Question

How do you convert $1.54\left( 54 \right)$ being repeated?

Answer 1

Hint: Consider the given number as $x$, and multiply both LHS and RHS by $100,$then by separating the like terms determine the fractional value of the given number.
Consider $x=1.5454.54.......$
Apply this to determine the fractional value of a given number.

Complete step by step solution:
As per data given in the question,
Let the value of $x=1.54....$ where $54$ are repeating continuously.
So,
$x=1.54.....$
Multiplying both side left hand side and Right hand side by $100,$
 We will get,
$100\times x=100\times 1.5454...$
$\Rightarrow 100x=154.5454....$
Converting RHS in form of $\left( x+c \right)$ where $x=1.54....$ and $c$ is any value of constant.
We will get,
$100x=153+1.545454......$
$\Rightarrow 100x=x+153$
Now separating the like terms,
We will get,
$100-x=153$
$99x=153$
$\Rightarrow x=$$\Rightarrow x=\dfrac{153}{99}=\dfrac{17}{11}$

Hence, when $1.54$ where $54$ being continuously repeated converted in form of fraction results to $\dfrac{17}{11}.$

Additional Information:
There are two types of numbers,
Rational numbers:
Rational numbers are those numbers which can be converted in form of fraction, or in form of $\dfrac{P}{Q},$where $P$ is value of numerator and $Q$ is value of denominator, where $Q\ne 0.$ when rational numbers are converted into a decimal form then either they are completely divisible or in a repeating decimal form.
Irrational numbers:
Irrational numbers are those numbers which are non-terminating non-repeating numbers.
Examples of rational numbers are, $2,3,\dfrac{1}{4}$ etc., while examples of irrational numbers are $\sqrt{2},\sqrt{3}$etc.
Here, in the Question given number is a rational numbers, as the given number can be converted in form of $\dfrac{P}{Q}.$

Note: Here, repeating of $1.54$ means that both the digits which are after decimals repeats continuously, so it will be $1.5454....,$ Here, repeating does not mean that only one digits are repeating. When we represent any value in form of fraction, then it is unit less, means, there is no unit of fraction.
Fraction means a part, like $\dfrac{3}{5}$ means $3$ parts out of total $5$ parts. Fraction are when converted into decimals, then it’s value will be either less than $1$ or $1$, but it can’t be more than $1.$