Repeating decimal

A repeating decimal is a decimal number where one or more of the digits repeats indefinitely. For example, when you convert the fraction 1/3 to a decimal, you get 0.333333 . . . where the 3s repeat indefinitely. That makes the decimal representation of 1/3 a repeating decimal. It can be written with a line above the number or numbers that are repeated, like this:
0.3 = 0.333333 . . . Repeated numbers can be more than a single digit.
For example, if you divide 5 by 7 you get 0.714285 714285 714285 . . . , which can be represented as
0.714285 Note: All repeating decimals are the result of dividing one whole number by another.
In other words, by converting a fraction to a decimal.

It is possible, since all repeating decimals are fractions, to reconstruct the fraction, given the repeating decimal. This is how you do it:

Multiply your number by a power of 10 in order to get the repeating part of the decimal to the LEFT of the decimal point.
For example, in the repeating decimal 0.55555555 . . . multiply by 10 to get:
5.5555555 . . .
Calling the original repeating decimal X, this new value is 10X

Subtract the smaller value from the larger value:

10X = 5.55555555555 . . .
- X = 0.55555555555 . . .
9X = 5
X = ⁵⁄₉

This last step has gotten all the repeating decimals to cancel each other, except for the first one(s).
Now, it's easy! 10X - X = 5, so 9X = 5 and the fraction is ⁵⁄₉
(divide 5 by 9 on your calculator and see if this is true!)

Now here are some problems for you (use your calculator):

Convert the following fractions to decimals.
They will result in repeating decimals. First find the decimal representation of the fraction and then write them with the line above the repeating part
(and only the repeating part!):

1/12 = _____________ = ______
7/9 = _____________ = ______
11/13 = ____________ = ______

Convert 0.4 back into a fraction:

Expand it to a repeating decimal:
0.4 = _____________ . . .<= this is X
Multiply it by 10:
__________________ . . .<= this is 10X
Subtract the original expanded decimal from it:
10X: ___________ . . .
- X: ___________ . . .
9X: ____________, so X = ______

Converting decimals with leading non-repeating numbers How do you convert a decimal like 1.042424242 . . . that has leading non-repeating numbers?
Well, its more complicated, but here's how:

1). Multiply the original number by a sufficient power of 10 to get the repeating part totally to the LEFT of the decimal.	1.04242424242 . . . times 1000 = 1042.424242 ... This is 1000 X Note that the repeating part, 42, is now on the LEFT side of the decimal point.
2). Multiply the original number by a sufficient power of 10 to get ONLY the repeating part to the RIGHT of the decimal.	1.042424242 ... times 10 = 10.424242 . . . This is 10 X Note that the repeating part, 42, is now on the RIGHT side of the decimal.
3). Subtract the second value from the first	1000X 1042.424242 . . . - 10X 10.424242 . . . 990X = 1032 so X = 1032/990 Try this on your calculator to see if it is true!

OK, here are a couple of problems for you. In the first I will give you the powers of 10 for each multiplication, but in the second you're on your own!

Problem 1: convert 0.538383838 . . . to a fraction.

1). Multiply the original number by a sufficient power of 10 to get the repeating part totally to the LEFT of the decimal.	0.538383 . . . times 1000 = ________ ... This is 1000 X Note: make sure that the repeating part is now on the LEFT side of the decimal point.
2). Multiply the original number by a sufficient power of 10 to get ONLY the repeating part to the RIGHT of the decimal.	0.5383838 ... times 10 = _______ . . . This is 10 X Note: make sure that the repeating part is now on the RIGHT side of the decimal.
3). Subtract the second value from the first	X = ______ Try this on your calculator to see if it is true!

Problem 2: convert 22.33333 . . . to fraction form.

1). Multiply the original number by a sufficient power of 10 to get the repeating part totally to the LEFT of the decimal.
2). Multiply the original number by a sufficient power of 10 to get ONLY the repeating part to the RIGHT of the decimal.
3). Subtract the second value from the first	X = ______ Try this on your calculator to see if it is true!