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Course: Class 9 (Old)>Unit 1

Lesson 2: Real numbers and their decimal expansions

Converting repeating decimals to fractions (part 2 of 2)

Repeated decimals can be converted into fractions by shifting the decimal to the right and subtracting the decimals. To do this, multiply the number by 10 to the second power, then subtract. For example, 0.363636 repeating is 4/11 and 0.7141414 repeating is 707/990. Another example is 3.257257257 repeating, which is 3257/999. This calculation can be done in the head or by borrowing. After the subtraction, the numerator and denominator can be reduced and the fraction can be simplified. Created by Sal Khan.

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• what is 0.333333333333333 in a fraction
• Here's a little table of repeating decimals. Notice that they all follow a pattern:

1/9 = 0.111111111111111...
2/9 = 0.222222222222222...
3/9 = 0.333333333333333...
4/9 = 0.444444444444444...
5/9 = 0.555555555555555...
6/9 = 0.666666666666666...
7/9 = 0.777777777777777...
8/9 = 0.888888888888888...

Because 3/9 = 1/3 and 6/9 = 2/3, the following are also true:

1/3 = 3/9
1/3 = 0.333333333333333...

2/3 = 6/9
2/3 = 0.666666666666666...

Hope this helps!
• Why would the repeating decimal 0.714141414... which equals x be multiplied by 100 instead of 1000 or 10?
• so you will have 71 left over and you will get rid of the other numbers that are after the decimal.
• How would you solve a number like 5.345555555...?
• You can do it in this way too...
x=5.345555555...
10x=53.455555...
100x=534.5555...
1000x=5345.555...
Since the decimal part is same, we can subtract 100x from 1000x.
So,
1000x=5345.555...
- 100x=-534. 555...
________________________
900x=4811
________________________
x=4811/900
Since x=5.345555...
5.345555...=4811/900
• In the second example, where Sal gets a decimal numerator, I found it easier to use 1000x and 10x instead. Why 1000x and 10x? You make a subtraction where the minuend (top part) is the number multiplied by 10 as many times as you need to move the decimal point the to the right side of the repeating part. For 0.714 with 14 repeating, you multiply by 1000 to get to 714.14 with the last 14 repeating.

For the subtrahend (bottom part of a subtraction), you multiply the number by 10 as many times as you need to get the decimal point to the left side of the repeating part. For 0.714 with 14 repeating, use 10x to get to 7.14 with 14 repeating. Now the repating part is directly after the decimal point in both minuend and subtrahend, so they cancel out nicely.
• Even I was confused as to why was he complicating it. I found using 1000x and 10x easier as well.
• I'm actually losing my mind
• you are not the only one trust me-
• 2 Minutes into studying and my brain abandoned skull
• okk but how do I solve the repeated beatings I get for not getting all A's
• You can solve the beatings by doing simple arithmetic. First, number all beatings so far. For you, I would assume around 143 which is the yearly average number of absolute whoops in the US. Then take 143 (the magic number) divided by 77, (Because it's my favorite number) and add the answer to the angel your parents whoop you from. Next, show your parents the results of your math equation and have them beat you some more because you messed up this easy equation.