Question

Express the recurring decimal \[0.777\] in ${p}/{q}\;$ form.

Answer 1

Hint: In order to find a solution to this problem, we shall denote the recurring decimal by a variable. Then, we will multiply by $10$ on both sides to get only the recurring digits after the decimal point. Then, finally we will subtract the resulting equations to get the required fraction i.e. in ${p}/{q}\;$ form.

Complete step by step answer:
Let us denote the given recurring decimal by a variable $x$.
That is, we get:
$x=0.777...\to \left( 1 \right)$
In the above decimal part of \[0.777...\] , there is only one digit that is recurring i.e., $7$.
In this case, we will multiply both sides of equation $\left( 1 \right)$ by $10$ to get a new recurring decimal.
Therefore, we get:
$10x=7.777...\to \left( 2 \right)$
Now that we have an equation that only contains the decimal part $7$.
Therefore, we will subtract equation $\left( 1 \right)$ from equation $\left( 2 \right)$.
With this, we will subtract the LHS of both equations on the LHS, and the RHS of both equations on the RHS. Hence, we get:
$10x-x=7.777...-0.777...$
On simplifying the expression on LHS, we get $9x$ and on simplifying the expression on RHS, we get $7$.
Therefore, we get:
$9x=7$
Now on shifting $9$ from LHS to the denominator in RHS, we get:
$x=\dfrac{7}{9}\to \left( 3 \right)$
We can see from equation $\left( 1 \right)$ and equation $\left( 3 \right)$ that the LHS is the same.
Therefore, we get:
$x=0.777...=\dfrac{7}{9}$

Therefore, the recurring decimal \[0.777...\] is represented as $\dfrac{7}{9}$ in the fractional form, that is, in ${p}/{q}\;$ form.

Note: While solving this problem, we should be careful to place the decimal point right before the repeating digits. If the number of recurring digits is $n$, then the recurring decimal should be multiplied by ${{10}^{n}}$. A recurring decimal number is a number in which the decimal part keeps repeating the same digit. It is also known as a non-terminating number.