Show that $1.272727 \ldots=1 . \overline{27}$ . can be expressed in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.
Show that $1.272727 \ldots=1 . \overline{27}$ . can be expressed in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.
Let $x=1.272727 \ldots$ since two digits are repeating, we multiply $x$ by $100$ to get
$100 x=127.2727 \ldots$
So, $100 x=126+1.272727 \ldots=126+x$
Therefore, $100 x-x=126,$ i.e., $99 x=126$
i.e., $x=\frac{126}{99}=\frac{14}{11}$
You can check the reverse that $\frac{14}{11}=1 . \overline{27}$
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