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

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 Equivalent Rational Numbers to HOME PAGE




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.



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.




Share this page: What’s this?

Recent Articles

  1. Word Problems on Dividing Money | Solving Money Division Word Problems

    Feb 13, 25 10:29 AM

    Word Problems on Dividing Money
    Read the questions given in the word problems on dividing money. We need to understand the statement and divide the amount of money as ordinary numbers with two digit numbers. 1. Ron buys 15 pens for…

    Read More

  2. Addition and Subtraction of Money | Examples | Worksheet With Answers

    Feb 13, 25 09:02 AM

    Add Money Method
    In Addition and Subtraction of Money we will learn how to add money and how to subtract money.

    Read More

  3. Worksheet on Division of Money | Word Problems on Division of Money

    Feb 13, 25 03:53 AM

    Division of Money Worksheet
    Practice the questions given in the worksheet on division of money. This sheet provides different types of questions on dividing the amount of money by a number; finding the quotient

    Read More

  4. Worksheet on Multiplication of Money | Word Problems | Answers

    Feb 13, 25 03:17 AM

    Worksheet on Multiplication of Money
    Practice the questions given in the worksheet on multiplication of money. This sheet provides different types of questions on multiplying the amount of money by a number; arrange in columns the amount…

    Read More

  5. Division of Money | Worked-out Examples | Divide the Amounts of Money

    Feb 13, 25 12:16 AM

    Divide Money
    In division of money we will learn how to divide the amounts of money by a number. We carryout division with money the same way as in decimal numbers. We put decimal point in the quotient after two pl…

    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