A toymaker is making miniature 3-wheel and 4-wheel all-terrain vehicles (ATVs). He has a total of wheels and sets of handlebars. How many 4-wheel ATVs can he make?
Source: NCTM Math Teacher 2008
If the toymaker makes 4-wheel ATVs, he uses wheels and sets of handlebars. The remaining wheel and handlebars are not enough to make any 3-wheel ATVs. If he makes 4-wheel, he uses wheels and handlebars. The remaining wheels and handlebars are enough to make 3-wheel ATVs leaving wheels and handlebars unused.
The following table summarizes the different combinations of ATVs
4-wheel 3-wheel wheels unused handlebars unused
Two possible solutions: 4-wheel and 3-wheel or 4-wheel and 3-wheel. Both solutions use all of wheels but the first one is better because it uses all of 17 sets of handlebars.
The following figure shows the graph of the Diophantine equation
Two points and on the graph are lattice points.