## Candy Bars

A school fund-raiser has students selling candy bars in two sizes: $\1.00$ and $\2.50$. Anthony collected $\42.00$ but forgot what he sold. If remembers selling at least $25$ candy bars and at least $1$ of the jumbo size, how many possible combinations could he have sold?
Source: NCTM Math Teacher 2008

SOLUTION
Let $x$ be the number of smaller size candy bars and $y$ the number of jumbo size.
$1x+2.5y=42$
$y$ must be even in order to make $42$ a whole dollar amount and not a fractional amount.
$y\quad x$
$2\quad 37$
$4\quad 32$
$6\quad 27$
$8\quad 22$
$10\: \: 17$
We stop here because of the constraint $x+y\geq 25$
Anthony could have sold $5$ possible combinations.

Alternative solution
The following figure shows the graph of the Diophantine equation $x+2.5y=42$

Five lattice points of the graph satisfy the constraints $x+y\geq 25$ and $y\geq 2$
$(17,10); (22,8); (27,6); (32,4); (37,2)$.