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
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
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
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
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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