Like Clockwork

The most important information to use here is that each wheel will rotate through the same amount of teeth! So when the 10 toothed gear completes its 48th rotation, all other wheels will have moved a total of (48)(10) = 480 teeth. Therefore, the 8 toothed gear will complete $\frac{480 \text{ teeth}}{8 \text{ teeth per rotation}} = 60 \text{ rotations}$

One rotation of 10 toothed gear rotates $\frac{10}{12}$ of the 12 toothed gear and $\frac{10}{8}$ of the 8 toothed gear.

So for 48 rotations of the 10 toothed gear, the 8 toothed gear will rotate $48\cdot\left(\frac{10}{8}\right) = 60$ times.

48 (10/12) (12/8) = 48*(10/8) = 60.

when 1 tooth moves every tooth moves, hence when 10 teeth moves of A 10 teeth of B also moves.Since B have 8 teeth it moves 1 revolution and 2 teeth. similarly when 48 revolution of 10 goes then 480 teeth i.e, 480/8 revolution of B

N1 T1=N2T2=N3T3=N4T4 whre N is the number of rotations and T is the number of teeth. here N1=48, T1=10 , T3=8

If first wheel has x no. of teeth n-th wheel has y no. of teeth:

1 rotation of first wheel = x/y rotations of n-th wheel.

Therefore, 48 rotations of first wheel = 48 * (x/y) rotations of n-th wheel.

Here: x=10; y=8, so,

48 rotations of first wheel = 48 *(10/8) rotations of 3rd wheel; or,

48 rotations of first wheel = 60 rotations of 3rd wheel.

I wonder why every body speak of gears while it is the teeth that mesh together. In gear system every meshing gear must have identical teeth. So the number of teeth engaged in the movement will be the same for every gear wheel. Number of teeth per wheel times the number of rotation gives the number of teeth engaged in the operation. It is same for every wheel and thus makes life simpler. Thanks and regards.

Just first of all forget about the rotation angles and position and side and bla bla bla...... Keep in mind the only thing that for single elemental rotation will rise only 1 rotational element to move in any star.

Now, here, 10 star wheel rotates 48 times.....means total elements rotated are 10x48=480. just divide this number by 8 to get how much rotations it is moving through i.e. 480/8=60. Thus, 60 rotations.

After the 10 toothed gear makes its 48th rotation, then each gear will have rotated through 480 teeth. This means that the 8 toothed gear will have gone through 480/8 or 60 rotations.

Note that (No. of teeth in gear) is inversely proportional to (No. of rotations) of the same gear. So 8 x R = 10 x 48; R = 60

$10 \times 48=480$

$So..$

$480 : 8=60$