# Corresponding numerator of the fraction

I am stuck at this question cant think of how to resolve this, sorry I did try to do working but can't think of any right solution. The question is

If the denominator is $9900$, then what is the corresponding numerator of the fraction with the recurring decimal $0.ab\overline{cd}$ when $a = 7$, $b = 6$, $c = 7$, $d = 6$? Don't forget that the denominator must be $9900$ and note that the bar is above $cd$ only.

Ok after some more googling, I came to know I have to do this:

Step 1, Substitute

$0.7676$

Step2, Solve

let $x = 0.7676$ --- (1)
$100x = 76.76$ --- (2)
then $(2) -(1)$
so answer will be $75.99$, correct?

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The point is that the $76$'s go on forever on the right. So your subtraction comes out $76$, not $75.99$ or $75.9944$ – Ross Millikan Sep 4 '12 at 14:18

$100x=76+x$ so $99x=76$, thus $x=76/99=7600/9900$

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my tots exactly i just got that on my calculator and was wondering if i am correct..tks for the confirmation – JackyBoi Sep 4 '12 at 14:27

Informally, the bar means your decimal is $0.76767676767676\ldots$. The first two digits could be different in $0.ab\overline{cd}$ but in this case you have $a=c=7, b=d=6$. To convert these to fractions observe that if $x=0.76767676767676\ldots, 100x=76.76767676767676\ldots$ Subtracting $99x=76$ (this can be made formal if needed). So your fraction is what?

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Oh (frantically raises hand).. so u mean to say the fraction is 76/99? then according to the question my answer is 76? – JackyBoi Sep 4 '12 at 14:21
@JackyBoi: Because the denominator is specified as $9900$, the question wants $7600$, but you have understood correctly. – Ross Millikan Sep 4 '12 at 14:58

For the general case, $10000\cdot 0.ab\overline{cd} = abcd.\overline{cd}$, $100\cdot 0.ab\overline{cd} = ab.\overline{cd}$, hence $9900\cdot 0.ab\overline{cd} = abcd-ab$.

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