# Finding the necessary diameter for a pulley to match the ratio of other pulleys

I have an analog odometer for a bicycle. It works with two pulleys connected by a small rubber drive belt. One pulley is affixed to the hub, and the other pulley is on the odometer unit itself. To compensate for differing wheel sizes, there are two different pulley sizes available on the odometer unit:

• The pulley for a $650mm$ wheel is $19.5mm$
• The pulley for a $700mm$ wheel is $18.5mm$

The drive pulley is $40mm$.

Assuming we use the pulley intended for a $650mm$ wheel, but actually have a $700mm$ wheel, what size would the drive pulley have to be to keep the established ratio approximately correct?

(For what it's worth, the drive pulley does not fit my hub correctly, and I'm trying to figure out if I can kludge this together by finding a different sized pulley for it.)

Thanks so much for your help. I hope this is an appropriate question to ask here! Math has never been a strength of mine, and I wouldn't know where to even begin to figure this out on my own. I apologize if this is shamefully elementary.

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For a 700 mm wheel you want a ratio of $\frac {40}{18.5}$ so you want $\frac {40\cdot 19.5}{18.5}$ mm