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