Question

To reduce a rational number to its standard form , we divide its numerator and denominator by their ?

Answer 1

Hint: In order to solve the given question , we should know the important concepts related to the question that is the term used here “ standard form” . When we say Standard form, it refers that in a rational number ( the number which can be expressed as in the form of $\dfrac{a}{b}$, where $a$ is numerator and $b$is denominator also $b \ne 0$ . ) is in the standard form if $b$ (denominator) is positive, and the integers a (numerator) and $b$ (denominator) have no common divisor other than 1. Here we are going to use the concept of Greatest Common Divisor (GCD) which is also known as Highest common Factor (HCF). Greatest common factor itself describes the number which has a common factor , by using this concept we will be able to answer the question.

Complete step-by-step answer:
We calculate GCDs in order to reduce rational numbers to lowest terms or we can say to its standard form .
There is a method to find the HCF of two integers and to factor them and search for common factors . For example –
Lets calculate the standard form of $\dfrac{{16}}{{24}}$.
For that we need to calculate their HCF .
For calculating HCF we need to factor them and search for common factors , So by doing prime factorization of 16 and 24 we get –
$
  \dfrac{{16}}{{24}} \\
  16 = 2 \times 2 \times 2 \times 2 \\
  24 = 2 \times 2 \times 2 \times 3 \;
 $
Now here , 2 is in common three times , so we get our HCF = $HCF = 2 \times 2 \times 2 = 8$
Now HCF of $\dfrac{{16}}{{24}} = 8$
in order to reduce rational numbers to its standard form, we will divide the rational number $\dfrac{{16}}{{24}}$ by its HCF or GCD , we get –
$\dfrac{{16 \div 8}}{{24 \div 8}} = \dfrac{2}{3}$
Now , the rational number has no common factor other than 1 this means the rational number is already in its standard form .
So, the rational number is reduced to its standard form by dividing its numerator and denominator by their HCF or GCD .
Therefore , To reduce a rational number to its standard form, we must divide both the numerator and the denominator by their GCD.
So, the correct answer is “GCD”.

Note: If the rational number is having no common factor other than 1 this means the rational number is already in its standard form .
To divide one rational number by the other non-zero rational number, we multiply the first rational number by the reciprocal of the other.
Always try to understand the mathematical statement carefully and keep things distinct .
Remember the properties and apply appropriately .
Choose the options wisely , it's better to break the question and then solve part by part .
Cross check the answer and always keep the final answer simplified .