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A Simple Way To turn Repeating Decimals Into Fractions
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Feb 19, 2025
Do you ever get frustrated when trying to work with repeating decimals? You're not alone! Many students find them difficult to understand. But there's a simple way to turn a repeating decimal into a fraction. With this method, you'll be able to quickly and easily work with any repeating decimal.
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all right so now we are going to learn
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how to write
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repeating decimals as fractions okay let
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me get rid of that g here it's an extra
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G and what am I saying by that let's say
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you have
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zero point six six this is a repeating
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decimal number okay this is basically
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saying that this is
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0.66666
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same thing so how can you turn this
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decimal number with repeating digits as
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a fraction how can you do that that's a
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simple technique you're gonna learn how
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to do it okay
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so let's assume that this number is a
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let's call it a okay n equals
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0.66
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okay now what you want to do is you want
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to turn this into a fraction okay so
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let's say we assume that our multiply
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this by ten okay it's a simple technique
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so I multiply this by ten that means I'm
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multiplying this by ten right so if I
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multiplying by 10 you're going to move
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this one place to the right so this is
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going to turn into six which six six six
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six and all that all right so now the
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next step is critical basically you are
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trying to solve you are trying to turn
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this into a fraction so if you recognize
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here they have the same ending okay n is
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the same thing n has 0.666 and then 10 n
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as 6.666 so I want to get rid of these
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decimal portion so how do I do that I
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want to subtract it so if I do 10 n
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minus a
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that gives me
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6.666
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right minus
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0.666 right so what's gonna happen is
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this this is going to take care of this
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ending okay this is going to be gone
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right that goes away so you're left with
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10 and
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minus n equal to 6 because 6 minus 0 is
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6 right now 10 n minus n is just nine n
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because n is just one n so you're left
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with six nines nine n equals to six and
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the solve for n
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we have nine and
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equals six that becomes just a simple uh
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equation you want to divide by nine
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because you're trying to isolate your
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n so n equals to six over nine and again
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last time we talked about simplifying
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fractions what's the GCF between six and
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nine meaning what goes into six that
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goes into nine three so six can be
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written as three
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times two and nine is three times three
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so what happens this cancels out you're
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left with two third okay so basically
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zero
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0.66
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is equal to
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2
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there okay so this is easy when the
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numbers are the same so we're going to
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work on a separate example let's assume
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that I have 0.42 okay let me write to
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0.42 here
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so let's try and turn
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0
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and 42
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into
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a fraction
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okay let's try to do that real quick
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so I'm going to start the same way I'm
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going to assume that
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X is equal to 0.42 okay STP in numbers
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so remember I'm trying to get rid of the
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decimal portion right so I have 4 2 so
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that means this is also equal to 0.42
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42 42 42
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that's what's happening right so now the
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key is this you're trying to get rid of
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this 42 42 42 42s so I'm going to
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multiply this by a hundred okay I'm
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going to multiply this by 100 so that's
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going to happen here so I'm going to do
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100
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times x is equal again we're moving two
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places to the right
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42 142
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42 42
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and the goal as you can recognize is to
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get rid of this portion okay we're
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trying to get rid of this
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decimal portion you see that
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so if I do 100x
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minus X is going to give me 42 because
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42 minus 0 is 42. now I have 99 x
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equals to 42 and to solve for x I'm just
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going to divide by 99
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so that gives me x equals 42 over 99.
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now there's no number let me try and see
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oh we can see let's try to see if you
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can simplify this okay let's try and see
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we might be able to
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so
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I have 99 and now 42. so here's what I
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do here I want to see if there's a GCF
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here a greatest common factor now 99 9
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plus 9 is 18 right so I know that they
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are both divisible by 3 why because 4
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plus 2 is 6. four plus two is six and
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six is a multiple of three and then nine
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plus nine is eighteen and eighteen is
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also a multiple of three which means
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that these numbers are both divisible by
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three so if I divide 42 by 3 I get 14
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if I divide 99 by 3 I get 33 so this
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becomes 14 over 33. all right so I hope
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that you recognize the pattern so if you
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have two digits that are repeating that
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are
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uh two digits that are like repeating
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but are not the same you multiply this
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by 100 if you have a three digits that
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are repeating but are not the same again
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you multiply by a thousand so this is
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the pattern all right so this is how you
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turn a repeating decimal into a fraction
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all right and I'll see you next time for
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the next one thank you
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