Remember your first tricycle? The one with a tiny front wheel with the pedals directly attached to the axle? You had to pedal furiously to get anywhere. Now imagine an old-style penny-farthing bicycle featuring a huge front wheel and a similar pedal arrangement. You'd probably have trouble getting started, but once underway, one turn of the crank would take you a long way. A typical modern bike has 27-inch wheels and at least 12 gears. Those different gears give you the sensation of riding bikes with 12 different sized wheels. In fact, a common way of measuring a gear ratio is to express it in gear-inches; the effective diameter1 of the wheel for a given combination of front and rear sprockets. The gear ratio is simply the number of teeth on the currently selected front chain-ring divided by the number of teeth on the rear sprocket. Ratios can go as low as one, where the front and rear cogs are the same, to four or higher, where the front cog is four times the size of the rear. A low ratio is an easy gear, suitable for steep hills. A high ratio is for fast riding on the flat or down hill.
Why so many Gears?
Just as a car engine operates to its optimum best by utilising a narrow range of revolutions per minute, so do your legs. To get started and to ride up a hill you need a low gear. Once you are rolling on the flat, your legs would be going too fast. That's why you need to change up to a higher gear. The turn-over speed of your pedals is called the cadence. As you have ten - or even 21 gears depending on your choice of bike - you can always pedal at an efficient cadence regardless of terrain. A Tour de France (the world's toughest professional cycling race) rider will pedal at 80 to 90rpm everywhere except up the steepest mountain.
A Common Mistake
Road bike and mountain bike riders often make the same mistake. For example, you see an otherwise competent rider, on the flat, in a pack, at about 30km/hr using 42:13. That is, the smallest cog at the front and the smallest at the back. They probably have a cadence they are comfortable in, but their chains are whirring like buzz saws. The chain is at the most extreme angle possible, from the innermost sprocket at the front to the outermost at the rear.
Now assuming 27-inch wheels and a ratio of 42:13, our riders are getting 87.23 gear-inches. They could get pretty close to that - 84.18 gear-inches - with 53:17. The chain would be almost straight back and forth and a lot quieter and probably more efficient.
The chain angle isn't the only consideration. Suppose our riders wanted to make a subtle change to a higher gear in one easy movement, they've only got one choice and that is to shift to 53:13. It's not at all subtle. You are changing from 87.23 to 110.8 gear-inches - a difference of over 20%. You'd have to shift down at the rear too to get something in between. However, if you were riding with 53:17 and shifted just once at the rear you'd be at 53:15, or 95.40 gear-inches. That's a jump of only approximately 12%.
Count Your Teeth
If you're really curious you can count all your teeth, front and rear2. Calculate the ratios and sort them and you'll see some duplication and overlap. You'll also see the perfect sequence to shift from the lowest to the highest gear. You may also be surprised to see that your 14-speed is effectively only a ten-speed if you ignore some of the close duplicates and don't want to back track as you shift upwards (the italicised entries in the table below). For example, the fourth gear is very close to the fifth, but to change in perfect sequence you'd have to shift up at the front and down two at the rear to reach it. This combination is also one that results in the aforementioned unfavourable chain angle.
Simply put, if you want to shift from the lowest to the highest gears in one smooth progression, this is what you have to do.
Start with the smallest chain-ring at the front and the largest cog at the rear.
Shift the rear deraillier, one gear at time, until around the middle of the range (the 6th or 7th).
Then shift once at the front.
Finally, exhaust the rest of the smaller cogs at the rear.
This is all good in theory. In a race, or in a panic you may still find yourself with an awkward combination. There's no need to correct it right away, but make sure the next gear change you make - up or down - brings your chain into better alignment
Points Of Clarification
Current Shimano cassette and chain-ring numbers were used (12, 13, 15, 17, 20, 23, 26 and 53/42). There are closer ratios and actual experiences may vary.