Equivalent Algebraic Fractions

What are Algebraic Fractions?

Algebraic fractions are factions that contain both letters and numbers e.g. 

\( \frac{x}{5}, \frac{2y}{3}, \frac{3x^2}{5y}, \frac{a \: + \: b}{x \: – \: 2y}, \scriptsize \: etc \)

Equivalent Algebraic Fractions:

Equivalent algebraic fractions can be obtained by multiplying or dividing both the numerator and denominator by the same quantity. 

Example 4.1.1: 

Copy and complete these equivalent fractions. 

(a) \( \frac{3x}{7} = \frac{?}{21} \)

(b) \( \frac{1}{x} = \frac{3}{?} \)

(c) \( \frac{a}{b} = \frac{?}{11bx} \)

Solution

(a) \( \frac{3x}{7} = \frac{?}{21} \)

7 multiplied by what number will give you 21?

7 × ? = 21

7 × 3 = 21

Remember: Equivalent algebraic fractions can be obtained by multiplying or dividing both the numerator and denominator by the same quantity. 

∴ \( \frac{3x}{7} = \frac{3x \: \times \: 3}{7 \: \times \: 3} = \frac{9x}{21} \)

∴ \( \normalsize \frac{3x}{7} \scriptsize \: is \: equivalent \: to \: \normalsize \frac{9x}{21} \)

(b) \( \frac{1}{x} = \frac{3}{?} \)

1 multiplied by what number will give you 3?

1 × ? = 3

1 × 3 = 3

∴ \( \frac{1}{x} = \frac{1 \: \times \: 3}{x \: \times \: 3} = \frac{3}{3x} \)

∴ \( \normalsize \frac{1}{x} \scriptsize \: is \: equivalent \: to \: \normalsize \frac{3}{3x} \)

(c) \( \frac{a}{b} = \frac{?}{11bx} \)

b multiplied by what will give 11bx?

b × ? = 11bx

b × 11x = 11bx

∴ \( \frac{a}{b} = \frac{a \: \times \: 11x}{b \: \times \: 11x} = \frac{11ax}{11bx} \)

∴ \( \normalsize \frac{a}{b} \scriptsize \: is \: equivalent \: to \: \normalsize \frac{11ax}{11bx} \)

Example 4.1.2:

Write any three equivalent fractions of these fractions 

(a) \( \frac{2}{x}\)

(b) \( \frac{3a}{5}\)

Solution

Let’s multiply the numerators and denominators by 2, 3 and 4

(a)

i. \( \frac{2}{x} = \frac{2 \: \times \: 2}{x \: \times \: 2} = \frac{4}{2x} \)

ii. \( \frac{2}{x} = \frac{2 \: \times \: 3}{x \: \times \: 3} = \frac{6}{3x} \)

iii. \( \frac{2}{x} = \frac{2 \: \times \: 4}{x \: \times \: 4} = \frac{8}{4x} \)

(b)

i. \( \frac{3a}{5} = \frac{3a \: \times \: 2}{5 \: \times \: 2} = \frac{6a}{10} \)

ii. \( \frac{3a}{5} = \frac{3a \: \times \: 3}{5 \: \times \: 3} = \frac{9a}{15} \)

iii. \( \frac{3a}{5} = \frac{3a \: \times \: 4}{5 \: \times \: 4} = \frac{12}{20} \)

Evaluation Questions: