How do you turn a repeating decimal into a fraction?

There are different methods. Algebraically as follows:

ex: 0.272727.....

write a simple equation x = 0.272727... notice 2 digits are repeating so multiply by 100 (1 with 2 zeros because 2 digits repeat)

100 x = 27.272727.... write the original equation below this and subtract

x = 0.272727...

99x = 27 notice all the decimals cancel. now divide by the coefficient but don't really divide just write as a fraction

99x/99 = 27/99 reduce the fraction

x = 3/11

I will give you an example that my son had in one of his tests and he found it difficult, but nothing is difficult if you apply your knowledge to solve it:

13.0123454545...

= 13.0123 + 0. 0000454545...

= 13.0123 + 0.454545... x 1/10000 now try to write 0.454545... as a fraction

let x = 0.454545... (1) multiply by 100 both sides

100x = 45.454545... (2) subtract (1) from (2)

99x = 45 divide both sides by 99

x = 45/99

Thus, 0.454545... = 45/99 replace it into the expression above

= 130123/10000 + 45/990000

= [(99)(130123) + 45]/990000

= (12882177 + 45)/990000

= 12882222/990000

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1. take the repeating decimal and count how many digits repeat.
2. multiply it by a power of 10 that has the same number of zeros as repeating digits; eg if 2 digits repeat multiple by 100
3. subtract the original from the multiplied version.
4. put the result over the multiplier in step 2 less 1
5. if there is a decimal point in the numerator, multiply both top and bottom by the same power of 10 to get rid of the decimal point
6. simplify the fraction.

Using the examples above:

0.272727.... has 2 digits repeating, so multiply by 100

0.272727... × 100 = 27.2727...

Subtract to get: 27.272727... - 0.272727... = 27

Put over 100 -1 = 99: 27/99

simplify: 27/99 = (9×3)/(9×11) = 3/11

0.1666... has 1 repeating digit so multiply by 10

0.1666 × 10 = 1.666...

subtract to get 1.666... - 0.1666... = 1.5

Put over 10 - 1 = 9: 1.6/9

As the numerator contains 1 decimal place multiply top and bottom by 10 to get rid of the decimal in the numerator: 1.5/9 × 10/10 = 15/90

Simplify: 15/90 = (15×1)/(15×6) = 1/7

13.0123454545... has 2 repeating digits, so multiply by 100

13.0123454545... × 100 = 1301.234545...

Subtract to get: 1301.234545... - 13.0123454545... = 1288.2222

Put over 100 - 1 = 99: 1288.2222/99

As numerator contains 4 decimal places, multiply top and bottom by 10000 to get rid of the decimal in the numerator: 1288.2222/99 × 10000/10000 = 12882222/990000

Simplify: 12882222/990000 = (18×715679)/(18×55000) = 715679/55000 = 13 679/55000