Equality of Rational Numbers with Common Denominator

We will learn about the equality of rational numbers with common denominator.

How to determine whether the two given rational numbers are equal or not with the common denominator?

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 with the same denominator.

In this method, denominators of the given rational numbers are made equal by using the following steps: 

Step I: Obtain the two numbers. 

Step II: Multiply the numerator and denominator of the first number by the denominator of the second number. 

Step III: Multiply the numerator and denominator of the second number by the denominator of the first number. 

Step IV: Check the numerators of the two numbers obtained in steps II and III.  If their numerators are equal, then the given rational numbers are equal, otherwise they are not equal.

Solved examples:

1. Are the rational numbers \(\frac{-9}{12}\) and \(\frac{21}{-28}\) equal?

Solution:

Multiplying the numerator and denominator of \(\frac{-9}{12}\) by the denominator of \(\frac{21}{-28}\) i.e. by -28, we get

\(\frac{-9}{12}\) = \(\frac{(-9)  ×  (-28)}{12  ×  (-28)}\) = \(\frac{252}{-336}\)

Multiplying the numerator and denominator of \(\frac{21}{-28}\) by the denominator of \(\frac{-9}{12}\) i.e., by 12, we get

\(\frac{21}{-28}\) = \(\frac{21  ×  12}{(-28)  ×  12}\) = \(\frac{252}{-336}\)

Clearly, the numerators of the above obtained rational numbers are equal.

Therefore, the given rational numbers \(\frac{-9}{12}\) and \(\frac{21}{-28}\) are equal.

2. Show that the rational numbers \(\frac{-6}{8}\) and \(\frac{10}{-15}\) are not equal.

Solution:

Multiplying the numerator and denominator of \(\frac{-6}{8}\) by the denominator of \(\frac{10}{-15}\) i.e. -15, we get

\(\frac{-6}{8}\) = \(\frac{(-6)  ×  (-15)}{8  ×  (-15)}\) = \(\frac{90}{-120}\)

Multiplying the numerator and denominator of \(\frac{10}{-15}\) by the denominator of \(\frac{-6}{8}\) i.e. 8, we get

\(\frac{10}{-15}\) = \(\frac{10  ×  8}{(-15)  ×  8}\) = \(\frac{80}{-120}\)

We find that the numerators of rational numbers \(\frac{90}{-120}\) and \(\frac{80}{-120}\) are unequal.

Therefore, the given rational numbers \(\frac{-6}{8}\) and \(\frac{10}{-15}\) are unequal.

● 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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