Equivalent Rational Numbers
We will learn how to find the equivalent rational numbers by multiplication and division.
Equivalent rational numbers by multiplication:
If \(\frac{a}{b}\) is a rational number and m is a non-zero integer then \(\frac{a × m}{b × m}\) is a rational number equivalent to \(\frac{a}{b}\).
For example, rational numbers \(\frac{12}{15}\), \(\frac{20}{25}\), \(\frac{-28}{-35}\), \(\frac{-48}{-60}\) are equivalent to the rational number \(\frac{4}{5}\).
We know that if we multiply the numerator and denominator of a fraction by the same positive integer, the value of the fraction does not change.
For example, the fractions \(\frac{3}{7}\) and \(\frac{21}{49}\) are equal because the numerator and the denominator of \(\frac{21}{49}\) can be obtained by multiplying each of the numerator and denominator of \(\frac{3}{7}\) by 7.
Also, \(\frac{-3}{4}\) = \(\frac{-3 × (-1)}{4 × (-1)}\)
= \(\frac{3}{-4}\), \(\frac{-3}{4}\) = \(\frac{-3 × 2}{4 × 2}\)
= \(\frac{-6}{8}\), \(\frac{-3}{4}\) = \(\frac{-3 × (-2)}{4 × (-2)}\)
= \(\frac{6}{-8}\) and so on …….
Therefore, \(\frac{-3}{4}\) = \(\frac{-3 × (-1)}{4 × (-1)}\) = \(\frac{-3 × 2}{4 × 2}\) = \(\frac{(-3) × (-2)}{4 × (-2)}\) and so on …….
Note: If the denominator of a rational number is a negative integer, then by
using the above property, we can make it positive by multiplying its numerator
and denominator by -1.
For example, \(\frac{5}{-7}\) = \(\frac{5 × (-1)}{(-7) × (-1)}\) = \(\frac{-5}{7}\)
Equivalent rational numbers by division:
If \(\frac{a}{b}\) is a rational number and m is a common divisor of a and b, then \(\frac{a ÷ m}{b ÷ m}\) is a rational number equivalent to \(\frac{a}{b}\).
For example, rational numbers \(\frac{-48}{-60}\), \(\frac{-28}{-35}\), \(\frac{20}{25}\), \(\frac{12}{15}\) are equivalent to the rational number \(\frac{4}{5}\).
We know that if we divide the numerator and denominator of a fraction by a common divisor, then the value of the fraction does not change.
For example, \(\frac{48}{64}\) = \(\frac{48 ÷ 16}{64 ÷ 16}\) = \(\frac{3}{4}\)
Similarly,
we have
\(\frac{-75}{100}\) = \(\frac{(-75) ÷ 5}{100 ÷ 5}\) = \(\frac{-15}{20}\) = \(\frac{(-15) ÷ 5}{20 ÷ 5}\) = \(\frac{-3}{4}\), and \(\frac{42}{-56}\) = \(\frac{42 ÷ 2}{(-56 ) ÷ 2}\) = \(\frac{21}{-28}\) = \(\frac{21 ÷ (-7)}{(-28) ÷ (-7)}\) = \(\frac{-3}{4}\)
Solved examples:
1. Find the two rational numbers equivalent to \(\frac{3}{7}\).
Solution:
\(\frac{3}{7}\) = \(\frac{3 × 4}{7 × 4}\) = \(\frac{12}{28}\) and
\(\frac{3}{7}\) = \(\frac{3 × 11}{7 × 11}\) = \(\frac{33}{77}\)
Therefore, the two rational numbers equivalent to \(\frac{3}{7}\) are \(\frac{12}{28}\) and \(\frac{33}{77}\)
2. Determine the smallest equivalent rational number of \(\frac{210}{462}\).
Solution:
\(\frac{210}{462}\) = \(\frac{210 ÷ 2}{462 ÷ 2}\) = \(\frac{105}{231}\) = \(\frac{105 ÷ 3}{231 ÷ 3}\) = \(\frac{35}{77}\) = \(\frac{35 ÷ 7}{77 ÷ 7}\) = \(\frac{5}{11}\)
Therefore, the least equivalent rational number of \(\frac{210}{462}\) is \(\frac{5}{11}\)
3. Write each of the following rational numbers with positive denominator:
\(\frac{3}{-7}\), \(\frac{11}{-28}\), \(\frac{-19}{-13}\)
Solution:
In order to express a rational number with positive denominator, we multiply its numerator and denominator by -1.
Therefore,
\(\frac{3}{-7}\) = \(\frac{3 × (-1)}{(-7) × (-1)}\) = \(\frac{-3}{7}\),
\(\frac{11}{-28}\) = \(\frac{11 × (-1)}{(-28) × (-1)}\) = \(\frac{-11}{28}\),
and \(\frac{-19}{-13}\) = \(\frac{(-19) × (-1)}{(-13) × (-1)}\) = \(\frac{19}{13}\)
4. Express \(\frac{-3}{7}\) as a rational number with numerator:
(i) -15; (ii) 21
Solution:
(i) In order to -3 as a rational number with numerator -15, we first find a number which when multiplied by -3 gives -15.
Clearly, such number is (-15) ÷ (-3) = 5
Multiplying the numerator and denominator of \(\frac{-3}{7}\) by 5, we have
\(\frac{-3}{7}\) = \(\frac{(-3) × 5}{7 × 5}\) = \(\frac{-15}{35}\)
Thus, the required rational number is \(\frac{-15}{35}\).
(ii) In order to express \(\frac{-3}{7}\) as a rational number with numerator 21, we first find a number which when multiplied with -3 gives 21.
Clearly, such a number is 21 ÷ (-3) = -7
Multiplying the numerator and denominator of \(\frac{-3}{7}\) by (-7), we have
\(\frac{-3}{7}\) = \(\frac{(-3) × (-7)}{7 × (-7)}\) = \(\frac{21}{-49}\)
These are the above examples on equivalent rational numbers.
● 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
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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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