KS4. Algebra. Algebraic Fractions - Maths with David - Free Printable

Let’s simplify each expression as far as possible. We’ll go one by one.

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1. $\frac{25}{35}$
→ GCF of 25 and 35 is 5.
$\frac{25 \div 5}{35 \div 5} = \boxed{\frac{5}{7}}$

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2. $\frac{84}{96}$
→ GCF of 84 and 96 is 12.
$\frac{84 \div 12}{96 \div 12} = \frac{7}{8}$
✔ $\boxed{\frac{7}{8}}$

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3. $\frac{5y^2}{y}$
→ Cancel one $y$: $\frac{5y \cdot y}{y} = 5y$
✔ $\boxed{5y}$

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4. $\frac{y}{2y}$
→ Cancel $y$ (assuming $y \ne 0$): $\frac{1}{2}$
✔ $\boxed{\frac{1}{2}}$

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5. $\frac{8x^2}{2x^2}$
→ Cancel $x^2$ (assuming $x \ne 0$), then 8/2 = 4
✔ $\boxed{4}$

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6. $\frac{2x}{4y}$
→ Simplify numerator/denominator by 2: $\frac{x}{2y}$
✔ $\boxed{\frac{x}{2y}}$

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7. $\frac{6y}{3y}$
→ Cancel $y$ (≠0), 6/3 = 2
✔ $\boxed{2}$

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8. $\frac{5ab}{10b}$
→ Cancel $b$ (≠0), 5/10 = 1/2 → $\frac{a}{2}$
✔ $\boxed{\frac{a}{2}}$

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9. $\frac{8ab^2}{12ab}$
→ Cancel $a$, one $b$, and divide 8/12 by 4: $\frac{2b}{3}$
✔ $\boxed{\frac{2b}{3}}$

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10. $\frac{7a^2b}{35ab^2}$
→ Cancel $a$, one $b$, 7/35 = 1/5 → $\frac{a}{5b}$
✔ $\boxed{\frac{a}{5b}}$

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11. $\frac{(2a)^2}{4a}$
→ $(2a)^2 = 4a^2$, so $\frac{4a^2}{4a} = a$ (cancel 4, one a)
✔ $\boxed{a}$

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12. $\frac{7yx}{8xy}$
→ $yx = xy$, so cancel $xy$ → $\frac{7}{8}$
✔ $\boxed{\frac{7}{8}}$

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13. $\frac{5x + 2x^2}{3x}$
→ Factor numerator: $x(5 + 2x)$, denominator: $3x$
→ Cancel $x$ (≠0): $\frac{2x + 5}{3}$
✔ $\boxed{\frac{2x + 5}{3}}$

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14. $\frac{9x + 3}{3x}$
→ Factor numerator: $3(3x + 1)$, denominator: $3x$
→ Cancel 3: $\frac{3x + 1}{x}$
✔ $\boxed{\frac{3x + 1}{x}}$

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15. $\frac{25 + 7}{25}$
→ Numerator: 32 → $\frac{32}{25}$ (already simplified)
✔ $\boxed{\frac{32}{25}}$

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16. $\frac{4a + 5a^2}{5a}$
→ Factor numerator: $a(4 + 5a)$, denominator: $5a$
→ Cancel $a$: $\frac{5a + 4}{5}$
✔ $\boxed{\frac{5a + 4}{5}}$

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17. $\frac{3x}{4x - x^2}$
→ Factor denominator: $x(4 - x)$
→ $\frac{3x}{x(4 - x)} = \frac{3}{4 - x}$ (cancel $x$, ≠0)
✔ $\boxed{\frac{3}{4 - x}}$

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18. $\frac{5ab}{15a + 10a^2}$
→ Factor denominator: $5a(3 + 2a)$
→ Numerator: $5ab$
→ Cancel $5a$: $\frac{b}{3 + 2a}$
✔ $\boxed{\frac{b}{2a + 3}}$ *(or same as above)*

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19. $\frac{5x + 4}{8x}$
→ No common factors → already simplified
✔ $\boxed{\frac{5x + 4}{8x}}$

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20. $\frac{12x + 6}{6y}$
→ Factor numerator: $6(2x + 1)$, denominator: $6y$
→ Cancel 6: $\frac{2x + 1}{y}$
✔ $\boxed{\frac{2x + 1}{y}}$

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21. $\frac{5x + 10y}{15xy}$
→ Factor numerator: $5(x + 2y)$, denominator: $15xy = 5 \cdot 3xy$
→ Cancel 5: $\frac{x + 2y}{3xy}$
✔ $\boxed{\frac{x + 2y}{3xy}}$

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22. $\frac{18a - 3ab}{6a^2}$
→ Factor numerator: $3a(6 - b)$, denominator: $6a^2 = 3a \cdot 2a$
→ Cancel $3a$: $\frac{6 - b}{2a}$
✔ $\boxed{\frac{6 - b}{2a}}$

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23. $\frac{4ab + 8a^2}{2ab}$
→ Factor numerator: $4a(b + 2a)$, denominator: $2ab$
→ Cancel 2a: $\frac{2(b + 2a)}{b} = \frac{2b + 4a}{b}$
Or leave as $\frac{2(2a + b)}{b}$
✔ $\boxed{\frac{2(2a + b)}{b}}$ or $\boxed{\frac{4a + 2b}{b}}$

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24. $\frac{(2x)^2 - 8x}{4x}$
→ Expand: $(2x)^2 = 4x^2$, so numerator: $4x^2 - 8x$
→ Factor: $4x(x - 2)$, denominator: $4x$
→ Cancel $4x$ (≠0): $x - 2$
✔ $\boxed{x - 2}$

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## ✔ Final Answers:

1. $\frac{5}{7}$
2. $\frac{7}{8}$
3. $5y$
4. $\frac{1}{2}$
5. $4$
6. $\frac{x}{2y}$
7. $2$
8. $\frac{a}{2}$
9. $\frac{2b}{3}$
10. $\frac{a}{5b}$
11. $a$
12. $\frac{7}{8}$
13. $\frac{2x + 5}{3}$
14. $\frac{3x + 1}{x}$
15. $\frac{32}{25}$
16. $\frac{5a + 4}{5}$
17. $\frac{3}{4 - x}$
18. $\frac{b}{2a + 3}$
19. $\frac{5x + 4}{8x}$
20. $\frac{2x + 1}{y}$
21. $\frac{x + 2y}{3xy}$
22. $\frac{6 - b}{2a}$
23. $\frac{2(2a + b)}{b}$ or $\frac{4a + 2b}{b}$
24. $x - 2$

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Let me know if you want step-by-step work for any specific one!