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 

From Equality of Rational Numbers using Cross Multiplication to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Method of H.C.F. |Highest Common Factor|Factorization &Division Method

    Apr 13, 24 05:12 PM

    HCF by Short Division Method
    We will discuss here about the method of h.c.f. (highest common factor). The highest common factor or HCF of two or more numbers is the greatest number which divides exactly the given numbers. Let us…

    Read More

  2. Factors | Understand the Factors of the Product | Concept of Factors

    Apr 13, 24 03:29 PM

    Factors
    Factors of a number are discussed here so that students can understand the factors of the product. What are factors? (i) If a dividend, when divided by a divisor, is divided completely

    Read More

  3. Methods of Prime Factorization | Division Method | Factor Tree Method

    Apr 13, 24 01:27 PM

    Factor Tree Method
    In prime factorization, we factorise the numbers into prime numbers, called prime factors. There are two methods of prime factorization: 1. Division Method 2. Factor Tree Method

    Read More

  4. Divisibility Rules | Divisibility Test|Divisibility Rules From 2 to 18

    Apr 13, 24 12:41 PM

    Divisibility Rules
    To find out factors of larger numbers quickly, we perform divisibility test. There are certain rules to check divisibility of numbers. Divisibility tests of a given number by any of the number 2, 3, 4…

    Read More

  5. Even and Odd Numbers Between 1 and 100 | Even and Odd Numbers|Examples

    Apr 12, 24 04:22 PM

    even and odd numbers
    All the even and odd numbers between 1 and 100 are discussed here. What are the even numbers from 1 to 100? The even numbers from 1 to 100 are:

    Read More

Rational Numbers - Worksheets

Worksheet on Rational Numbers

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Word Problems on Rational Numbers

Worksheet on Operations on Rational Expressions

Objective Questions on Rational Numbers