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

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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