For the objective function $Z=ax+by$, where $a,b>0$, the maximum value occurs at both $(2,3)$ and $(4,1)$. What is the ratio of $a$ to $b$?

Solution

1. Given the objective function $Z=ax+by$, the slope of the line $ax+by=Z$ is $-\frac{a}{b}$.
2. The slope of the line passing through the points $(2,3)$ and $(4,1)$ is calculated as $$\begin{array}{rl}m& =\frac{1-3}{4-2}\\ & =\frac{-2}{2}\\ & =-1\end{array}$$.
3. Since the maximum occurs at both points, the slope of the objective function must equal the slope of the line segment: $-\frac{a}{b}=-1$.
4. Solving for the ratio, we find $\frac{a}{b}=1$, which corresponds to the ratio $a:b=1:1$.
5. Hence the answer is (C).