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.
● 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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