside the parentheses:

\[
1.02 \times 0.95 = 0.969
\]

So the balance formula becomes:

\[
B = 16,000 \times (0.969)^t
\]

We want to find the smallest number of months, $ t $, such that the balance $ B $ is below $400. This means we need to solve the inequality:

\[
16,000 \times (0.969)^t < 400
\]

### Step 1: Divide both sides by 16,000

\[
(0.969)^t < \frac{400}{16,000}
\]

Simplifying the right-hand side:

\[
(0.969)^t < 0.025
\]

### Step 2: Take the natural logarithm of both sides

Taking the natural logarithm of both sides helps to solve for $ t $:

\[
\ln((0.969)^t) < \ln(0.025)
\]

Using the logarithmic property $ \ln(a^b) = b \ln(a) $, we get:

\[
t \ln(0.969) < \ln(0.025)
\]

### Step 3: Solve for $ t $

Now divide both sides by $ \ln(0.969) $. Since $ \ln(0.969) $ is negative, we must reverse the inequality:

\[
t > \frac{\ln(0.025)}{\ln(0.969)}
\]

Using a calculator:

\[
\ln(0.025) \approx -3.6889
\]
\[
\ln(0.969) \approx -0.0315
\]

Now compute $ t $:

\[
t > \frac{-3.6889}{-0.0315} \approx 117.07
\]

Since $ t $ must be an integer, the smallest number of months is $ t = 118 $.

### Conclusion:

You will need to make **118 payments** to get the balance below $400.

Would you like more details on the steps, or do you have any further questions?

Here are 5 related questions to consider:
1. What happens if you increase your monthly payment slightly?
2. How does the interest rate (APR) affect the total time to pay off the debt?
3. How would the formula change if the minimum payment were a fixed dollar amount rather than a percentage?
4. What is the total amount of money paid over 118 months?
5. How does compound interest factor into long-term debt repayment?

**Tip:** Paying more than the minimum monthly payment will significantly reduce the time it takes to pay off a debt.

Suppose you owe $16,000 on a credit card that carries an APR of 24%. Because the balance is so high, you choose to stop charging and pay off the card. You can afford to make only the minimum monthly payment, which is 5% of the balance. Then your balance after t months is given by B = 16,000(1.02 ✕ 0.95)^t dollars. How many payments will you have to make to get the balance below $400? (Enter the smallest such number of payments as an integer.)

Find out how many payments are required to reduce a $16,000 credit card balance with an APR of 24% and minimum monthly payments of 5%. This detailed solution uses exponential decay and logarithms, perfect for high school students (Grades 10-12).

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