Convert recurring decimal to fraction?

Question

Convert the recurring decimal 2.136 (with only the 3 and the 6 being the recurring numbers with the dot on the top) to a mixed number, in its simplest form.
Please show and explain steps carefully

Answer 1

Start with the fixed part... 2.1 is 21/10.

The rest of the number is 0.036363636...
Let's call that x.

x = 0.036363636...
Notice what happens if you multiply by 100. I picked 100 because there are two repeating decimals. If there were 1 you would pick 10, if 3, you would pick 1000, if 4 pick 10000, etc.
100x = 3.6363636...
.......x = 0.0363636...

Subtract and the repeating decimals are cancelled:
99x = 3.6

Multiply by 10 to get rid of the last remaining decimal:
990x = 36

Now solve for x:
x = 36/990

Now you just need to add your two fractions:
21/10 + 36/990

Get a common denominator:
21*99/990 + 36/990

= (2079 + 36)/990
= 2115/990

Now reduce:
= 235 / 110
= 47 / 22
= 2 3/22

So your answer is 2 and 3/22.

You can also do this in one step, without having to do the two parts:
100x = 213.63636...
.......x = .....2.13636...

Subtracting:
99x = 211.5

Multiply by 10 to remove the decimal:
990x = 2115

Solve for x:
x = 2115/990
x = 47/22
x = 2 3/22

A final way, if you now understand the underlying math is to just do the following:

Pull out the part in front of the decimal (2)... just keep that for later.

The remaining part is 0.136363636...
Since we have two repeating decimals, use two nines (99). However, since it is offset by one fixed digit, add a trailing zero (990).
Take the fixed digit (1) and subtract it from your number (136, ignoring anything but the first repeat). This leaves 135.

So the fraction should be 135/990. This reduces to 3/22.

So it becomes 2 and 3/22.

Always check that the final result makes sense. Obviously the first part of the mixed fraction should match the decimal (2) and it does. If you put 3/22 in your calculator, you get 0.1363636...

There's 3 different methods and they all give you 2 and 3/22.

Answer 2

Converting Recurring Decimals To Fractions

Answer 3

1 non-repeating digit after the decimal = one 0.
2 repeating digits after the decimal = 2 9's.

Subtract the non-repeating part from the rest:
136 - 1 = 135

So the fraction is 135 / 990
(990 from the 2 9's and 1 0.)
Simplify to get
= 27 / 198
= 3 / 22
and add the 2 to get
2 3/22

Answer 4

x=2.136 recurring
=2.1363636363636363636
multiply by 10
10x=21.36363636363636363636...(1)
multiply by 1000
1000x=2136.363636363636363636...(2)
(2)-(1)
990x=2115
x=2115/990
=423/198
=141/66
=47/22

Answer 5

let x = 2.1363636....
1000x=2136.363636..
999x=2134.23; this is the important step because it gets rid of the reoccurring digits.
99900x=213423
Simplify

Answer 6

28.91