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