2530.73 – Strange Salary

Many years ago, a teacher’s salary was based on the number of students in the teacher’s class. Suppose a salary of $60 per month was to be paid for a class of 50 students, and $50 per month was to be paid for a class of 30 students. If the actual enrollment was 45 students, how much should the teacher be paid each month?

Solution

We’ll treat salary as a linear function $f(x)$ where $x$ is the number of students and use the points (50,60) and (30,50) to find the equation for $f(x)$ :

$f(x)=mx+b$

There is a formula for the slope $m$ :

$m = \frac{\Delta y}{\Delta x} = \frac{60-50}{50-30} = \frac{10}{20}=0.5$

So,

$f(x)=0.5x+b$

We substitute a known point to find b:

$f(x)=0.5x+b \rightarrow 60=0.5(50)+b \rightarrow 60=25+b \rightarrow b=35$

This means that

$f(x)=0.5x+35$

All that’s left is to substitute the number of students into the equation to find the appropriate salary:

$f(x)=0.5x+35 \rightarrow f(x)=0.5(45)+35 \rightarrow f(x)=22.5+35=57.5$

The teacher should be paid $57.50 per month for a class of 45 students.