Question

Express $\text{12}\text{.}\overline{\text{28}}$ in the rational form $\left( \dfrac{p}{q} \right)$ .

Answer 1

Hint: Here the given decimal number is a repeating decimal number. So to express the given decimal number into rational form$\left( \dfrac{p}{q} \right)$, we will multiply then given decimal number with 100 and then subtracting it from the original decimal number then we will get the number in the rational form.

Complete step-by-step answer:
Let $x\text{ }=\text{ }12.\overline{28}$ ………………… $\left( 1 \right)$
Now, Multiply equation$\left( 1 \right)$ with 100;
$\Rightarrow$ $100x\text{ }=\text{ }1228.\overline{28}$ ………….. $\left( 2 \right)$
Now, by subtracting equation$\left( 1 \right)$ from equation $\left( 2 \right)$we get;
$\Rightarrow$ $100x\text{ }-\text{ }x\text{ }=\text{ }1228.\overline{28}\text{ }-\text{ }12.\overline{28}$
$\Rightarrow$ $99x\text{ }=\text{ }1216$
$\Rightarrow$ $x\text{ }=\text{ }\dfrac{1216}{99}$
$\therefore $ $\left( \dfrac{p}{q} \right)$ form of decimal number $\text{12}\text{.}\overline{\text{28}}$ is $\text{ }\dfrac{1216}{99}$. Where it is satisfying the condition of rational number i.e. $q\ne 0$

Additional information: A rational is a number that can be expressed in the form $\left( \dfrac{p}{q} \right)$ where $q\ne 0$ .
Since q can be 1 so we can say that every integer is a rational number. Zero is the smallest rational number.

Note: Bar notation is a process where we use bar over the repeating decimals or repeating digits.
Given question is the example of bar notation
It is easier to write the repeating decimal by putting a bar over the repeating digits of the decimal number rather than writing those digits again and again.