How to find the repeating decimal 0.82 with 82 repeated as a fraction?

Question

Rhys Murphy · Accepted Answer

Let me introduce you to some notation in case you've never seen it:
A bar over the repeating pattern of decimals can be used to represent a repeating decimal.
So instead of writing

0.828282 ...

, you can write

0 . \bar{82}

If we multiply

0 . \bar{82}

by

(100 - 1)

we get an integer:

(100 - 1) 0 . \bar{82} = 100 \cdot 0 . \bar{82} - 1 \cdot 0 . \bar{82}

(100 - 1) 0 . \bar{82} = 82 . \bar{82} - 0 . \bar{82}

(100 - 1) 0 . \bar{82} = 82

Notice that the

100

shifts the number two places to the left - the length of the repeating pattern. Then the

- 1

cancels out the repeating tail.
Next divide both ends by

(100 - 1)

and simplify:

0 . \bar{82} = \frac{82}{100 - 1} = \frac{82}{99}

82

and

99

have no common factor, so this is in simplest terms.

Alternative method
An alternative method recognises that:

0 . \bar{82} = 0.82 + 0.0082 + 0.000082 + ...

is a geometric series, with initial term

a = 0.82

and common ratio

r = \frac{1}{100}

.
This has sum given by the formula:

s_{\infty} = \frac{a}{1 - r} = \frac{0.82}{1 - \frac{1}{100}} = \frac{0.82}{\frac{99}{100}} = \frac{0.82 \cdot 100}{99} = \frac{82}{99}

How to find the repeating decimal 0.82 with 82 repeated as a fraction?

Answered question

Answer & Explanation

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