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 $m$ :

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

So, $f(x)=0.5x+b$ . We substitute a known point to find b:

$\begin{align*}f(x) &= 0.5x+b \\\implies 60& = 0.5(50)+b \\60 &= 25+b \\b &= 35\end{align*}$

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:

$\begin{align*}f(x) &= 0.5x+35 \\f(45) &= 0.5(45)+35 \\&= 22.5+35 \\&=57.5\end{align*}$

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