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

Recent Articles

• 

1. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

   May 18, 25 04:33 PM

   
   In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

   Read More

2. Worksheet on Average | Word Problem on Average | Questions on Average

   May 17, 25 05:37 PM

   In worksheet on average interest we will solve 10 different types of question. Find the average of first 10 prime numbers. The average height of a family of five is 150 cm. If the heights of 4 family

   Read More

3. How to Find the Average in Math? | What Does Average Mean? |Definition

   May 17, 25 04:04 PM

   
   Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

   Read More

4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

   May 17, 25 03:47 PM

   Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

   Read More

5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

   May 16, 25 11:13 AM

   
   Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

   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