Z7–I–4 2018 MO Betka

Karel was playing with gears assembled into a gear train. When he turned one wheel, all the others turned too. The first wheel had 32 teeth and the second had 24 teeth. When the third wheel (which is in the middle of the gear train) made exactly eight full rotations, the second wheel made five rotations and part of a sixth, and the first wheel made four rotations and part of a fifth.
How many teeth did the third wheel have?

Final Answer:

c =  17

Step-by-step explanation:

$$\begin{gathered} a=32 \\ b=24 \\ 8\cdot c=5,xx\cdot b=4,yy\cdot a \\ {c}_0=5\cdot b/8=5\cdot 24/8=15 \\ {b}_0=8\cdot c_0/b=8\cdot 15/24=5 \\ {a}_0=8\cdot c_0/a=8\cdot 15/32=\frac{15}{4}=3\frac{3}{4}=3.75 \\ {c}_1=c_0+1=15+1=16 \\ {b}_1=8\cdot c_1/b=8\cdot 16/24=\frac{16}{3}=5\frac{1}{3}\doteq 5.3333 \\ {a}_1=8\cdot c_1/a=8\cdot 16/32=4 \\ {c}_2=c_1+1=16+1=17 \\ {b}_2=8\cdot c_2/b=8\cdot 17/24=\frac{17}{3}=5\frac{2}{3}\doteq 5.6667 \\ {a}_2=8\cdot c_2/a=8\cdot 17/32=\frac{17}{4}=4\frac{1}{4}=4.25 \\ {c}_3=c_2+1=17+1=18 \\ {b}_3=8\cdot c_3/b=8\cdot 18/24=6 \\ {a}_3=8\cdot c_3/a=8\cdot 18/32=\frac{9}{2}=4\frac{1}{2}=4.5 \\ c=c_2=17 \end{gathered}$$

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