Question

Express $0.\overline{6}$ to the form $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

Answer 1

Hint: Consider $x=0.666$, after that multiply by $10$ and name it equation (2). Then subtract equation (1) and (2). Simplify it, you will definitely get the answer.

Complete step by step solution: Here we are given $0.\overline{6}$ and so we have to express it in $\dfrac{p}{q}$ form where it is mentioned that $p$ and $q$ are integers and $q\ne 0$.
Now let us consider $x=0.666$.
$x=0.666$ ……….. (1)
So here multiply equation (1) by $10$.
$10\times x=10\times 0.666$
Simplifying in the simple form we get.
$10x=6.666$ ……………… (2)
Now let us subtract equation (1) from equation (2) we get,
$10x-x=6.666-0.666$
Now simplifying we get,
$9x=6$
Now divide the above whole equation by $9$ we get,
$\dfrac{9x}{9}=\dfrac{6}{9}$
Again simplify in simple form.
$x=\dfrac{6}{9}$
$x=\dfrac{2}{3}$
So we get the equation in $\dfrac{p}{q}$ form.
Now let us check whether we are correct or not.
Now here we can compare the above equation with $\dfrac{p}{q}$.
We can see that $x=\dfrac{2}{3}=\dfrac{p}{q}$, so here $p=2$ and $q=3$.
So it satisfy the condition that $p=2$ and $q=3$ are integers.
We are correct.
So we get $x=0.\overline{6}=\dfrac{2}{3}=\dfrac{p}{q}$.

Additional information:
Our favourite numbers with numerators and denominators, fractions. When we talk about rational numbers, we are talking about numbers that can be represented as fractions, or parts of a whole.
This includes $0$ (which can be represented as $\dfrac{0}{1}$, or $\dfrac{0}{2}$, and so on), decimals like $0.25$ or $0.75$ (which are equivalent to the fractions $\dfrac{1}{4}$ and $\dfrac{3}{4}$, respectively), integers ($5=\dfrac{5}{1}$, $-1=\dfrac{-1}{1}$, etc), and numbers with infinitely repeating digits ($0.\overline{3}=\dfrac{1}{3}$, $0.6=\dfrac{2}{3}$, and so on). All other numbers, that don't have this pattern, are called irrational.
Finally, note that when you take the square root of a non-square number, the result is never a rational number. That's because you cannot express the result as a fraction with an integer numerator and denominator.

Note: First of all you should be familiar with $\dfrac{p}{q}$ form. Here in the above problem you must know the conditions and apply it in the proper way as in the above problem we can see equation (1) is multiplied by $10$ and then subtracted. You must know the steps completely