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Converting repeating decimals to fractions (part 1 of 2)

Learn how to convert the repeating decimals 0.77777... and 1.22222... to fractions. Created by Sal Khan.
Video transcript
In this video I want to talk about how we can convert repeating decimals into fractions. So let's give ourselves a repeating decimal. So let's say I had the repeating decimal zero point seven and sometimes it'll be written like that. [bar above the seven] Which just means that the 7 keeps on repeating. So this is the same thing as zero point seven seven seven seven And I could just keep going on and on and on, forever with those sevens. So the trick to converting these things into fractions is to essentially set this equal to a variable. And we will sort of do it step by step. So let set this equal to a variable, let me call this x. So x is equal to zero point seven and the seven repeats on an on for ever. Now what would ten x be? Well let's think about this, ten x would just be ten times this so it would be, we can even think of it right over here. it would be, if we multiplied this by ten. We would be moving the decimal one to the right it would be seven point seven seven seven, on and on and on forever. Or we could say it is seven point seven repeating. Now this is the trick here. Let me make these equal to each other. So we know what x is, it is point seven seven repeating forever. Ten x is this. And it is another repeating thing. Now the way we can get rid of the repeating decimals is if we subtracting x from ten x, right? Because x has all these repeating point seven seven seven. If you subtract that from seven point seven seven seven, you are just going to be left with seven. So let's do that. Let me rewrite it here. Ten, ten x is equal to seven point seven repeating. Which is equal to seven point seven seven seven on and on forever. As we established earlier that x is equal to zero point seven repeating; which is equal to seven point seven seven seven on and on and on forever. Now what happens when you subtract x from ten x. So we are going to subtract the yellow from the green. Well ten of something minus one of something is just going to be nine of that something. And then that is going to be equal to: What's seven point seven seven repeating, minus point seven seven, going on and on, forever repeating? Well it is just going to be seven. These parts are going to cancel out, you are just left with seven or we could say, these two parts cancel out and you are left with seven. So you get nine x is equal to seven. To solve for x you just divide both sides nine. Well I could do all three sides, although these are all saying the same thing and you get x is equal to seven ninths. [7/9] Let's do another one. I will leave this one here so you can refer to it. So let's say I have the number one point two repeating. So this is the same as one point two two two and on and on. Whatever the bar is on top of, that is the part that repeats forever. So just like we did over here, lets set this equal to x. And let's say ten x -- let's multiply this by ten. So ten x is equal to twelve point two repeating. Which is the same thing as twelve point two two two on and on and on. Then we can subtract x from ten x. And you don't have to rewrite it but I rewrite it here, just so we don't get confused. So we have x is equal to one point two repeating. And if we subtract x from ten x what do we get. On the left hand side we get x minus, ten x minus x is equal to to nine x and this is going to be equal to: Well the two repeating parts cancel out. This cancels with that. If two repeating minus two repeating that's just a bunch of zeros. Twelve minus one is eleven. You have nine x is equal to eleven. Divide both sides by nine, and you get left with x is equal to eleven over nine. [x=11/9]