For most purposes, cyclists determine speed ratio from gear size. Analysing gear ratios mathematically is too complicated. On all bikes, the best gear ratio for speed is to have the chain on the biggest gear in front, and the smallest gear on the back. The gear size is usually stamped on the front or backside of the gear. The more distance between numbers, the more speed.
Most popular road bikes have what is called a "compact double." This means that the bike only has two gears on the front, the biggest gear usually having 50 teeth. It could have 8 or 10 gears in the back, with the smallest gear having 10 or 12 teeth. The best ratio for speed on this combination would be for the chain to ride on the 50-tooth gear in front and the 10-tooth gear in back. Some road bikes have three gears in front with a larger 52-tooth gear in front. If the bike has a 52-tooth gear or bigger, then the 52-tooth gear would be faster than the 50.
Mountain bikes are very similar to road bikes in their gear configurations. Mountain bikes almost always have three gears in front, with the biggest gear having 40 to 42 teeth in front. They may have six to 10 gears in the back, with the smallest gear having only nine or 10 teeth. The fastest combination would be a 42-tooth gear in front and a nine-tooth gear in back.
Touring bikes are set up much like road bikes. They are much slower than road bikes, but speed is not their intent. Their gear ratio calculations are for towing and tolerating more weight. Touring bikes also come equipped with either double or triple gears on the front. They can have either a 48-tooth gear on the front or a 50 like road bikes. Most common touring bikes have a 22 to 32 tooth for the smallest gear on back. The best combination for speed would be for the chain to be on the 22 on back and the 50 in front.
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