Equivalent form of Rational Numbers

We will learn how to find the equivalent form of rational numbers expressing a given rational number in different forms and the equivalent form of the rational numbers having a common denominator.

1. Express \(\frac{-54}{90}\) as a rational number with denominator 5.

Solution:

In order to express \(\frac{-54}{90}\) as a rational number with denominator 5, we first find a number which gives 5 when 90 is divided by it. 
Clearly, such a number = (90 ÷ 5) = 18

Dividing the numerator and denominator of \(\frac{-54}{90}\) by 18, we have 
\(\frac{-54}{90}\) = \(\frac{(-54)  ÷  18}{90  ÷  18}\) = \(\frac{-3}{5}\)

Hence, expressing \(\frac{-54}{90}\) as a rational number with denominator 5 is \(\frac{-3}{5}\).

2. Fill in the blanks with the appropriate number in the numerator: \(\frac{5}{-7}\) = \(\frac{.....}{35}\) = \(\frac{.....}{-77}\).

Solution:

We have, 35 ÷ (-7) = - 5

Therefore, \(\frac{5}{-7}\) = \(\frac{5  ×  (-5)}{(-7)  ×  (- 5)}\) =  \(\frac{-25}{35}\)

Similarly, we have (-77) ÷ (-7) = 11
Therefore, \(\frac{5}{-7}\) = \(\frac{5  ×  11}{(-7)  ×  11}\) = \(\frac{55}{-77}\)

Hence, \(\frac{5}{-7}\) = \(\frac{-25}{35}\) = \(\frac{55}{-77}\)

More examples on equivalent form of rational numbers:

3. Find an equivalent form of the rational numbers \(\frac{2}{9}\) and \(\frac{5}{6}\) having a common denominator.

Solution:

We have to convert \(\frac{2}{9}\) and \(\frac{5}{6}\) into equivalent rational numbers having common denominator. 

Clearly, such a denominator is the LCM of 9 and 6.

We have, 9 = 3 × 3 and 6 = 2 × 3

Therefore, LCM of 9 and 6 is 2 × 3 × 3 = 18

Now, 18 ÷ 9 = 2 and 18 ÷ 6 = 3

Therefore, \(\frac{2}{9}\) = \(\frac{2  ×  2}{9  ×  2}\) = \(\frac{4}{18}\) and \(\frac{5}{6}\) = \(\frac{5  ×  3}{6  ×  3}\) = \(\frac{15}{18}\).

Hence, the given rational numbers with common denominator are \(\frac{4}{18}\) and \(\frac{15}{18}\).

4. Find an equivalent form of the rational numbers \(\frac{3}{4}\), \(\frac{7}{6}\) and \(\frac{11}{12}\) having a common denominator.

Solution:

We have to convert \(\frac{3}{4}\), \(\frac{7}{6}\) and \(\frac{11}{12}\) into equivalent rational numbers having common denominator. 

Clearly, such a denominator is the LCM of 4, 6 and 12.

We have, 4 = 2 × 2, 6 = 2 × 3 and 12 = 2 × 2 × 3

Therefore, LCM of 4, 6 and 12 is 2 × 2 × 3 = 12

Now, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1

Therefore, \(\frac{3}{4}\) = \(\frac{3  ×  3}{4  ×  3}\) = \(\frac{9}{12}\), \(\frac{7}{6}\) = \(\frac{7  ×  2}{6  ×  2}\) = \(\frac{12}{12}\) and \(\frac{11}{12}\) = \(\frac{11  ×  1}{12  ×  1}\) = \(\frac{11}{12}\)

Hence, the given rational numbers with common denominator are \(\frac{9}{12}\), \(\frac{14}{12}\) and \(\frac{11}{12}\).

● 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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