swerve wrote:An auto dealer sells each car at either $20,000 or $30,000. Some cars are marked up at 20% of the cost price and the remaining are marked up at 30% of the cost price. If the dealer sells 50 cars in total, what is the total profit of the dealer?

(1) 20 cars were sold for $20,000
(2) 30 cars were marked up 30%

Source: e-GMAT

Both "grids" (double-matrix) below are viable (check that!) and each one gives a different total profit (our FOCUS) to the dealer.
(No need to calculate explicitly: in the second scenario the dealer sells a larger quantity of cars with lower cost and larger markup...)

\[\left( {1 + 2} \right)\,\,\,{\text{BIFURCATES}}:\]
\[\begin{array}{*{20}{c}}
{}&{{\text{markup}}\,\,{\text{20\% }}}&{{\text{markup}}\,\,{\text{30\% }}}&{{\text{total}}} \\
{{\text{\$ }}\,{\text{20,000}}}&{20}&0&{20} \\
{{\text{\$ }}\,{\text{30,000}}}&0&{30}&{30} \\
{{\text{total}}}&{20}&{30}&{50}
\end{array}\]

\[\begin{array}{*{20}{c}}
{}&{{\text{markup}}\,\,{\text{20\% }}}&{{\text{markup}}\,\,{\text{30\% }}}&{{\text{total}}} \\
{{\text{\$ }}\,{\text{20,000}}}&0&{20}&{20} \\
{{\text{\$ }}\,{\text{30,000}}}&{20}&{10}&{30} \\
{{\text{total}}}&{20}&{30}&{50}
\end{array}\]

We are done!

This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.