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!
3 Answers
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 $S=0.\overline{34}=0.343434\cdots$
$100S=34.3434\cdots$
$\implies 100S-S=34.3434\cdots-0.343434\cdots=34$
$\implies 1.2\overline{34}=1.2+0.01\cdot\frac{34}{99}$
$1.2\overline{34}= 1\frac{2\frac{34}{99}}{10}=1\frac{232}{990}=1\frac{116}{495}$
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
