Comparing Two Changing Quantities

A rideshare app offers two pricing plans.

Plan A charges a fixed booking fee plus a per mile rate. For a trip of $x$ miles, the cost in dollars is

$$y=2x+10$$

Plan B skips the booking fee but charges more per mile. Its cost is

$$y=5x$$

If you travel only a short distance, that $+10$ in Plan A can dominate and make it more expensive. If you travel far, the cheaper per mile rate in Plan A can catch up and eventually win. The question is where the switch happens and how to predict it without guessing.

In these equations, the number multiplying $x$ is the rate per mile. It tells how fast the dollars increase as miles increase. The constant term is the starting cost at $x=0$. It tells what you pay before you go anywhere.

For Plan A, the starting cost is $10 and each mile adds $2. For Plan B, the starting cost is $0 and each mile adds $5. The break even point is the trip length where both plans cost the same. Before that point one plan is cheaper. After that point the other plan is cheaper.

Try comparing the two plans at a few mileages and predict which plan is cheaper on different ranges of $x$.

Break even as a graph intersection

When two costs depend on miles, a graph turns the comparison into a picture. Put miles on the horizontal axis and dollars on the vertical axis. Each plan becomes a line.

A useful way to read the picture is to fix an $x$ value and look straight up. The line that is lower at that $x$ gives the cheaper cost. The break even mileage is the $x$ value where the two lines meet. At that point, both plans have the same $y$ value, meaning the same number of dollars.

Graph both lines and locate the intersection point $(x,y)$. Then interpret both coordinates in context.