Question

Joseph is going on a trip and he needs to rent a car. He looks online and finds two companies that offer different pricing options for car rentals.

Company A charges $0.25 per mile plus a $50 rental fee.
Company B charges $0.45 per mile plus a $20 rental fee.
What is the minimum number of miles that Joseph must drive in order for Company A to be a better buy? The company charges only for whole number mileage (not fractional increments of miles driven).

Answer 1

To determine the minimum number of miles that Joseph must drive for Company A to be a better buy than Company B, we can set up the cost equations for both companies based on the number of miles driven, \( x \).

The cost for Company A can be expressed as:
\[
\text{Cost}_A = 0.25x + 50
\]

The cost for Company B can be expressed as:
\[
\text{Cost}_B = 0.45x + 20
\]

We need to find the smallest integer \( x \) such that:
\[
\text{Cost}_A < \text{Cost}_B
\]
This gives us the inequality:
\[
0.25x + 50 < 0.45x + 20
\]

To solve for \( x \), we first isolate the terms involving \( x \):
\[
50 - 20 < 0.45x - 0.25x
\]
\[
30 < 0.20x
\]

Now, we divide both sides by 0.20 to solve for \( x \):
\[
x > \frac{30}{0.20}
\]
\[
x > 150
\]

This means \( x \) must be greater than 150. Since we are looking for the minimum number of miles, the smallest whole number that satisfies this inequality is:
\[
x = 151
\]

To confirm, we should compare the costs at \( x = 151 \):
- For Company A:
\[
\text{Cost}_A = 0.25(151) + 50 = 37.75 + 50 = 87.75
\]
- For Company B:
\[
\text{Cost}_B = 0.45(151) + 20 = 67.95 + 20 = 87.95
\]

Since \( 87.75 < 87.95 \), Company A is indeed the better option for 151 miles.

Thus, the minimum number of miles Joseph must drive for Company A to be a better buy is:
\[
\boxed{151}
\]