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Decimal to Fraction

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Step 11: Using the formula for "pure" repeating decimals

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The trick here is to expand the decimal number as a the sum of its integer part, the non-repeating part, and whatever is left over. The whatever-is-left-over part will contain a pure repeating decimal times a power of 10. Then I use the formula for a pure repeating decimal to find the equivalent fraction f(p) = p/(10n-1) Then I substitute that fraction back into the previous expression.

Let's revisit one of the repeating decimals from Step 7.
r = 3.45[6]
From inspection this is an impure repeating decimal because it has a non-repeating part. It also has an integer part, which is another reason why it is not pure.

Expand this decimal as a the sum of its integer part, the non-repeating part, and whatever is left over.
r = 3 + 0.45 + 0.00[6]
The middle term is equal to 45/100 and the last term is equal to (1/100)*0. [6]
r = 3 + 45/100 + (1/100)*0.[6]
Using the formula for a pure repeating decimal, f(p) = p/(10n - 1), find 0. [6] =6/9 =2/3. Substitute this into the previous expression and get:
r = 3 + 45/100 + (1/100)*(2/3)
Find a common denominator for the two fractional terms, and add these together. This gives an answer in mixed fraction form.
r = 3 + 135/300 + 2/300 = 3 + 137/300
Convert this to an improper fraction, just to check that it is the same result as the one found in the example in Step 7.
r = 3 + 137/300 = 900/300 + 137/300 = 1037/300

Another example:
r = 0.00[45]
This decimal has zero integer part, and non-repeating part consisting of two zeros. The repeating part is not "touching" the decimal point, so this repeating decimal cannot be pure.

However it is equal to (1/100)*0. [45], and 0. [45] = 45/99 is a pure repeating decimal.
r = (1/100)*0.[45] = (1/100)*(45/99) = 45/9900 = 1/220
 
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