Question

Write an equation and interpret the solution to the following problem: Rental car A has a rate of an initial $100, plus $10 per day. Rental car B has a rate of an initial $50, plus $20 per day. After how many days are the rental car prices equal?

Answer 1

Let \( x \) represent the number of days for which the cars are rented.

For rental car A, the total cost can be represented by the equation:
\[
\text{Cost of A} = 100 + 10x
\]

For rental car B, the total cost can be represented by the equation:
\[
\text{Cost of B} = 50 + 20x
\]

To find the number of days after which the rental car prices are equal, we set the two equations equal to each other:
\[
100 + 10x = 50 + 20x
\]

Now, let's solve for \( x \):

1. Subtract \( 10x \) from both sides:
\[
100 = 50 + 10x
\]

2. Subtract 50 from both sides:
\[
50 = 10x
\]

3. Divide both sides by 10:
\[
x = 5
\]

### Interpretation of the Solution:
The solution \( x = 5 \) means that after 5 days, the cost of renting car A will be the same as the cost of renting car B. At this point, both rental cars will have the same total rental charge. For any rental period shorter than 5 days, car B will be cheaper, while for any rental period longer than 5 days, car A will become the more economical option.