Question

Now that you have converted a terminating decimal number into a fraction, try converting a repeating decimal number into a fraction. Repeating decimal numbers are more difficult to convert into fractions. The first step is to assign the given decimal number to be equal to a variable, . For the decimal number , that means . If , what does 10x equal?

Answer 1

The first step is to assign the decimal number to a variable.

For the repeating fraction 0.111_1, this would look like
.. x = 0.111_1 . . . . . . . . . . where we use an underscore to identify the following digit(s) as repeating

The next step is to multiply that value by 10 to a power equal to the number of repeating digits. If there is one repeating digit (as here), then you want (10^1)x = 10x.
.. 10x = 1.111_1

The third step is to subtract x from this.
.. 10x -x = 1.111_1 -0.111_1 = 1
.. 9x = 1

And the final step is to divide by the coefficient of x.
.. x = 1/9 . . . . . . this is the value of the repeating decimal fraction.

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Here's one that's a little more complicated. It is done the same way.
.. x = 3.254545_45
.. 100x = 325.454545_45
.. 100x -x = 99x = 322.2
.. x = 322.2/99 = 3222/990 = 179/55