Save time with this short cut method for converting a repeating decimal to a fraction. It even works on a crazy repeating decimal like,
Welcome to MooMooMath where we upload a new Math video everyday. In this video I would like to show a shortcut for converting repeating decimals to a fraction.
First the rules.
First, with the numerator, if you only have repeating numbers like .222 you write the repeating number as your numerator.
If you have a mixed recurring decimal like .16666 subtract
Notice the 1 does not repeat like the 6.
The non-repeating number from the repeating number. Make sure the repeating number has the same number of digits as the repeating number.
The denominator rules for non-repeating numbers, add a 0 ,and a 9 for repeating numbers.
Once you write the fraction you then reduce.
Lets work a couple of examples.
Lets start with .333 which is a repeating digit without any non-repeating numbers.
I will write 3 as the numerator, and nine for the repeating number.
I can reduce this down to 1/3
Another example .666
Write 6 which is the repeating number, you have one repeating number so write a nine.
This will reduce to 2/3
Lets go with .833 which has a non-repeating number.
What we will do is subtract the repeating number 83 minus 8 because that is the non-repeating number.
83-8 equals 75, which becomes your numerator.
We have 1 non-repeating number so I will write a 0 and a 9 for the repeating number. 75/90 reduces to 5/6
How about this example?
.1818 we have a repeating number with two digits so I will write 18 and then for the denominator I will write a 9 for each digit which becomes 99 and 18 over 99 can be reduced to 2/11
Lets work a couple more.
.16 we have a non-repeating number.
I will take the repeating number minus the non-repeating number which is 1 and subtract which equals 15
15 over 0 for the non-repeating number and a 9 for the repeating number which becomes 15 over 90 and you can reduce to 1/6
I took 16 minus 1 which is your non-recurring which equals 15 over 90 which reduces to 1/6
The one exception I have found is this,
Take .4166 and you may write the repeating number as 416 but because you have two non-repeating numbers you need at least two repeating numbers. Instead of writing 416 write 4166. From there is would be 4166 minus 41 equals 4125 and 4125 over two non-repeating and two repeating and I cheated and put this in a fraction reduce, which reduces to 5/12
Let me go through two more just to show how this works.
Take .251818 repeating.
Take the repeating fraction of .2518 minus the non-repeating and you have two non-repeating minus 25 because that is the non-repeating portion.
2518 - 25 equals 2493 and you have two non-repeating so write 00 and two repeating numbers so write 99 and simplify to 277/1100 and if you divide it equals .2518
I will show you a crazy number like .70643889889
The repeating is 889 you can see it repeating twice.
I will first write the repeating decimal 70643889 minus the non-repeating of 70643 and when you subtract you get 70573246 and I will write it again 70573246 and use the same rules for the denominator.
For the non-repeating I add a 0 and I have 1 2 3 4 5 zeros.
1 2 3 4 5 and three repeating numbers and three non-repeating 999 and that is your fraction I will not reduce, but test it out and if you divide the numbers it will equal the repeating fraction. It does work
Hope this helps with converting repeating decimals to a fraction
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