Converting repeating decimals to fractions
Converting Repeating Decimals to Fractions 2 Converting more interesting repeating decimals to fractions
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- In the last video we did some example were we had one digit repeating on and on forever and we were able to convert those
- into fraction and in this video we want to tackle something a little bit more interesting which is multiple digit repeating on and on forever
- so let say I had zero point three six repeating, which is the same thing as zero point, since the bar over
- the three and six both of those repeat, three six... three six... three six... and just keeps going 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 ten , ten would only shift it one over
- we want to shift it over enough so that we can kind of... so that
- the when we line them up, the decimal parts will still line up with each other.
- And To do that, we want to actual shift the decimal space two to the right
- And to shift it two to the right we have to multiply it by 100, or 10 to the second power
- So, 100 X, 100X is going to equal to what.. we are 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, its going to be Thirty Six point three six.. three six. three six.. on and on and on forever.
- And let me re write X over here, we are going to subtract that
- from the 100X, 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, the three's and the six's still line up with each other.
- when we line the decimals, you want to make sure that the decimals line up
- appropriately, and the reason why this is valuable is now that when we subtract
- X from the 100X, this . the repeating parts will cancel out
- so, lets subtract, let us subtract these two things,
- So, the left hand side we have 100X minus X, so that gives us 99X
- And then we get on the right hand side, this part cancels out with that card,
- So, we are just left with Thirty Six, we are just left with Thirty Six
- We can divide, both sides by 99, and we are left with, we are left with
- X is equal to Thirty Six over Ninety Nine and both the numerator and the denominator
- is divisible by Nine., so we can reduce this, we divide the numerator by Nine we get Four
- The denominator by Nine, we get Eleven, So Point Three Six, three six, three six forever and forever repeating
- is four elevenths, Now lets do another interesting one
- Lets say I have, and I will just set it equal to X, well lets say we have the number, lets say we have
- the number Zero point seven one four and the one four is repeating,
- so this the same thing, so notice the seven one four isnt 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..
- So lets set this equal to X, now you might be tempted to multiply this by 1000, to get the decimal all
- the way clear from seven one four to get the decimal all the way clear of seven one four
- But you actually dont want to do that, you want to shift it just enough so that the repeating
- pattern can be right under itself, when you do the subtraction
- So again in this situation, even though we have three number behind the decimal
- point because only two of them are repeating we only want to multiply it by Ten to the second power
- So once again, you want to multiply it by a hundred
- so you get 100X is equal to removing the decimal two to the right
- One .. two,, so its going to be Seventy one point four one, four one, on and on and on
- So, its going to be Seventy one point four one four one four one four and on and on
- and let me rewrite X right below this, we have X is equal to Zero point seven one four
- one four, one four .. and notice one four's one four's one four's
- are lined up right below each other. so it is going to work out when we subtract
- So, lets subtract these things, 100X minus X is 99X, and this is going to be
- equal to these one four's one four's are going to cancel those one four's
- Now we have seventy one point four minus point seven, we can do this in our head
- or we can borrow if you like this could be fourteen, this could be zero
- so you have point four, fourteen minus seven is seven and seventy minus zero
- so, you have 99X is equal to Seventy point seven
- and then we can divide both sides by Ninety nine, you could see all sort of something strange
- happening because we still have decimal, but we can fix that up in the end
- So, lets divide both sides by ninety nine
- lets divide both sides by ninety nine, you get X is equal to seventy point seven over nintey nine
- now, obviously we have not converted this into a pure fraction yet, we still have a decimal in the numerator
- but thats pretty easy to fix, you just have to multiply the numerator and the denominator
- by ten to get rid of decimal, so lets multiply the numerator by ten
- and the denominator by ten, and so we get
- Seven hundred and seven, seven hundred and seven over
- Nine hundred and ninety
- Lets do one more example over here, so lets say
- lets say we have something like one , let me write this
- Three point two five seven repeating , and we want to covert this into a fraction
- so ones again, we set this equal to X, and know this is going to be three point two five seven
- two five seven. two five seven, two five seven's going to repeat on and on and on
- Since we have three digit are repeating.. we want to multiply this, we want to
- think about a thousand X, ten to the third power times X, so and
- that will let us shift it just right so that the repeating parts and cancel out
- so we can get, so one thousand X, one thousand X is going to be equal to what
- we can shift the decimal to the right, one two three, so its going to three thousand
- two hundred and fifety seven point then the two five seven keeps repeating,
- two five seven, two five seven, two five seven keeps going on and on forever
- and then we are going to subtract X from that so here is
- X , X is equal to three, you want to make sure that decimal sign is lined up
- three point two five seven, two five seven, two five seven dot dot dot
- keep going on forever, notice when we multiply it by thousand it allowed us to line up
- the two five and seven's so that when we subtract the repeating part cancels out
- So lets do that subtraction, on the left hand side Thousand of something minus One of something
- you are left with Nine hundred and ninety nine of that something is equal to
- this part is going to be cancel out with that part, is going to be equal to
- this is seven minus three is four, and then you have this five and the two and the three
- so you have Nine hundred and ninety nine X is equal to three thousand two hundred and fifty four
- And then you can divide numerator, err.. divide both side of this by Nine hundred and ninety nine
- divide both sides by nine hundred and Ninety Nine
- and you are left with X is equal to three thousand two hundred and Fifty four
- over nine hundred and ninety nine, and so obviously this is 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 try to
- figure out what the , the two, the point two five seven repeating forever is equal to
- and just have three being the whole number part of the mixed fraction
- or you can just divide Nine nine nine into three thousand two hundred and fifty four
- actually we can do that pretty straight forward, it goes into three times, and the remainder,
- let me .. let me do it , so just do , just do the whole division
- nine nine nine goes to Three thousand two hundred and fifty four
- it will go into three times, now we know that because this is originally three point two five seven
- so we just going to find out the remainder so, three times nine is twenty seven
- three times nine is twenty seven
- we have to add the two;s so we have twenty nine
- three times nine is twenty seven we have to add two's so we have twenty nine
- so we are left with, if we subtract if we re group, or borrow or however we want to call it
- this could be fourteen, this could be a four, let us a new colour
- this would be a four, and then the four is still smaller than this nine, so we need to re group again
- so this could be fourteen and this could be one, but this is still smaller then this nine right over here, so we
- regroup it again , this would be eleven and then this is a two
- fourteen minus seven is seven, fourteen minus nine is five
- eleven minus nine is two, so we are left with, we are left with
- can do that right, yeah, so this is going to be equal to three and two hundred and fifty seven
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