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Converting repeating decimals to fractions (part 1 of 2)

CCSS.Math:

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 add the repeating decimal zero point seven and sometimes it'll be written like that which just means that the seven 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 7s so the trick to converting these things into fractions is to essentially set this equal to a variable and we'll just sort of do it step by step so let me set this equal to variable let me call this X so X is equal to zero point seven and then the seven repeats on and on forever now what would ten X be well let's think about this 10 X 10 X would just be ten times this so it would be and we could even think of it right over here would be if we multiply this times 10 you'd be moving the decimal one over to the right it would be seven point seven seven seven on and on and on and on forever or you could say it is seven point seven repeating now this is the trick here so we've let me make these equal to each other so we know what X is X is this this point seven seven seven repeating forever ten X is this and this is another repeating thing now the way that we can get rid of the repeating decimals is if we subtract x from 10x right because x has all these point seven seven seven seven if you subtract that from seven point seven seven seven seven seven then you're just going to be left with seven so let's do that so let me let me rewrite it here just looks a little bit neater 10 ten X is equal to seven point seven repeating which is equal to seven point seven seven seven on and on forever and we established earlier earlier that X is equal to zero point seven repeating which is equal to zero point seven seven seven on and on and on forever now what happens if you subtract X from 10x so we're going to subtract the yellow from the green well 10 at ten of something minus one of something is just going to be nine of that something and then that's going to be equal to what seven point seven seven seven seven seven repeating minus Oh point seven seven seven and 77 what going on and on and forever repeating what's just going to be seven these parts are going to cancel out you're just left with seven or you could say these two parts cancel out you are just left with seven so you get 9x is equal to seven to solve for X you just divide both sides by nine let's divide both sides by nine well I can do all three sides although these are really saying the same thing and you get X is equal to X is equal to seven ninths let's do another one let's do another one leave this one here so you can refer to it so let's say let's say I have the number one point to one point two repeating so this is the same thing as one point two two two two on and on and on whatever the bar is on top of that's the part that repeats on and on forever so just like we did over here let's set this equal to X and then let's say ten X let's multiply this by 10 so 10 X is equal to it would be 12 point two repeating which is the same thing as twelve point two two two on and on and on and on and then we can subtract X from 10 X and you don't have to rewrite it but I'll rewrite it here just so we don't get confused so we have X is equal to one point two repeating and so if we subtract X from 10x what do we get on the left hand side we get X minus 10x minus X is 9x 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 zero so it's 12 minus one is 11 and you have nine X is equal to 11 divide both sides by nine you get X is equal to eleven over nine