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).

Answer: 5

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About mvtrinh

Retired high school math teacher.
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