In this video, I wantto talk about how we can convert repeatingdecimals into fractions. So let's give ourselvesa repeating decimal. So let's say I had therepeating decimal 0.7. And sometimes it'llbe written like that, which just means thatthe 7 keeps on repeating. So this is the samething as 0.7777 and I could just keep goingon and on and on forever with those 7s. So the trick to convertingthese things into fractions is to essentially setthis equal to a variable. And we'll just showit, do it step-by-step. So let me set thisequal to a variable. Let me call this x. So x is equal to0.7, and then the 7 repeats on and on forever. Now what would 10x be? Well, let's think about this. 10x. 10x would just be 10 times this. And we could even thinkof it right over here. It would be, if wemultiplied this times 10, you'd be moving the decimal1 over to the right, it would be 7.777, on andon and on and on forever. Or you could say itis 7.7 repeating. Now this is the trick here. So let me make theseequal to each other. So we know what x is. x is this,just 0.777 repeating forever. 10x is this. And this is anotherrepeating thing. Now the way that we can getrid of the repeating decimals is if we subtract x from 10x. Right? Because x has all these 0.7777. If you subtractthat from 7.77777, then you're just goingto be left with 7. So let's do that. So let me rewrite it here justso it's a little bit neater. 10x is equal to 7.7repeating, which is equal to 7.777on and on forever. And we establishedearlier that x is equal to 0.7 repeating,which is equal to 0.777 on and on and on forever. Now what happens if yousubtract x from 10x? So we're going to subtractthe yellow from the green. Well, 10 of somethingminus 1 of something is just going to be9 of that something. And then that'sgoing to be equal to, what's 7.7777repeating minus 0.77777 going on and onforever repeating? Well it's just going to be 7. These parts aregoing to cancel out. You're just left with 7. Or you could say thesetwo parts cancel out. You're just left with 7. And so you get 9x is equal to 7. To solve for x, you justdivide both sides by 9. Let's divide both sides by nine. I could do all three sides,although these are really saying the same thing. And you get x is equal to 7/9. Let's do another one. I'll leave this one hereso you can refer to it. So let's say I have thenumber 1.2 repeating. So this is the same thingas 1.2222 on and on and on. Whatever the bar ison top of, that's the part that repeatson and on forever. So just like we did over here,let's set this equal to x. And then let's say 10x. Let's multiply this by 10. So 10x is equal to, itwould be 12.2 repeating, which is the same thing as12.222 on and on and on and on. And then we cansubtract x from 10x. And you don't haveto rewrite it, but I'll rewrite it here justso we don't get confused. So we have x is equalto 1.2 repeating. And so if we subtract xfrom 10x, what do we get? On the left-hand side,we get 10x minus x is 9x. And this is going to beequal to, well, the 2 repeating parts cancel out. This cancels with that. If 2 repeatingminus 2 repeating, that's just a bunch of 0. So it's 12 minus 1 is 11. And you have 9x is equal to 11.
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Divide both sides by 9. You get x is equal to 11/9.