Simplifying Fractions

To simplify a fraction means to reduce it to its simplest form.

Example 4.2.1:

Simplify each of the following fractions: 

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

(b) \( \frac{16a^3}{4a^2} \)

(c) \( \frac{x^2}{x^4} \)

(d) \( \frac{8e^3f}{10ef^2} \)

(e) \( \frac{10y}{15z} \)

Solution 

(a) \( \frac{2b}{2a} \\ = \frac{2 \: \times \: b }{2\: \times \: a}\\ = \frac{\not{2} \: \times \: b }{\not{2} \: \times \: a} \\ = \frac{b}{a}\)

(b) \(\frac{16a^3}{4a^2} \\ =\frac{16\: \times \:a \: \times \:a\: \times \:a}{4\: \times \: a\: \times \:a}\\ =\frac{16\: \times \:a \: \times \: \not{a}\: \times \: \not{a}}{4\: \times \: \not{a}\: \times \:\not{a}} \\ = \frac{16\: \times \:a }{4} \\ = \scriptsize 4 \: \times \: a \\ = \scriptsize 4a \)

(c) \( \frac{x^2}{x^4} \\ = \frac{x \: \times \: x}{x \: \times \: x \: \times \: x \: \times \: x} \\ = \frac{\not{x} \: \times \: \not{x}}{\not{x} \: \times \: \not{x} \: \times \: x \: \times \: x} \\ = \frac{1}{x \: \times \: x} \\ = \frac{1}{x^2} \)

(d) \( \frac{8e^3f}{10ef^2} \\ = \frac{8\: \times \:e\: \times \:e\: \times \:e\: \times \:f}{10\: \times \:e\: \times \:f\: \times \:f} \\ = \frac{8\: \times \:e\: \times \:e\: \times \:\not{e} \: \times \:\not{f}}{10\: \times \:\not{e}\: \times \:f\: \times \: \not{f}} \\ = \frac{8\: \times \:e\: \times \:e}{10\: \times \: f } \\ = \frac{ \not{8}^{4} \: \times \:e\: \times \:e}{\not{10}^{5}\: \times \: f } \\ = \frac{4\: \times \:e\: \times \:e}{5\: \times \: f } \\ = \frac{4e^2}{5f} \)

(e) \( \frac{10y}{15z} \\ = \frac{10\: \times \: y}{15\: \times \:z} \\ = \frac{2\: \times \: y}{3\: \times \:z} \\ = \frac{2y}{3z} \)

Evaluation Questions: