Question

what is the rational number equivalent to 3 point 12 with a bar over 12? 3 and 4 over 33 3 and 8 over 33 3 and 10 over 39 3 and 5 over 39

Answer 1

3 4/33

Explanation:

`3.\overline{12}=3(12)/(99)=3(4)/(33)`

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Without going through the whole derivation, a repeating decimal of n digits can be written as those n digits over the same number of 9s.

Here, we have the repeating decimal .121212... so its equivalent fraction is 12/99. That can be reduced to 4/33.

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Comment on the above

If the repeating digits don't start at the decimal point, the fraction must be scaled by the appropriate power of 10. For example, ...

0.83333... = 0.8 + 0.03333... = 0.8 + 0.1·(0.3333...) = 0.8 + 0.1·(3/9)

= 8/10 +(1/10)(1/3) = 8/10 + 1/30

= 24/30 + 1/30 = 25/30

= 5/6