Chain Maths

Maths time kids. Not my strongest subject - help me out here.

Question 1.
I have a bike with a gear ratio of 51 x 18, or 76.5". I want to lower the ratio, and end up with my wheel in the same place in the dropouts, by taking some links out of the chain.

What size chainring do I need to mount, assuming I don’t change the sprocket?


  1. Chains links have one roller every 1/2 inch. So if I remove one pair of links, I take 1" out of the chain.

  2. Chainring teeth are presumably also spaced every 1/2 inch. So if I remove two chainring teeth, this should match the 1" shorter chain.

So why is it that when I swapped the 51 for a 49, and took out a pair of links, it didn’t work out? The position of the wheel in the track ends needed to be about 10mm further forward than I expected, i.e. my chain was now too short. I gather it’s to do with the way the chain wraps around the chainring and cog, but is there some way to calculate this?

I’ve read somewhere about the distance an axle will move in the dropouts with different tooth counts, but can’t remember where and google is failing me.

This might be of use

That’s pretty much exactly what I was after. Thanks.