Example 11 from Precalculus, 2.1 Linear Functions

Using a Linear Function to Calculate Salary Plus Commission

Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya’s weekly income, $I,$ depends on the number of new policies, $n,$ he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for $I (n),$ and interpret the meaning of the components of the equation.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The given information gives us two input-output pairs: $(3,760)$ and $(5,920) .$ We start by finding the rate of change.

$\begin{array}{l} m = \frac{920 - 760}{5 - 3} \\ = \frac{$160}{2 policies} \\ = $80 per policy \end{array}$
Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is $80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold during the week.

We can then solve for the initial value.

$\begin{array}{l} I (n) = 80 n + b \\ 760 = 80 (3) + b & when n = 3, I (3) = 760 \\ 760 - 80 (3) = b \\ 520 = b \end{array}$
The value of $b$ is the starting value for the function and represents Ilya’s income when $n = 0,$ or when no new policies are sold. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold.

We can now write the final equation.

$I (n) = 80 n + 520$

Our final interpretation is that Ilya’s base salary is $520 per week and he earns an additional $80 commission for each policy sold.

How did it go?

Solution

Solution