Question

Show that $1.272727.... = 1.\overline {27} $ can be expressed in the form of $\dfrac{p}{q}$.

Answer 1

Hint: In this question, we need to convert $1.\overline {27} $ (with $27$ repeating) into the form of $\dfrac{p}{q}$. Here, we will consider $1.\overline {27} $ as x. So, we multiply and divide the decimal by $10$ to get the repeating entity just after the decimal point. Then, we will subtract the two equations so that the recurring numbers in the decimal expansion get cancelled and we obtain a terminating decimal. Then we find the value of $x$ by using a method of transposition.

Complete step by step answer:
In this question, we need to convert $1.\overline {27} $ to a fraction.Let $x$ be that fraction.
Here, consider the given value as $x = 1.\overline {27} - - - - - \left( 1 \right)$.
Let us consider this as the equation $\left( 1 \right)$. Now, let us multiply both sides of the equation by $100$. So, we get,
$100x = 1.\overline {27} \times 100$
Expanding the recurring decimal expression, we get,
$ \Rightarrow 100x = 1.27272727..... \times 100$
Simplifying the expression,
$ \Rightarrow 100x = 127.27272727.....$
$ \Rightarrow 100x = 127.\overline {27} - - - - - \left( 2 \right)$

Subtracting the equation $\left( 1 \right)$ from equation $\left( 2 \right)$, we get,
$ \Rightarrow 100x - x = 127.\overline {27} - 1.\overline {27} $
Simplifying the calculations,
$ \Rightarrow 99x = 127.272727.... - 1.272727....$
$ \Rightarrow 99x = 126.0000....$
Dividing both sides of equation by $99$, we get,
$ \Rightarrow x = \dfrac{{126}}{{99}}$
Cancelling common factors in numerator and denominator, we get,
$ \therefore x = \dfrac{{14}}{{11}}$

Hence, the decimal expansion $1.272727.... = 1.\overline {27} $ can be represented as $\dfrac{{14}}{{11}}$ in the form of $\dfrac{p}{q}$.

Note: In this question it is important to note that, here we have multiplied and divided $1.\overline {27} $ by $100$, then subtracted both the equations to determine the value of x as in this question we have a repetition of a repetition of two digits in $1.\overline {27} $. The scenario may be different in each question depending on the situation as the decimal may have more number of digits as its repeating entity.