Equality of Rational Numbers using Cross Multiplication
We will learn about the equality of rational numbers using cross multiplication.
How to determine whether the two given rational numbers are equal or not using cross multiplication?
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 cross multiplication.
In this method, to determine the equality of two rational numbers a/b and c/d, we use the following result:
\(\frac{a}{b}\) = \(\frac{c}{d}\)
⇔ a × d = b × c
⇔ Numerator of first × Denominator of second = Denominator of first × Numerator of second
Solved
examples on equality of rational numbers using
cross multiplication:
1. Which of the following pairs of
rational numbers are equal?
(i) \(\frac{-8}{32}\) and \(\frac{6}{-24}\)
(ii) \(\frac{-4}{-18}\) and \(\frac{8}{24}\)
Solution:
(i) The given rational numbers are \(\frac{-8}{32}\) and \(\frac{6}{-24}\)
Numerator of first × Denominator of second = (-8) × (-24) = 192 and, Denominator of first × Numerator of second = 32 × 6 = 192.
Clearly,
Numerator of first × Denominator of second = Denominator
of first × Numerator of second
Hence, \(\frac{-8}{32}\) = \(\frac{6}{-24}\)
Therefore, the given rational numbers \(\frac{-8}{32}\) and \(\frac{6}{-24}\) are equal.
(ii) The given rational numbers are \(\frac{-4}{-18}\) and \(\frac{8}{24}\)
Numerator of first × Denominator of second = -4 × 24 = -96 and, Denominator of first × Numerator of second = (-18)
× 8 = -144
Clearly,
Numerator of first × Denominator of second ≠ Denominator of first × Numerator of second
Hence, \(\frac{-4}{-18}\) ≠ \(\frac{8}{24}\).
Therefore, the given rational numbers \(\frac{-4}{-18}\) and \(\frac{8}{24}\) are not equal.
2. If \(\frac{-6}{8}\) = \(\frac{k}{64}\), find the value of k.
Solution :
We know that \(\frac{a}{b}\) = \(\frac{c}{d}\) if ad = bc
Therefore, \(\frac{-6}{8}\) = \(\frac{k}{64}\)
⇒ -6 × 64 = 8 × k, [Numerator of first × Denominator of second = Denominator of first × Numerator of second]
⇒ -384 = 8k
⇒ 8k = -384
⇒ \(\frac{8k}{8}\) = \(\frac{-384}{8}\), [Dividing both sides by 8]
⇒ k = -48
Therefore, the value of k = -48
3. If \(\frac{7}{m}\) = \(\frac{49}{63}\), find the value of m.
Solution:
In order to write \(\frac{49}{63}\) as a rational number with numerator 7, we first find a number which when divided 49 gives 7.
Clearly, such a number is 49 ÷ 7 = 7.
Dividing the numerator and denominator of 49/63 by 7, we have
\(\frac{49}{63}\) = \(\frac{49 ÷ 7}{63 ÷ 7}\) = \(\frac{7}{9}\)
Therefore, \(\frac{7}{m}\) = \(\frac{49}{63}\)
⇒ \(\frac{7}{m}\) = \(\frac{7}{9}\)
⇒ m = 9
4. Fill in the blank: \(\frac{-7}{15}\) = \(\frac{.....}{135}\)
Solution:
In order to fill the required blank, we have to express -7 as a rational number with denominator 135. For this, we first find an integer which when multiplied with 15 gives us 135.
Clearly,
such an integer is 135 ÷ 15 = 9
Multiplying the numerator and denominator of \(\frac{-7}{15}\) by 9, we get
\(\frac{-7}{15}\) = \(\frac{(-7) × 9}{15 × 9}\) = \(\frac{-63}{135}\)
Therefore, the required
number is -63.
● 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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