Answer
$6,667$
Work Step by Step
Let $s_1(x)$ be the monthly salary of option 1 and $x$ be the sale amount. Given that the commission rate of this option is $ 4\% $ and the at the base salary is $200$ dollars, the monthly salary can be modelled as a linear function with a slope of $m= \frac{4}{100} = 0.04$ and a y-intercept of $200$. Hence, we have: $$ s_1(x) = 0.04x+200. $$ In a similar way, we can model the second option as $$ s_2(x) = 0.055x+100.$$ The sale amount can be found by setting the two functions equal and solving for $x$. $$\begin{aligned}
s_2(x)& = s_1(x) \\\\
0.055x+100 & = 0.04x+200\\
0.055x-0.04x& = 200-100\\
0.015x& = 100 \\
x&= \frac{100}{0.015}\\
&\approx 6667.
\end{aligned}$$ Check $$\begin{aligned}
s_2(6667) &= 0.055\cdot 6667+100 \\
& = \$466.68\\
s_1(6667) &= 0.04\cdot 6667+200 \\
& = \$466.68.
\end{aligned}$$ Hence, the sale amount is $6,667$ units.
