A toymaker is making miniature 3-wheel and 4-wheel all-terrain vehicles (ATVs). He has a total of 61 wheels and 17 sets of handlebars. How many 4-wheel ATVs can he make?
Source: NCTM Math Teacher 2008

If the toymaker makes 15 4-wheel ATVs, he uses 15\times 4=60 wheels and 15 sets of handlebars. The remaining 1 wheel and 2 handlebars are not enough to make any 3-wheel ATVs. If he makes 14 4-wheel, he uses 14\times 4=56 wheels and 14 handlebars. The remaining 5 wheels and 3 handlebars are enough to make 1 3-wheel ATVs leaving 2 wheels and 2 handlebars unused.
The following table summarizes the different combinations of ATVs
4-wheel    3-wheel    wheels unused    handlebars unused
15\qquad\quad\:\: 0\qquad\qquad 1\qquad\qquad\qquad\:\:\: 2
14\qquad\quad\:\: 1\qquad\qquad 2\qquad\qquad\qquad\:\:\: 2
13\qquad\quad\:\: 3\qquad\qquad 0\qquad\qquad\qquad\:\:\: 1
12\qquad\quad\:\: 4\qquad\qquad 1\qquad\qquad\qquad\:\:\: 1
11\qquad\quad\:\: 5\qquad\qquad 2\qquad\qquad\qquad\:\:\: 1
10\qquad\quad\:\: 7\qquad\qquad 0\qquad\qquad\qquad\:\:\: 0
9\qquad\quad\:\:\:\: 8\qquad\qquad 1\qquad\qquad\qquad\:\:\: 0
8\qquad\quad\:\:\:\: 9\qquad\qquad 2\qquad\qquad\qquad\:\:\: 0
Two possible solutions: 10 4-wheel and 7 3-wheel or 13 4-wheel and 3 3-wheel. Both solutions use all of 61 wheels but the first one is  better because it uses all of 17 sets of handlebars.

Alternative solution
The following figure shows the graph of the Diophantine equation 4x+3y=61

Two points (10,7) and (13,3) on the graph are lattice points.

Answer: 10


About mvtrinh

Retired high school math teacher.
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