Question 2
Class 6A has 18 boys and 23 girls planting trees. A pupil working alone plants one tree. Groups work according to these rules:
• A group of $n$ boys plants $n+\left\lfloor(n+1)/4\right\rfloor$ trees.
• A group of $k$ girls plants $k+\left\lfloor(k+1)/6\right\rfloor$ trees.
• A mixed group of $p$ boys and $q$ girls plants $p+q+\left\lfloor(p+q+2)/5\right\rfloor$ trees, where $p,q>0$.
How should the class combine these methods to plant the greatest possible number of trees? Find the maximum and explain why. The symbol $\lfloor a/b\rfloor$ means the integer part of $a/b$; for example, $\lfloor5/3\rfloor=1$ and $\lfloor10/5\rfloor=2$.
Model clarification: each pupil participates once, either alone or in one group. More than one group of the same type is allowed. The alternative restriction of at most one group of each type is discussed separately in the solution.
• A group of $n$ boys plants $n+\left\lfloor(n+1)/4\right\rfloor$ trees.
• A group of $k$ girls plants $k+\left\lfloor(k+1)/6\right\rfloor$ trees.
• A mixed group of $p$ boys and $q$ girls plants $p+q+\left\lfloor(p+q+2)/5\right\rfloor$ trees, where $p,q>0$.
How should the class combine these methods to plant the greatest possible number of trees? Find the maximum and explain why. The symbol $\lfloor a/b\rfloor$ means the integer part of $a/b$; for example, $\lfloor5/3\rfloor=1$ and $\lfloor10/5\rfloor=2$.
Model clarification: each pupil participates once, either alone or in one group. More than one group of the same type is allowed. The alternative restriction of at most one group of each type is discussed separately in the solution.