Question

In the following exercises, translate to a system of equations and solve.

Ashanti has been offered positions by two phone companies. The first company pays a salary of \($22,000$plus a commission of \)$100$for each contract sold. The second pays a salary of \($28,000$ plus a commission of \)$25$ for each contract sold. How many contract would need to be sold to make the total pay the same?

Short answer

A total of$580$contracts would need to be sold to make the total pay the same.

Step-by-step solution

Step 1. Identify and name what we are looking for. 

The objective is to find how many contracts would need to be sold to make the total pay the same.

Let $\mathrm{x}$represents the number of contracts to be sold and $y$represents the total pay.

Step 2. Form the equations

The first company pays a salary of $$22,000$plus a commission of $$100$for each contract sold. So for $x$contracts sold the total pay for the first company is given by the equation

role="math" localid="1644733667540" $y=22000+100x...\left(1\right)$

The second pays a salary of $$28,000$plus a commission of $$25$for each contract sold. So for $x$contracts sold the total pay for the second company is given by the equation

$y=28000+25x...\left(2\right)$

Step 3. Compare the total pay of the two companies

As the total pay of the two companies is the same, so on comparing the $y$values of the two equations we get

role="math" localid="1644733883985" $\begin{array}{rcl}22000+100x& =& 28000+25x\\ 100x-25x& =& 28000-22000\\ 75x& =& 6000\\ x& =& 580\end{array}$

So on selling role="math" localid="1644733890439" $580$contracts both companies will have the same total pay.