# How many times does the smaller wheel turn compared with the box when it turns once around the bigger wheel?

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There are two cogwheels in a box. The bigger one has 24 teeth and it is tightly attached to the center of the box (it doesn’t turn or move). The smaller one has 8 teeth and it rotates around the bigger wheel. How many times does the smaller wheel turn compared with the box when it turns once around the bigger wheel?

pnikam Expert Asked on 21st August 2015 in

The smaller wheel turns around 4 times.

Imagine the two wheels turning next to each other. While bigger wheel turns once (24 teeth) small one turns three times (3 x 8 teeth) to the other side.

Now we want the big wheel to stop and the small wheel to turn around the big one. There it makes one additional turn on the way around the big wheel. If you can’t imagine that you can try with two coins. If you hold one and roll the other around the first one it doesn’t turn once but two times.

Still didn’t get it? Disassemble some clockwork and try it.

anikam Expert Answered on 21st August 2015.

If we say that the radius of the big wheel is R than the radius of the small wheel will be 1/3R. The Perimeter of the circle that the small wheel does is 2*pi*(R+1/3R)=2*pi*4/3R. The perimeter of the small wheel is 2*pi*1/3R so if we divide the them we get 4 turns.

Moshe Expert Answered on 25th January 2023.

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