One kind of cake requires $200 g m$ of flour and $25 g m$ of fat, and another kind of cake requires $100 g m$ of flour and $50 g m$ of fat. Find the maximum number of cakes which can be made from $5 k g$ of flour and $1 k g$ of fat assuming that there is no shortage of other ingredients used to making the cakes.

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Solution

Let $x$ be number of cakes of one kind and $y$ be number of cakes of other kind . Now we have to maximize $x + y$
The constraints are $200 x + 100 y \leq 5000$ , $25 x + 50 y \leq 1000$
which implies $2 x + y \leq 50, x + 2 y \leq 40$
The point of intersection of lines $2 x + y = 50$ and $x + 2 y = 40$ is $(20, 10)$
Therefore maximum number of cakes which can be made is $20 + 10 = 30$

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