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.