Question

How can you change a repeating decimal $0.25$ into a percentage?

Answer 1

Hint: (1) we know that , $0.25....$ means, $0.252525....,$
(2) Let $x=0.25...$ and then multiply both L.H.S. and R.H.S. by $100,$ and solve the expression to get fractional value of $0.252525...$
We will multiply the obtained fraction by $100$ in order to convert it into percentage.

Complete step-by-step solution:
According to question,
We have to convert repeating decimal $0.25,$
As repeating decimal means that the value of $\left( 0.25 \right)$ here $25$ repeats continually.
So, $0.25=0.252525....$
Here,
Let, $x=0.252525.....$
Hence multiplying both R.H.S. and R.H.S. by $100$
We will get,
$100x=100\left( 0.252525... \right)$
$\Rightarrow 100x=25.252525....$
$\Rightarrow 100x=25+0.252525...$
$\Rightarrow 100x=25+x$
$\Rightarrow 100x-x=25$
$\Rightarrow 99x=25$
$\Rightarrow x=\dfrac{25}{99}$
As we know that,
Here we have to convert numbers in percentage.
So,
For converting number into percentage we multiply it by $100$
 So,
$\dfrac{25}{99}$ in term of percentage will be
$\dfrac{25}{99}\times 100$
$=\dfrac{2500}{99}%$

Hence, repeating decimal of $0.25$ in term of percentage will be, $\dfrac{2500}{99}%$

Note: There are two types of numbers,
Rational numbers: rational numbers are those numbers which can easily represented in form of p/q, where, p is value of numerator of the fraction and q is the denominator of the fraction, where (denominator of fraction i.e. value of q will never equal to zero.)
Those numbers which are in form of repeating or terminating decimal form comes under the category of rational number, as such decimals numbers whose digits are repeating after certain interval can easily be converted in form of $\dfrac{P}{Q},$where $Q\ne 0$
While, irrational numbers, are those particular numbers which cannot be represented in form of $\dfrac{P}{Q},$where $Q\ne 0$
Those decimal numbers which are in form of non-terminating or non-repeating form comes under the category of irrational number, as such numbers cannot be represented in form of $\dfrac{P}{Q},$where $Q\ne 0$
(1) Here in $0.252525...,$ the two digit $25$ are called as recurring terms,
(2) The number $0.252525...$ is a rational number as it can be converted form of $\dfrac{P}{Q},$where $Q\ne 0$
(3) If the number written after decimals never repeats, then it will be considered as an irrational number.
(4) Non-terminating and non-terminating numbers are considered as irrational numbers.