Equality of Rational Numbers using Standard Form

We will learn about the equality of rational numbers using standard form.

How to determine whether the two given rational numbers are equal or not using standard form?

We know there are many methods to determine the equality of two rational numbers but here we will learn the method of equality of two rational numbers using standard form.

In order to determine the equality of two rational numbers, we express both the rational numbers in the standard form. If they have the same standard form they are equal, otherwise they are not equal. 

Solved examples on equality of rational numbers using standard form:

1. Are the rational numbers \(\frac{14}{-35}\) and  \(\frac{-26}{65}\) equal?

Solution:  

First we express the given rational numbers in the standard form.

\(\frac{14}{-35}\)

The denominator of \(\frac{14}{-35}\) is negative. So, we first make it positive.

Multiplying the numerator and denominator of \(\frac{14}{-35}\) by -1, we get

= \(\frac{14  ×  (-1)}{(-35)  ×  (-1)}\)

⇒ \(\frac{14}{-35}\) = \(\frac{-14}{35}\) ← Standard form

The greatest common divisor of 14 and 35 is 7.

Dividing the numerator and denominator by the greatest common divisor of 14 and 35 i.e. 7, we get

⇒ \(\frac{14}{-35}\) = \(\frac{(-14)  ÷  7}{35  ÷  7}\)

⇒ \(\frac{14}{-35}\) = \(\frac{-2}{3}\)

and, \(\frac{-26}{65}\) is already in the standard from.

The greatest common divisor of 26 and 65 is 13.

Dividing the numerator and denominator by the greatest common divisor of 26 and 65 i.e., 13

⇒ \(\frac{-26}{65}\) = \(\frac{(-26)  ÷  13}{65  ÷  13}\)

⇒ \(\frac{-26}{65}\) = \(\frac{-2}{3}\)

Clearly, the given rational numbers have the same standard form.

Hence, \(\frac{14}{-35}\) = \(\frac{-26}{65}\)

Therefore, the given rational numbers \(\frac{14}{-35}\) and \(\frac{-26}{65}\) are equal.

2. Are the rational numbers \(\frac{-12}{40}\) and \(\frac{24}{-54}\) equal?

Solution:

In order to test the equality of the given rational numbers, we first express them in the standard form.

\(\frac{-12}{40}\) is already in the standard from.

The greatest common divisor of 12 and 40 is 4.

Dividing the numerator and denominator by the greatest common divisor of 12 and 40 i.e. 4, we get

\(\frac{-12}{40}\) = \(\frac{(-12)  ÷  4}{40  ÷  4}\)            

⇒ \(\frac{-12}{40}\) = \(\frac{-3}{10}\)

and \(\frac{24}{-54}\) is not in standard from so, we first express them in the standard form.

The denominator of \(\frac{24}{-54}\) is negative. So, we first make it positive.

Multiplying the numerator and denominator of \(\frac{24}{-54}\) by -1, we get

⇒ \(\frac{24}{-54}\) = \(\frac{24  ×  (-1)}{(-54)  ×  (-1)}\)

⇒ \(\frac{24}{-54}\) = \(\frac{-24}{54}\) ← Standard form

The greatest common divisor of 24 and 54 is 6.

Dividing the numerator and denominator by the greatest common divisor of 24 and 54 i.e. 6, we get

⇒ \(\frac{-24}{54}\) = \(\frac{(-24)  ÷  6}{54  ÷  6}\)

⇒ \(\frac{-24}{54}\) = \(\frac{-4}{9}\)

Clearly, the standard forms of two rational numbers are not same.

Therefore, the given rational numbers \(\frac{-12}{40}\) and \(\frac{24}{-54}\) are not equal.

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● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Addition of Rational Numbers

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers

8th Grade Math Practice 

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