Converting Reapting Decimals to Fractions?

Question

I would like to know how you turn a reapting decimal to a fraction, for example 0.09 with the 9 with a line over (0.09999999...)

Also can someone explain how to convert a number with a line over 2 numbers? (0.4545454545454545454545)

Answer 1

The answers above are great for cases where the repeating part is right at the beginning as shown in the examples below:

0.22222222... = 2/9
0.54545454... = 54/99
0.298298298... = 298/999
0.142857142857... = 142857/999999 = 1/7
Division by 9s causes the repeating pattern. Notice how the number of 9s is equal to the length of the repeat. Don't forget to reduce the fraction if you can.

There are other cases though that you asked about. For example if there are zeroes preceeding the repeating decimal, then you need to convert as follows:

0.022222222... = 2/90
0.099999999... = 9/90 = 1/10 (same as 0.1 by the way)
0.00054545454... = 54/99000
0.00298298298... = 298/99900
Note that adding zeroes to the denominator adds an equivalent number of zeroes before the repeating decimal.

To convert a decimal that begins with a non-repeating part, such as 0.21456456456456456..., to a fraction, write it as the sum of the non-repeating part and the repeating part.
0.21 + 0.00456456456456456...

Next, convert each of these decimals to fractions. The first decimal has a divisor of power ten. The second decimal (which repeats) is converted according to the pattern given above.
21/100 + 456/99900

Now add these fraction by expressing both with a common divisor
20979/99900 + 456/99900

and add.
21435/99900

Finally simplify it to lowest terms
1429/6660

and check on your calculator or with long division.
= 0.2145645645...

That should cover all the cases you might have.

Answer 2

very easy... e.g.
X = 0.123451234512345..... =
(12345 + 0.123451234512345...) /100000 = (12345 + X)/100000
so 100000*X = 12345+X
so 99999X = 12345
so X=12345/99999

similar: X = 0.999999... = (9+0.9999...)/10 = (9+X)/10
so 10X = 9+X, so 9X=9, so X=1 (exactly, 0.99999... = 1)
0.09999... = 0.9999.../10 = 1/10

similar: X = 0.454545... = (45+X)/100
so: 100X = 45+X, so 99X=45, so X = 45/99 = 5/11
I hope you got the idea.

In general,
0.aaaa... = a/9
0.abababab... = ab/99
0.abcabcabc... = abc/999
etc etc etc.

Answer 3

It's all algebra...

Lets say your decimal is .45 repeating...
Set the decimal equal to x.

x = .4545454545.....

Multiply by a mulitple of 10, in order to make the number have one set of the repeating number to the left of decimal...

In this case, .454545... * 100 = 45.45454545...

so we have...

x = .4545454545...
100x = 45.45454545...

Subtracting gives us...

99x = 45
x = 45/99

This can be used for any repeating decimal, and is used as an algebraic proof that .99999999999.... = 1.

Hope this helps :)

Answer 4

well i do it like dis!
u take ur decimal 0.99999
find wats alway being repeated, this time its 9
so u put da first 9 over 10
put the second 9 over 100 an so on
take these fractions an apply them 2 dis formula

a over (1-r)

a is equal to ur first fraction ie 9 over 10
r is equal to wat u would multipe the first fraction by to get the second ie 1 over 10

so ur fraction wud be 9/10 over 1-1/10

the same ting goes for 0.45454545
just this time a wud be equal to 45/100 and r wud be 1/100

hope dat helps! if nt contact me and il try again!

Answer 5

Let x = .454545.... (1)
100x = 45.454545....(2)

(2) - (1) =
99x = 45
There fore x = 45/99