Challenging +1.2 This is a standard linear programming formulation question requiring students to translate verbal constraints into inequalities and eliminate one variable. While it involves multiple constraints including a percentage condition and ingredient ratios, the techniques are routine for FD1: substituting z = 48 - x - y, converting ratios to inequalities, and handling percentage constraints. The question is methodical rather than requiring novel insight, making it moderately above average difficulty for A-level but typical for Further Maths Decision content.
3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins.
Donald decides to bake 48 brownies and muffins in total.
Donald decides to bake at least 5 brownies for every 3 flapjacks.
At most \(40 \%\) of the cakes will be muffins.
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.
Donald plans to sell each brownie for \(\pounds 1.50\), each flapjack for \(\pounds 1\) and each muffin for \(\pounds 1.25\) He wants to maximise the total income from selling the cakes.
Let \(x\) represent the number of brownies, let \(y\) represent the number of flapjacks and let \(z\) represent the number of muffins that Donald will bake.
Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients.
You should not attempt to solve the problem.
3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins.
Donald decides to bake 48 brownies and muffins in total.\\
Donald decides to bake at least 5 brownies for every 3 flapjacks.\\
At most $40 \%$ of the cakes will be muffins.\\
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.\\
Donald plans to sell each brownie for $\pounds 1.50$, each flapjack for $\pounds 1$ and each muffin for $\pounds 1.25$ He wants to maximise the total income from selling the cakes.
Let $x$ represent the number of brownies, let $y$ represent the number of flapjacks and let $z$ represent the number of muffins that Donald will bake.
Formulate this as a linear programming problem in $x$ and $y$ only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients.
You should not attempt to solve the problem.\\
\hfill \mbox{\textit{Edexcel FD1 AS 2021 Q3 [9]}}