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Show that $$1.272727... =1.\overline {27}$$ can be expressed in the form $$\frac {p}{q},$$ where $$p$$ and $$q$$ are integers and $$q\neq 0.$$
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QuestionMathsClass 9

Show that $$1.272727... =1.\overline {27}$$ can be expressed in the form $$\frac {p}{q},$$ where $$p$$ and $$q$$ are integers and $$q\neq 0.$$

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 $$\frac{14}{11}=1 . \overline{27}.$$