This is the single shared fractions page used across Grade 10, Grade 11, and Grade 12.
Each section follows this progression:
- Example 1 = Grade 10 core fluency
- Example 2 = Grade 11 bridge skill
- Example 3 = Grade 12 exam-level use
1) Multiplying Fractions#
Rule:
$$\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$$Example 1 (Grade 10 core)#
$$\frac{2}{3}\times\frac{5}{7}=\frac{10}{21}$$Example 2 (Grade 11 bridge)#
$$\frac{6x^2y}{14y^3}\times\frac{21y^2}{9x}=x$$Example 3 (Grade 12 exam-level)#
$$\frac{3x^2}{14y}\times\frac{21y^3}{10x^3}=\frac{9y^2}{20x}$$2) Dividing Fractions#
Rule: divide by a fraction = multiply by its reciprocal.
$$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$$Example 1 (Grade 10 core)#
$$\frac{3}{4}\div\frac{1}{2}=\frac{3}{2}$$Example 2 (Grade 11 bridge)#
$$\frac{\sin\theta}{\frac{\sin\theta}{\cos\theta}}=\cos\theta$$Example 3 (Grade 12 exam-level)#
$$\frac{3x}{2y}\div\frac{9x^2}{10y^3}=\frac{5y^2}{3x}$$3) Adding and Subtracting Fractions#
Rule: use a common denominator first.
$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$Example 1 (Grade 10 core)#
$$\frac{2}{3}+\frac{1}{4}=\frac{11}{12}$$Example 2 (Grade 11 bridge)#
$$\frac{\cos\theta}{1-\sin\theta}+\frac{\cos\theta}{1+\sin\theta}=\frac{2}{\cos\theta}$$Example 3 (Grade 12 exam-level)#
$$\frac{2}{x-1}-\frac{3}{x+2}=\frac{7-x}{(x-1)(x+2)}$$4) Splitting a Fraction#
You may split only when the numerator is a sum/difference over a single denominator:
$$\frac{a+b+c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$$Example 1 (Grade 10 core)#
$$\frac{18+6}{3}=\frac{18}{3}+\frac{6}{3}=8$$Example 2 (Grade 12 direct application)#
$$\frac{x^3-4x^2+2}{x^2}=x-4+2x^{-2}$$Example 3 (Grade 12 first-principles application)#
$$\frac{(x+h)^3-x^3}{h}=3x^2+3xh+h^2$$Never do this#
$$\frac{a}{b+c}\neq\frac{a}{b}+\frac{a}{c}$$5) Complex Fractions#
Method:
- Make the top one fraction.
- Make the bottom one fraction.
- Divide fractions using reciprocal.
Example 1 (Grade 10 core)#
$$\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}$$Example 2 (Grade 11 bridge)#
$$\frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x}+\frac{1}{y}}=\frac{y-x}{y+x}$$Example 3 (Grade 12 exam-level)#
$$\frac{\frac{2}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}=\frac{2y+x}{y-x}$$6) Cancelling Correctly#
You may cancel only common factors, not terms in sums.
Correct (Grade 10-12 essential rule)#
$$\frac{x^2-9}{x-3}=\frac{(x-3)(x+3)}{x-3}=x+3\quad(x\neq3)$$Wrong (common all-grade trap)#
$$\frac{x+3}{x}\neq3$$With restrictions (Grade 12 emphasis)#
$$\frac{2x^2-8x}{x^2-16}=\frac{2x}{x+4}\quad(x\neq\pm4)$$7) Calculator Habits#
- Never round in the middle of a calculation.
- Use brackets aggressively.
- Enter full numerators and denominators as grouped expressions.
Common Mistakes#
- Adding numerators and denominators directly.
- Splitting the denominator.
- Cancelling terms instead of factors.
- Ignoring denominator restrictions.