every recurring decimal can be written as a fraction so we've got a question here that says express 0.2 recurring the dot above the two means recurring as a fraction so 0.2 recurring means 0.2222 [Music] and it goes on forever it goes on forever so 0.2 going on forever so express 0.2 recurring as a fraction so what we're going to do here is we're going to take 0.2 recurring 2 0.22222 and so on and we're going to call that x and what i need is i need something else that ends in the 2 2 2 2 2 the 2 recurring so what i'm going to do is i'm going to multiply this by 10. so 0.2 recurring multiplied by 10 gives us 2.2 recurring and that will be 10 lots of x so 10 x now what i can do is i can take away 1x from 10x so if i've got 10 x's and i take away one of them that leaves me with 9 x's and if i've got 2.2 recurring so if i've got point two two two two two two two two two and i take away zero point two two two two two two two two two two and so on that just leaves me with two so i can say 2 is equal to 9 x and if i divide both sides by 9 i get x is equal divide both sides by 9 x is equal to 2 9. so x is 0.2 recurring and x is also 2 9. so 0.2 recurring 0.2 recurring is equal to 2 9. okay so there's a question for you to try so express 0.5 recurring as a fraction so give it a go press pause and press play when you're ready to carry on so we've got 0.5 recurring and we're going to call that x so how can i get something else ending in the 5 recurring i'm going to multiply it by 10 again so 5.5 recurring is 10x and if i do 10x take away 1x that's 9 x 5.5 recurring take away 0.5 recurring is just 5 and divide both sides by 9 which leaves me with x is equal to five ninths so zero point five occurring is five nines okay here's a different question we've got express 0.45 recurring as a fraction so the dot is above the 4 and the 5 which means this is 0.454545 four five going on forever so zero point four five four five four five four five and so on so it's the same principle we take 0.45 recurring and we call that x now we're looking for something that ends in 4 5 recurring and to do that we're going to have to multiply by 100 this time if we multiply by 10 we'd get 4.545454 so we need something that ends in four or five recurring so the only way to do that is to times it by 100 which gives us 45.45 recurring now we're going to take away 1x from 100x 100 lexus take away 1x is 99 x's and 45.45 recurring take away 0.45 recurring is 45. so we divide both sides by 99 so x is 45 over 99 we can simplify it further they're both in the three times table so that's 15 divide the top by three and divide the bottom by three so 15 over 33 and again they're in the three times table so that's five elevenths so zero point four five recurring is equal to 5 11. and there's a question for you to try so express 0.15 recurring as a fraction pause the video give it a go so we've got 0.15 recurring so 0.1515151515 we're going to call that x we need something else that ends in the one five one five one five so we're gonna multiply by a hundred which will give me fifteen point one five recurring if i take away 1 take 100x take away 1x that's 99x 15.15 recurring take away 0.15 recurring is 15 and we divide both sides by 99 to get x by itself so x is 15 over 99 and they're both in the three times table so divide top and bottom by three which gives us five over 33 okay next question so we've got this time we've got express 0.215 all recurring as a fraction so when we've got recurring numbers we only ever use two dots so if there's a dot at the start of what recurs and there's a dot at the end of what recurs so this means 0.215 five two one five and so on so we do the same thing we're going to call 0.215 recurring x and we need something else that ends in two one five recurring so we can't multiply by ten that will give me two point one five two point one five two one five two one five if we multiply by a hundred we get twenty one point five two one five two one five two one we need to multiply by one thousand to get two hundred and fifteen recurring so 1000 x and we're still going to take away like we did before so 1000x takeaway 1x is 999 x 215.215 occurring take away 0.215 recurring is 215. so x is 215 over 999 i don't think that can be simplified any further so that is our answer okay so there's a question for you to try express 0.105 all recurring as a fraction pause the video give it a go and press play when you're ready to carry on so 0.105 that's x we need something else that ends in 105 recurring so we're going to multiply by a thousand again to give us 105.105 recurring we're going to take away so 1000x take away 1x is 999x and 105.105 recurring take away 0.105 recurring is 105 divide both sides by 999 so we've got 105 over 999 they're both in the three times table so we've got 35 over 333 and i don't think it simplifies any further so there is our answer and we've got another two questions here so express 0.215 as a fraction so this time we only have the 1 5 recurring so the 2 is not recurring so we've got 0.2 and then one five one five one five one five and so on so we need to express this as a fraction so if we say zero point two one five the one five recurring is x we need things that end in one five recurring so we're not going to get anything else that ends in two one five occurring so if we multiply by ten we will get 10x is 2.15 recurring now i need something else that ends in 1 5 recurring so if i multiply my original by 1000 i'll get 215.15 recurring is 1 000 x so now i've got two things that end in 1 5 recurring so i can take them away and get rid of the recurring bit so 1000x take away 10x is 990x and 215.15 recurring takes away 2.15 recurring so the 1-5 recurrings cancel out 215 takeaway 2 is 213. so 213 equals 990x divide both sides by 990 so x is 213 over 990 they're both in the three times table so we can simplify so we've got 71 over 330 and there's our answer okay question for you to try pause the video give it a go and press play when you're ready to go through the answer okay so we've got 0.15 recurring so zero point one and then five five five five five five five five five we're gonna call that x if we multiply by ten we get one point five recurring as ten x and if we multiply the original by 100 we'll get 15.5 recurring as 100x so we've got two things that end in the five recurring so we're going to take away 100x takeaway 10x is 90x 15.5 recurring takeaway 1.5 recurring so the five recurrings cancel out 15 take away 1 is 14. so 14 equals 90x divide both sides by 90. so x is 14 over 90. or 7 over 45 okay three more questions to end so have a go at these pause the video and press play when you're ready for the answers okay firstly 0.8 recurring so 0.8 recurring is x 8.8 recurring is 10x take away 10x take away 1x is 9x 8.8 recurring take away 0.8 recurring is eight and divide both sides by nine so x is eight ninths zero point eight one recurring so 0.81 recurring is x to get something else that ends in 81 recurring we're going to multiply by 100 so 81 81.81 recurring is 100x 100x take away 1x is 99x and 81.81 recurring takeaway 0.81 recurring is 81 and we divide both sides by 99 so x is 81 over 99 or so they're both in the nine times table nine elevenths and finally we've got zero 0.81 recurring so just the one recurring this time 0.81 recurring is x if we multiply by 10 we get 8.1 recurring as 10x and multiply by the original by 100 or 10x by 10 either one so 81.1 recurring is 100x so we take away 100x take away 10x is 90 x 81.1 recurring take away 8.1 recurring so 81 take away 8 and that's 73 and divide both sides by 90 so x equals 73 over 90. you