everyone! I came across a problem in math that dealt with bar notation. Does anyone know how, for instance, 1.234(with a bar notation over the 34) is expressed as a fraction? I know already how 1.22(with a bar notation over the 22) is expressed as 1 and 2/9. Any answer would be helpful. Thanks!
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2Let $x=1.2\overline{34}$. Then $100x=123.4\overline{34}$. Subtracting the former from the latter gives you $$100x-x=123.4-1.2=122.2$$ Can you take it from here? – Jyrki Lahtonen Aug 17 '13 at 18:03
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Hint: Multiply $1.2\overline{34}$ with 100 – peterwhy Aug 17 '13 at 18:04
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@JyrkiLahtonen, "with a bar notation over the 34" – lab bhattacharjee Aug 17 '13 at 18:08
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@JyrkiLahtonen, yes there is no automatic update on comments:) – lab bhattacharjee Aug 17 '13 at 18:13
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@lab: sorry about the hassle here. +1 to your answer. – Jyrki Lahtonen Aug 21 '13 at 18:32
3 Answers
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HINT:
Let $x=1.2\overline{34}$
$\implies 100x=123.4\overline{34}$
So, $100x-x=123.4\overline{34}-1.2\overline{34}=122.2$
Alternatively, $1.2\overline{34}=1.2+0.01\cdot 0.\overline{34}$
Now, let $S=0.\overline{34}=0.343434\cdots$
$100S=34.3434\cdots$
$\implies 100S-S=34.3434\cdots-0.343434\cdots=34$
$\implies 1.2\overline{34}=1.2+0.01\cdot\frac{34}{99}$
lab bhattacharjee
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@user2516595, my pleasure. If there are $n$ digits inside the bar, we need to multiply by $10^n$ where integer $n\ge1$ – lab bhattacharjee Aug 17 '13 at 18:12
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@user2516595, we have $99x=\frac{1222}{10}=\frac{611}5,x=\frac{611}{5\cdot99}$ – lab bhattacharjee Aug 17 '13 at 18:19
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$1.2\overline{34}= 1\frac{2\frac{34}{99}}{10}=1\frac{232}{990}=1\frac{116}{495}$
John Douma
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let x=1.23434..... eq.1
so eq.1 mulyiply by 10
10x = 12.3434... eq.2
so eq.2 multiply by 100
1000x =1234.34..... eq.3
subtract eq.2 from eq.3
1000x-10x = 1234.34-12.34
990x =1222
x=1222/990=611/495
saransh
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