Originally Posted by Arjonius
The rule of 4 assumes you will see both the turn and the river. If you consider only the turn, there are 9 outs among 47 unknown cards, which is about 19%, which is roughly 2x9, not 4x9. The 4x comes about if you get to see the river, which gives you a second chance to hit if you miss the turn.
So it's actually 2x to improve on each street, so 4x on the turn to improve with 2 cards to come, or 2x to improve on the river with 1 card to come.
katy's approach is the proper way to calculate per-street odds, and is the common approach espoused by most mathy poker publications. The 4x/2x rules were just quick approximations to do at the table because for most people, simple multiplication is easier/faster to do in their head than division. So with a flush draw, it's quicker for most to calculate 9*4=36% with 2 cards to come, or 9*2=18% with 1 card to come, than it is to calculate (47-9)/9 = 4.2:1 against, or about 19% with one card to come.
As far as percentage<->ratio conversion, it's one of those things that look complicated to write out, but it comes pretty natural once you've done it a few times. For instance, assuming a x:1 ratio like 4:1, take the larger number (4), add one (4+1=5), divide it into 100 (100/5=20% = the value of the ":1" part). Then simply multiply by the original number (20*4 = 80%). So you get 4:1 = 80%/20%. If you have an odd ratio like 5:2, just reduce it to 2.5:1 first.
The route from percentage to odds is even easier: if you figure you have a 36% chance of improving with 2 cards to come, then convert 36% to a fraction (a little bigger than 1/3, call it 1/2.8). Subtract 1 from the bigger number and flip them around to get 1.8:1.
The important thing to remember is you don't have to be precise at the table. A close approximation is all that's needed. And using a chart like katy posted is a good shortcut for some folks starting out. But after awhile once you get used to the mental calculation, I find it quicker to do so than looking it up on a chart.