Converting repeating decimals to fractions
Converting Repeating Decimals to Fractions 1 Examples of how to convert basic repeating decimals to fractions
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- 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 I had the repeating decimal zero point seven and sometimes it'll be written like that. [bar above the seven]
- Which just means that the 7 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 sevens.
- So the trick to converting these things into fractions is to essentially set this equal to a variable.
- And we will sort of do it step by step.
- So let set this equal to a variable, let me call this x.
- So x is equal to zero point seven and the seven repeats on an on for ever.
- Now what would ten x be?
- Well let's think about this, ten x would just be ten times this
- so it would be, we can even think of it right over here.
- it would be, if we multiplied this by ten.
- We would be moving the decimal one to the right
- it would be seven point seven seven seven, on and on and on forever.
- Or we could say it is seven point seven repeating.
- Now this is the trick here.
- Let me make these equal to each other.
- So we know what x is, it is point seven seven repeating forever.
- Ten x is this. And it is another repeating thing.
- Now the way we can get rid of the repeating decimals
- is if we subtracting x from ten x, right?
- Because x has all these repeating point seven seven seven.
- If you subtract that from seven point seven seven seven,
- you are just going to be left with seven.
- So let's do that.
- Let me rewrite it here.
- Ten, ten x is equal to seven point seven repeating.
- Which is equal to seven point seven seven seven on and on forever.
- As we established earlier that x is equal to
- zero point seven repeating; which is equal to seven point seven seven seven on and on and on forever.
- Now what happens when you subtract x from ten x.
- So we are going to subtract the yellow from the green.
- Well ten of something minus one of something is just going to be nine of that something.
- And then that is going to be equal to:
- What's seven point seven seven repeating, minus point seven seven, going on and on, forever repeating?
- Well it is just going to be seven.
- These parts are going to cancel out, you are just left with seven or we could say,
- these two parts cancel out and you are left with seven.
- So you get nine x is equal to seven.
- To solve for x you just divide both sides nine.
- Well I could do all three sides,
- although these are all saying the same thing
- and you get x is equal to seven ninths. [7/9]
- Let's do another one.
- I will leave this one here so you can refer to it.
- So let's say I have the number one point two repeating.
- So this is the same as one point two two two and on and on.
- Whatever the bar is on top of, that is the part that repeats forever.
- So just like we did over here, lets set this equal to x.
- And let's say ten x -- let's multiply this by ten.
- So ten x is equal to twelve point two repeating.
- Which is the same thing as twelve point two two two on and on and on.
- Then we can subtract x from ten x.
- And you don't have to rewrite it but I rewrite it here, just so we don't get confused.
- So we have x is equal to one point two repeating.
- And if we subtract x from ten x what do we get.
- On the left hand side we get x minus,
- ten x minus x is equal to to nine x 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 zeros.
- Twelve minus one is eleven.
- You have nine x is equal to eleven.
- Divide both sides by nine,
- and you get left with x is equal to eleven over nine. [x=11/9]
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