A car is purchased for $31,000. Each year it loses 25% of its value. After how many years will the car be worth $6400 or less?

Sure, here is the step by step solution:

Step 1: Understand the problem. The car's value decreases by 25% each year. We need to find out when it will be worth $6400 or less.

Step 2: Set up the equation. The formula for depreciation is:

Value = Initial Value * (1 - Depreciation Rate) ^ Number of Years

In this case, the depreciation rate is 25% or 0.25, the initial value is $31,000, and we want to find out when the value will be$6400 or less.

So, we set up the equation as follows:

6400 >= 31000 * (1 - 0.25) ^ Number of Years

Step 3: Solve the equation. This is a logarithmic equation, so we'll need to use logarithms to solve it.

First, divide both sides by 31000 to simplify:

0.20645 >= (0.75) ^ Number of Years

Then, take the natural logarithm (ln) of both sides:

ln(0.20645) >= Number of Years * ln(0.75)

Finally, divide both sides by ln(0.75) to solve for the number of years:

Number of Years >= ln(0.20645) / ln(0.75)

Step 4: Calculate the result. Using a calculator, we find that:

Number of Years >= 7.99

Since we can't have a fraction of a year, we round up to the nearest whole number.

So, the car will be worth $6400 or less after 8 years.