Lesson 4, Topic 1
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Equivalent Algebraic Fractions

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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 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 x ? = 21

7 x 3 = 21

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

âˆ´ $$\frac{3x}{7} = \frac{3x \: \times \: 3}{7 \: \times \: 3} = \frac{9x}{21}$$

âˆ´ $$\normalsize \frac{3x}{7} \scriptsize \: is \: equivalent \: to \: \normalsize \frac{9x}{21}$$

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

1 multiplied by what number will give you 3

1 x ? = 3

1 x 3 = 3

âˆ´ $$\frac{1}{x} = \frac{1 \: \times \: 3}{x \: \times \: 3} = \frac{3}{3x}$$

âˆ´ $$\normalsize \frac{1}{x} \scriptsize \: is \: equivalent \: to \: \normalsize \frac{3}{3x}$$

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

b multiplied by what will give 11bx

b x ? = 11bx

b x 11x = 11bx

âˆ´ $$\frac{a}{b} = \frac{a \: \times \: 11x}{b \: \times \: 11x} = \frac{11ax}{11bx}$$

âˆ´ $$\normalsize \frac{a}{b} \scriptsize \: is \: equivalent \: to \: \normalsize \frac{11ax}{11bx}$$

Example 2

Write any three equivalent fractions of these fractions

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

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

Solution

Let’s multiply the numerator and denominator by 2, 3 and 4

(a)

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

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

ii. $$\frac{2}{x} = \frac{2 \: \times \: 4}{x \: \times \: 4} = \frac{8}{4x}$$

(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

1. Write down 3 equivalent fractions of these fraction

(a) $$\frac{5a}{3y}$$

(b) $$\frac{6x}{5}$$

(c) $$\frac{2a}{6y}$$

2. Find the missing number

(a) $$\frac{3b}{2} = \frac{?}{8a}$$

(b) $$\frac{10a}{1} = \frac{?}{4}$$

(c) $$\frac{?}{3a} = \frac{6x}{9ax}$$

c) $$\frac{4x}{?} = \frac{32x}{8ax}$$

1a. $$\frac{10a}{6y}, \frac{15a}{9y}, \frac{20a}{12y}$$

(b) $$\frac{12x}{10}, \frac{18x}{15}, \frac{24x}{20}$$

(c) $$\frac{4a}{2y}, \frac{6a}{3y}, \frac{8a}{4y}$$

2(a) 12ab

(b) 40a

(c) 2

(d) ax

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