Question

How do you convert \[0.55555...\] (\[5\] being repeated) to fraction?

Answer 1

Hint: In this type of questions first let this value be \[x\] then multiply it with \[{{10}^{n}}\] where \[n\] is the same as the number of digits that are being repeated, in this \[n=1\] so multiply \[x\] with \[10\] then subtract the initial value from the final value that is obtained after multiplication and then simplify you will get the answer.

Complete step by step answer:
Here the question is we need to convert \[0.\bar{5}\] to a fraction where a bar on \[5\] represents that it is repeating.
Let this value be \[x\]
\[\Rightarrow x=0.\bar{5}\]
\[\Rightarrow x=0.5555...--(1)\]
Now multiply both sides by \[10\]
\[\Rightarrow 10x=5.5555...--(2)\]
Now subtracting the \[Eq(1)\] from \[Eq\left( 2 \right)\]
\[\Rightarrow 10x-x=5.5555...-0.5555...\]
\[\Rightarrow 9x=5.0000...\]
\[\Rightarrow x=\dfrac{5}{9}\]

Hence \[0.\bar{5}\] in fractional form is \[\dfrac{5}{9}\]

Note: In this question as the repeating part is till infinity therefore we can’t directly convert into fractions we first need to remove that repeating part for this we have multiplied with the \[10\] raised to power with same as the repeating because when we subtract these values the part after decimal will be completely removed and hence we will have only the absolute part which can be converted easily into fractions by simplifying it.