I have a 21 speed bike (a hybrid commuting bike). But AFAIK, this does not mean that I have 21 different gear ratios. Is there any way to calculate the different gear ratios?
Excluding the two extremes (see Scott Robinson's answer), you have 19 ratios. Some of these may be very close to each other. The easiest way (although a little cumbersome) is to count the number of teeth (
n) on all gears (three in front, seven rear) counting from the centerline of the bike (outwards). The ratio is given as
n(front) / n(rear), e.g.:
1st = n(front1) / n(rear1) 2nd = n(front1) / n(rear2) ... 8th = n(front2) / n(rear1) ... 21st = n(front3) / n(rear7)
Now you may find identical (or nearly identical) ratios. The number of unique ratios is the number of effective gears.
Answers for a typical 21-speed commuter I could find specs on: 20, 19, 15, 12.
20 is the technically right answer, 19 is a slightly more technically right answer, 15 is if you take into account gears that are nearly identical, but 12 is the practical usable answer.
Really, it's going to depend on your exact gear setup.
At the very least, you really should avoid the big-front:big-back gear, since that's hardest on your derailer.
So, if you take @jensgram's answer and work it out on a typical bike similar to what you describe ... I'll take the Trek 7.1FX for my example. The chainrings in front are 48-38-28t. In the back it's 14-16-18-20-22-24-34r. (Numbers are the number of teeth). I don't know if it's typical on other hybrids, but that jump from 24 to 34 is a little unusual and they market it as a special "megarange" feature.
If you work out all the ratios (I wrote a sort of little computer program up real quick to do it), the only exact duplicate ratios this gear setup has is 28:14 and 48:24, one of which is one of those extreme ranges it's suggested to avoid. So this bike actually has 20 unique gear ratios, but since one of those unique ranges is 48:34 that you should avoid, it's 19 usable unique gear ratios.
If you round to only one digit after the decimal to give you the nearly identical ratios, you get 15 gears:
0.8 28:34 1.1 28:24 or 38:34 1.2 28:22 1.4 28:20 or 48:34 1.5 28:18 or 38:24 1.7 28:16 or 38:22 1.9 38:20 2.0 28:14 or 48:24 2.1 38:18 or 48:22 2.3 38:16 2.4 48:20 2.6 48:18 2.7 38:14 3.0 48:16 3.4 48:14
But look carefully at what front chainrings you use to get those... If you use the smallest front chainring with only the 3 biggest rear cogs, the middle front chainring with all but the biggest and smallest rear, and the big front chainring with only the four smallest rear cogs, you get your whole range of gears in a way you can comprehend, but it's 12 really usable gears (you skip 1.4, 2.0 and 2.7 in my above chart; all close enough to another gear)
How many rear cogs it makes sense to use on each front chainring will depend on the exact gearing setup. But if you play around with it a little you can probably just feel it. Switch the front chainring to the middle, switch the back to the smallest and feel that gear; then switch the front to the biggest, switch the back in by 2 or 3 gears and see if it feels about the same. Do the same thing with the low-end.
The way I visualize this is as follows:
Usually change to a smaller or larger chainring corresponds to roughly 2 jumps at the sprocket cassette (on a road bike "compact double" 34/50, it may be 3 jumps but on triple chainring setups it's almost always 2 jumps, excluding the ridiculous 13-speed cassettes and considering only reasonable 7 or 8 speed cassettes).
Thus, you have 7 ratios on the middle chainring. The small chainring adds 2 ratios and the big chainring adds 2 ratios. So you have 11 ratios.
In theory, you may have more distinct ratios if you want to be accurate, but most of these ratios are very close to each other, some may be actually identical. So in theory the answer "19 ratios" is the correct one, but in practice you have no easy way to access these ratios in sequence, and the jumps between these would not be constantly spaced.
So what matters in practice is how you operate the gears. You either shift on the rear one sprocket up or down, or if you run to the edge of the cassette (or the chainline becomes too crooked), you shift on the front, and then on the rear at the same time in the opposite direction to get effectively one jump. Calculating the practical number of ratios based on this usage pattern, you get 11 ratios.
It's better to visualize the system as consisting of three heavily overlapping ranges, with 7 ratios on each range (minus the ones where the chainline is so crooked you wouldn't use those). Because the ranges are heavily overlapping, you have 11 practically different ratios instead of 21 practically different ratios.
Here is an excellent gear calculator. You just need to know a few basic things like crank length, wheel size, etc. and then it will figure everything out for you.
I've heard that the you should only use all the rear chain rings when using the middle chain ring on the front. When using the largest front chain ring, you should only use the bottom 2 or 3 chain rings, and when using the smaller chain ring on the front, you should only use the top 2 or 3 chain rings. This is to prevent having too much of and angle between in the chain. Also, you generally don't need to use those gears anyway, since they are usually the same, or in between the range of the other gears, which means they won't help you much. Here's a good explaination of how to change gears.
So while you may effectively have 21 gears, you shouldn't use them all, as the extremes can cause your chain to strech. Mostly, you'll use only 3+7+3, giving you 13 gears. Some people even say to keep you chain really straight really straight, and you should only use 3+3+3 = 9. Just try to keep your chain as straight as possible. Since there is a lot of overlap, you'll want to shoot for a straight chain as much as possible, over trying to use all the gears.
There is just another thing to mention. You can't use all combinations else there is too much strain on the system. With my 3x8 I just ignore the extremes:
- Front 1: Back 1-6
- Front 2: Back 2-7
- Front 3: Back 3-8
You loose some combinations, but they are not very useful anyway.
Note: I once had an accident. While driving, the bike suddenly stopped with a metallic klunc. The rear derailleur was caught between the spokes. Ruining about everything. The bike just had its maintenance (replacement of chain and cassette) so it was probably a mistake of the bicycle repairshop. And they agreed). Athough they said that it was because of the extreme settings (Front 3, Back 1). But that was a result of the accident. So they fixed it for free.