Algebraic Fractions (B) worksheet from Cazoom Maths, designed for GCSE Higher Tier students, offering practice in simplifying and combining algebraic fractions.

Let's solve each problem in Section A and Section B step by step, simplifying where possible.

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🔷 Section A: Multiply or Divide Algebraic Fractions

#### 1) $\frac{ac}{b} \times \frac{b}{c}$

- Cancel $c$ in numerator and denominator.
- Cancel $b$ in numerator and denominator.
$$
= \frac{a\cancel{c}}{\cancel{b}} \times \frac{\cancel{b}}{\cancel{c}} = a
$$
✔ Answer: $a$

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#### 2) $\frac{a + b}{3ab} \times \frac{6a^2}{2a + 2b}$

- Factor denominator: $2a + 2b = 2(a + b)$
- Rewrite:
$$
= \frac{a + b}{3ab} \times \frac{6a^2}{2(a + b)}
$$
- Cancel $a + b$: one from top and bottom
- Simplify numbers: $6 / (3 \times 2) = 1$
- Simplify variables: $a^2 / ab = a / b$
$$
= \frac{1}{3ab} \times \frac{6a^2}{2} = \frac{6a^2}{6ab} = \frac{a}{b}
$$
✔ Answer: $\frac{a}{b}$

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#### 3) $\frac{x^2}{x^2 - 2x} \times \frac{3 + x}{x}$

- Factor denominator: $x^2 - 2x = x(x - 2)$
- Rewrite:
$$
= \frac{x^2}{x(x - 2)} \times \frac{x + 3}{x}
$$
- Cancel $x$ from numerator and denominator:
$$
= \frac{x}{x - 2} \times \frac{x + 3}{x} = \frac{x(x + 3)}{(x - 2)x} = \frac{x + 3}{x - 2}
$$
✔ Answer: $\frac{x + 3}{x - 2}$

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#### 4) $\frac{(3x + 2)^2}{6x} \times \frac{x^4}{6x + 4}$

- Factor $6x + 4 = 2(3x + 2)$
- Rewrite:
$$
= \frac{(3x + 2)^2}{6x} \times \frac{x^4}{2(3x + 2)}
$$
- Cancel one $(3x + 2)$:
$$
= \frac{(3x + 2)}{6x} \times \frac{x^4}{2} = \frac{(3x + 2)x^4}{12x}
$$
- Simplify $x^4 / x = x^3$:
$$
= \frac{(3x + 2)x^3}{12}
$$
✔ Answer: $\frac{x^3(3x + 2)}{12}$

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#### 5) $\frac{x^2y^2}{y} \div \frac{x^4y}{x}$

- First simplify both fractions:
- $\frac{x^2y^2}{y} = x^2y$
- $\frac{x^4y}{x} = x^3y$
- So: $x^2y \div x^3y = \frac{x^2y}{x^3y} = \frac{1}{x}$
✔ Answer: $\frac{1}{x}$

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#### 6) $\frac{b - 3}{2b^2 - 5b - 3} \div (2b - 6)$

- First factor denominator: $2b^2 - 5b - 3$
- Find two numbers: $-6$ and $+1$: $(-6)(1) = -6$, $-6 + 1 = -5$
- So: $2b^2 - 6b + b - 3 = 2b(b - 3) + 1(b - 3) = (2b + 1)(b - 3)$
- Now rewrite:
$$
= \frac{b - 3}{(2b + 1)(b - 3)} \div (2b - 6)
$$
- Cancel $b - 3$: $\frac{1}{2b + 1} \div (2b - 6)$
- $2b - 6 = 2(b - 3)$
- So: $\frac{1}{2b + 1} \div 2(b - 3) = \frac{1}{(2b + 1) \cdot 2(b - 3)} = \frac{1}{2(2b + 1)(b - 3)}$
✔ Answer: $\frac{1}{2(2b + 1)(b - 3)}$

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#### 7) $\frac{1}{y^2 + 3y + 2} \div \frac{2}{y^2 - 4}$

- Factor:
- $y^2 + 3y + 2 = (y + 1)(y + 2)$
- $y^2 - 4 = (y - 2)(y + 2)$
- So:
$$
= \frac{1}{(y + 1)(y + 2)} \div \frac{2}{(y - 2)(y + 2)} = \frac{1}{(y + 1)(y + 2)} \times \frac{(y - 2)(y + 2)}{2}
$$
- Cancel $(y + 2)$:
$$
= \frac{1}{(y + 1)} \times \frac{(y - 2)}{2} = \frac{y - 2}{2(y + 1)}
$$
✔ Answer: $\frac{y - 2}{2(y + 1)}$

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#### 8) $\frac{a^2(a - 2)}{y^2} \div \frac{a(a - 2)}{y^5}$

- Rewrite division as multiplication:
$$
= \frac{a^2(a - 2)}{y^2} \times \frac{y^5}{a(a - 2)}
$$
- Cancel $a$: $a^2 / a = a$
- Cancel $(a - 2)$
- $y^5 / y^2 = y^3$
- Result: $a \cdot y^3 = a y^3$
✔ Answer: $a y^3$

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🔷 Section B: Write as Single Fractions

We'll now work through these by finding common denominators.

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#### 1) $\frac{x}{5} + 4$

- Write 4 as $\frac{20}{5}$
- $\frac{x + 20}{5}$
✔ Answer: $\frac{x + 20}{5}$

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#### 2) $y - \frac{6}{y}$

- Write $y = \frac{y^2}{y}$
- $\frac{y^2 - 6}{y}$
✔ Answer: $\frac{y^2 - 6}{y}$

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#### 3) $4 - \frac{5}{x - 2}$

- $4 = \frac{4(x - 2)}{x - 2} = \frac{4x - 8}{x - 2}$
- Subtract: $\frac{4x - 8 - 5}{x - 2} = \frac{4x - 13}{x - 2}$
✔ Answer: $\frac{4x - 13}{x - 2}$

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#### 4) $\frac{1}{3w + 5} + 9$

- $9 = \frac{9(3w + 5)}{3w + 5} = \frac{27w + 45}{3w + 5}$
- Add: $\frac{1 + 27w + 45}{3w + 5} = \frac{27w + 46}{3w + 5}$
✔ Answer: $\frac{27w + 46}{3w + 5}$

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#### 5) $6 + \frac{3}{x + 2} + x$

- Combine $6 + x = x + 6$
- Now: $x + 6 + \frac{3}{x + 2} = \frac{(x + 6)(x + 2)}{x + 2} + \frac{3}{x + 2}$
- Expand: $(x + 6)(x + 2) = x^2 + 2x + 6x + 12 = x^2 + 8x + 12$
- Add 3: $x^2 + 8x + 15$
✔ Answer: $\frac{x^2 + 8x + 15}{x + 2}$

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#### 6) $\frac{3}{x} + \frac{5}{x} = \frac{8}{x}$

✔ Answer: $\frac{8}{x}$

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#### 7) $\frac{7a}{4} + \frac{11a}{4} = \frac{18a}{4} = \frac{9a}{2}$

✔ Answer: $\frac{9a}{2}$

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#### 8) $\frac{5b}{3} - \frac{9b}{2}$

- LCD = 6
- $\frac{10b}{6} - \frac{27b}{6} = \frac{-17b}{6}$
✔ Answer: $-\frac{17b}{6}$

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#### 9) $\frac{1}{2y} + \frac{7}{y} = \frac{1}{2y} + \frac{14}{2y} = \frac{15}{2y}$

✔ Answer: $\frac{15}{2y}$

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#### 10) $\frac{6}{4k} - \frac{5}{6k}$

- Simplify: $\frac{3}{2k} - \frac{5}{6k}$
- LCD = $6k$: $\frac{9}{6k} - \frac{5}{6k} = \frac{4}{6k} = \frac{2}{3k}$
✔ Answer: $\frac{2}{3k}$

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#### 11) $\frac{r}{p} - \frac{4p}{q}$

- LCD = $pq$
- $\frac{rq}{pq} - \frac{4p^2}{pq} = \frac{rq - 4p^2}{pq}$
✔ Answer: $\frac{rq - 4p^2}{pq}$

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#### 12) $\frac{x}{8} + \frac{1}{x + 2}$

- LCD = $8(x + 2)$
- $\frac{x(x + 2)}{8(x + 2)} + \frac{8}{8(x + 2)} = \frac{x^2 + 2x + 8}{8(x + 2)}$
✔ Answer: $\frac{x^2 + 2x + 8}{8(x + 2)}$

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#### 13) $\frac{2}{x + 1} + \frac{1}{x + 2}$

- LCD = $(x + 1)(x + 2)$
- $\frac{2(x + 2)}{(x + 1)(x + 2)} + \frac{1(x + 1)}{(x + 1)(x + 2)} = \frac{2x + 4 + x + 1}{(x + 1)(x + 2)} = \frac{3x + 5}{(x + 1)(x + 2)}$
✔ Answer: $\frac{3x + 5}{(x + 1)(x + 2)}$

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#### 14) $\frac{3}{y - 1} - \frac{1}{y + 2}$

- LCD = $(y - 1)(y + 2)$
- $\frac{3(y + 2)}{(y - 1)(y + 2)} - \frac{1(y - 1)}{(y - 1)(y + 2)} = \frac{3y + 6 - y + 1}{(y - 1)(y + 2)} = \frac{2y + 7}{(y - 1)(y + 2)}$
✔ Answer: $\frac{2y + 7}{(y - 1)(y + 2)}$

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#### 15) $\frac{6}{1 - 2b} - \frac{b}{3 + b}$

- Note: $1 - 2b = -(2b - 1)$, but keep as is.
- LCD = $(1 - 2b)(3 + b)$
- $\frac{6(3 + b)}{(1 - 2b)(3 + b)} - \frac{b(1 - 2b)}{(1 - 2b)(3 + b)} = \frac{18 + 6b - b + 2b^2}{(1 - 2b)(3 + b)} = \frac{2b^2 + 5b + 18}{(1 - 2b)(3 + b)}$
✔ Answer: $\frac{2b^2 + 5b + 18}{(1 - 2b)(3 + b)}$

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#### 16) $\frac{2}{x^2} - \frac{1}{x(x - 1)}$

- LCD = $x^2(x - 1)$
- $\frac{2(x - 1)}{x^2(x - 1)} - \frac{x}{x^2(x - 1)} = \frac{2x - 2 - x}{x^2(x - 1)} = \frac{x - 2}{x^2(x - 1)}$
✔ Answer: $\frac{x - 2}{x^2(x - 1)}$

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#### 17) $7 - \frac{x - 4}{4x(x - 2)}$

- $7 = \frac{7 \cdot 4x(x - 2)}{4x(x - 2)} = \frac{28x(x - 2)}{4x(x - 2)}$
- But better: write $7 = \frac{28x(x - 2)}{4x(x - 2)}$, but let’s do it carefully:
- LCD = $4x(x - 2)$
- $7 = \frac{7 \cdot 4x(x - 2)}{4x(x - 2)} = \frac{28x(x - 2)}{4x(x - 2)}$
- But this is messy. Instead:
$$
= \frac{7 \cdot 4x(x - 2) - (x - 4)}{4x(x - 2)} = \frac{28x(x - 2) - (x - 4)}{4x(x - 2)}
$$
- Expand: $28x^2 - 56x - x + 4 = 28x^2 - 57x + 4$
✔ Answer: $\frac{28x^2 - 57x + 4}{4x(x - 2)}$

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#### 18) $\frac{3y + 1}{(y - 2)^2} + \frac{y}{y - 2}$

- LCD = $(y - 2)^2$
- $\frac{3y + 1}{(y - 2)^2} + \frac{y(y - 2)}{(y - 2)^2} = \frac{3y + 1 + y^2 - 2y}{(y - 2)^2} = \frac{y^2 + y + 1}{(y - 2)^2}$
✔ Answer: $\frac{y^2 + y + 1}{(y - 2)^2}$

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#### 19) $\frac{w}{w^2 - 1} + \frac{w + 1}{w - 1}$

- Note: $w^2 - 1 = (w - 1)(w + 1)$
- So:
$$
= \frac{w}{(w - 1)(w + 1)} + \frac{w + 1}{w - 1} = \frac{w}{(w - 1)(w + 1)} + \frac{(w + 1)^2}{(w - 1)(w + 1)}
$$
- Add: $\frac{w + (w^2 + 2w + 1)}{(w - 1)(w + 1)} = \frac{w^2 + 3w + 1}{(w - 1)(w + 1)}$
✔ Answer: $\frac{w^2 + 3w + 1}{w^2 - 1}$

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#### 20) $\frac{1}{x^2 + 5x + 4} + \frac{1}{x^2 + 7x + 12}$

- Factor:
- $x^2 + 5x + 4 = (x + 1)(x + 4)$
- $x^2 + 7x + 12 = (x + 3)(x + 4)$
- LCD = $(x + 1)(x + 3)(x + 4)$
- First term: $\frac{1}{(x + 1)(x + 4)} = \frac{x + 3}{(x + 1)(x + 3)(x + 4)}$
- Second term: $\frac{1}{(x + 3)(x + 4)} = \frac{x + 1}{(x + 1)(x + 3)(x + 4)}$
- Add: $\frac{x + 3 + x + 1}{(x + 1)(x + 3)(x + 4)} = \frac{2x + 4}{(x + 1)(x + 3)(x + 4)} = \frac{2(x + 2)}{(x + 1)(x + 3)(x + 4)}$
✔ Answer: $\frac{2(x + 2)}{(x + 1)(x + 3)(x + 4)}$

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#### 21) $\frac{2}{2x^2 + 5x - 3} - \frac{3}{3x^2 + 11x + 6}$

- Factor:
- $2x^2 + 5x - 3$: try $2x^2 + 6x - x - 3 = 2x(x + 3) -1(x + 3) = (2x - 1)(x + 3)$
- $3x^2 + 11x + 6$: $3x^2 + 9x + 2x + 6 = 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3)$
- So:
$$
= \frac{2}{(2x - 1)(x + 3)} - \frac{3}{(3x + 2)(x + 3)}
$$
- LCD = $(2x - 1)(3x + 2)(x + 3)$
- First term: $\frac{2(3x + 2)}{(2x - 1)(3x + 2)(x + 3)}$
- Second term: $\frac{3(2x - 1)}{(2x - 1)(3x + 2)(x + 3)}$
- Subtract: $\frac{2(3x + 2) - 3(2x - 1)}{(2x - 1)(3x + 2)(x + 3)}$
- Expand: $6x + 4 - 6x + 3 = 7$
- So: $\frac{7}{(2x - 1)(3x + 2)(x + 3)}$
✔ Answer: $\frac{7}{(2x - 1)(3x + 2)(x + 3)}$

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✔ Final Answers Summary:

#### Section A
1. $a$
2. $\frac{a}{b}$
3. $\frac{x + 3}{x - 2}$
4. $\frac{x^3(3x + 2)}{12}$
5. $\frac{1}{x}$
6. $\frac{1}{2(2b + 1)(b - 3)}$
7. $\frac{y - 2}{2(y + 1)}$
8. $a y^3$

#### Section B
1. $\frac{x + 20}{5}$
2. $\frac{y^2 - 6}{y}$
3. $\frac{4x - 13}{x - 2}$
4. $\frac{27w + 46}{3w + 5}$
5. $\frac{x^2 + 8x + 15}{x + 2}$
6. $\frac{8}{x}$
7. $\frac{9a}{2}$
8. $-\frac{17b}{6}$
9. $\frac{15}{2y}$
10. $\frac{2}{3k}$
11. $\frac{rq - 4p^2}{pq}$
12. $\frac{x^2 + 2x + 8}{8(x + 2)}$
13. $\frac{3x + 5}{(x + 1)(x + 2)}$
14. $\frac{2y + 7}{(y - 1)(y + 2)}$
15. $\frac{2b^2 + 5b + 18}{(1 - 2b)(3 + b)}$
16. $\frac{x - 2}{x^2(x - 1)}$
17. $\frac{28x^2 - 57x + 4}{4x(x - 2)}$
18. $\frac{y^2 + y + 1}{(y - 2)^2}$
19. $\frac{w^2 + 3w + 1}{w^2 - 1}$
20. $\frac{2(x + 2)}{(x + 1)(x + 3)(x + 4)}$
21. $\frac{7}{(2x - 1)(3x + 2)(x + 3)}$

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