# Line and segment relationship in the circle

if the larger gear has 30 teeth and the smaller gear has 18, then the gear ratio(larger to smaller) is 5:3. when the larger gear rotates through and of 60°, through what angel measure does the smaller gear rotate?

• Think of it this way: the number of teeth that the big gear rotates by must equal the number of teeth that the small gear rotates by. If a $30$-tooth gear rotates by $60^{\circ}$, how many teeth has it rotated by? By what degree should an $18$-tooth gear rotate by in order to have rotated the same number of teeth that the big gear did? – 2012ssohn Nov 14 '16 at 5:00
• Thank You makes it much clearer – Edith Mendoza Nov 14 '16 at 5:04
• @EdithMendoza, You can also do like this: $\frac {30}{18}=\frac {\alpha}{60°}$, therefore $\alpha=100°$ – Seyed Nov 14 '16 at 18:19

The angle between two teeth in the lager gear is $\frac {360°}{30}=12°$ and for smaller gear is $\frac {360°}{18}=20°$. When the larger gear rotates for $60°$ it means $5$ teeth has moved $(\frac {60°}{12°}=5)$. Because these gears are connected then the smaller gear must move for $5$ teeth too, and $5$ teeth in smaller gear means $5\times 20°=100°$, so the smaller gear will rotate for $100$ degrees.