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# Converting repeating decimals to fractions (part 2 of 2)

CCSS.Math:

## Video transcript

in the last video we did some examples where we had one digit repeating on and on forever and we were able to convert those into fractions in this video we want to tackle something a little bit more interesting which is multiple digits repeating on and on forever so let's say I had zero point zero point three six repeating which is the same thing as zero point since the bars over the 3 and the 6 both of those repeat 3 6 3 6 3 6 and it just keeps going on and on and on like that forever now the key to doing this type of problem is instead of multiplying so like we did in the last video we said this is equal to X instead of just multiplying it by 10 10 would only shift it 1 over we want to shift it over enough so that we can kind of so that the the when we line them up the decimal parts will still line up with each other and to do that we want to actually shift the decimal space two to the right and if we to shift it two to the right we have to multiply by a hundred or ten to the second power so 100 X 100 X is going to be equal to what we're shifting this two to the right one two so 100x is going to be equal to is going to be equal to the decimal is going to be there now so it's going to be thirty six point three six three six three six on and on and on forever and then let me rewrite X over here we're going to subtract that from the hundred X X is equal to zero point three six three six three six repeating on and on forever and notice when we multiply it by 100 X the 3s and the six is still line up with each other when we line the decimals and you want to make sure you get the decimals lined up appropriately and the reason why this is valuable is now that when we subtract X from 100x this the repeating parts will cancel out so let's subtract let us subtract these two things so on the left hand side we have 100 X minus X so that gives us 99 X and then we get on the right hand side this part cancels out with that part we're just left with 36 we're just left with 36 we can divide both sides by 99 and we are left with we are left with X is equal to 36 over 99 and both the numerator and the denominator is divisible by nine so we can reduce this we divide the numerator by nine we get for the denominator by nine we get 11 so 0.36 three six three six forever and forever repeating is four 11s now let's do another interesting one let's say I have and I'll just set it equal to X well let's say we had the number let's say we have the number zero point seven one four and the one four is repeating and so this is the same thing so notice the seven one four isn't going to repeat just the one four is going to repeat so this is zero point seven one four one four one four on and on and on and on so let's set this equal to X now you might be tempted to multiply this by a thousand X to get the decimal all the way clear of one seven one four so get the decimal all the way clear of seven one four but you actually don't want to do that you want to shift it just enough so that you can so that the repeating pattern can be right under itself when you when you do the subtraction so again in this situation even though we have three numbers behind the decimal point because only two of them are repeating we only want to multiply by 10 to the second power so once again you want to multiply by 100 so you get 100 X is equal to we're moving the decimal two to the right one two so it's going to be seventy one point four one four one on and on and on and on so it's going to be seventy one point four one four one four one four and on and on and on and then let me rewrite X right below this we have X is equal to zero point seven one four one four one four and notice now the 1/4 is 1/4 1/4 so they're lined up right below each other so it's going to work out when we subtract so let's subtract these things 100 X minus X is 99 X and this is going to be equal to these 1 4 1 4 s are going to cancel with those 1/4 1/4 s and we have 71 point 4 minus point 7 we can do this in our head or borrow if you like this could be a 14 this is a zero so you have 0.4 14 minus 7 is 7 and then 70 minus 0 so you have 99 X is equal to 70 point 7 and then we can divide both sides by 99 you can see also something strange is happening because we still have a decimal but we can fix that up at the end so let's divide both sides by 99 let's divide both sides by 99 you get X is equal to 70 point 7 over 99 now obviously we haven't converted this into a pure fraction yet we still have a decimal in the numerator but that's pretty easy to fix you just have to multiply the numerator and the denominator by 10 to get rid of this decimal so let's multiply the numerator by 10 and the denominator by 10 and so we get seven hundred and seven seven hundred and seven over over nine hundred and ninety let's do one more example over here so let's say let's say we had something like 1 let me write it this way three point two five seven repeating and we want to convert this into a fraction so once again we set this equal to X and notice this is going to be three point two five seven two five seven two five seven the to five seven is going to repeat on and on and on since we have three digits are repeating we want to multiply this we want to think about a thousand X 10 to the third power times X so and that'll let us shift it just right so that the repeating parts can cancel out so we can get so 1000x1000 x is going to be equal to what we're going to shift the decimal three to the right one two three so it's going to be three thousand two hundred and fifty-seven point and then the two five seven keeps repeating two five seven two five seven two five seven keeps going on and on and on forever and then we're going to subtract X from that so here's X X is equal to three you want to make sure you have your decimals lined up it's three point two five seven two five seven two five seven dot-dot keeps going on forever notice when we multiplied it by a thousand it allowed us to line up the two fives so that when we subtract the repeating part cancels out so let's do that subtraction on the left-hand side a thousand of something minus 1 of that something you're left with 999 of that something is equal to this part is going to cancel out with that part it's going to be equal to let's see 7 minus 3 is 4 and then you have this the 5 the 2 and the 3 so you get 999 X is equal to three thousand two hundred and fifty four and then you can divide the numerator or divide both sides of this by 999 both sides by 999 and you are left with X is equal to three thousand two hundred and fifty-four over 999 and so obviously this is an improper fraction the numerator is larger than the denominator you could convert this to a proper fraction if you like one way you could have just tried to figure out what the the to the point two five seven repeating forever is equal to and just have three being the whole number part of a mixed fraction or you can just divide 999 into 3254 actually we could do that pretty straightforwardly it goes into it three times and the remainder well let me do it let me do it so let me just do it just to go through the motions so nine nine nine goes into 3254 it'll go into it three times and we know that because this is originally three point two five seven so we're just going to find the remainder so 3 times 9 is 27 3 times 9 is 27 but we have to add the two so it's 29 3 times 9 is 27 we have a 2 so it's 29 and so we are left with if we subtract if we regroup or borrow or however we want to call it this could be a 14 and then this could be a 4 let me do this in a new color this would be a 4 and then the 4 is still smaller than this 9 so we need to regroup again so then this could be a 14 and then this could be a 1 but this is smaller than this 9 right over here so we regroup again this would be an 11 and then this is a 2 14 minus 7 is 7 14 minus 9 is 5 11 minus 9 is too so we are left with we are left with did I do that right yep so this is going to be equal to three and 257 over nine nine nine and we're done