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# GCSE Maths | Algebraic Fractions
- URL: https://www.a-level-maths-tutor.com/gcse-maths-algebraic-fractions/
- Published: 2026-09-23T16:49:27.000Z
- Updated: 2026-09-23T16:49:46.000Z
- Author: Jaisul Naik
- Tags: GCSE Maths

Simplifying algebraic fractions

### Question 1

Simplify each of the following.

a) $\\dfrac{42xyz}{56}$

b) $\\dfrac{45ab}{60abc}$

c) $\\dfrac{16mn}{18n}$

d) $\\dfrac{40x^2y}{32xy}$

e) $\\dfrac{17cf}{34c^3}$

f) $\\dfrac{8x^4}{2x^2}$

g) $\\dfrac{33a^2b^2}{44a^3b}$

h) $\\dfrac{12x^3}{20x^7}$

Show Answers 

a) $\\dfrac{3xyz}{4}$

b) $\\dfrac{3}{4c}$

c) $\\dfrac{8m}{9}$

d) $\\dfrac{5x}{4}$

e) $\\dfrac{f}{2c^2}$

f) $4x^2$

g) $\\dfrac{3b}{4a}$

h) $\\dfrac{3}{5x^4}$

### Question 2

Simplify each of the following.

a) $\\dfrac{(x+6)(x+3)}{(x+3)}$

b) $\\dfrac{(x-1)(x+1)}{(x-1)}$

c) $\\dfrac{(x-3)}{(x-4)(x-3)}$

d) $\\dfrac{(x+7)^2}{(x+7)}$

e) $\\dfrac{(x-3)(x+2)}{(x+2)(x+9)}$

f) $\\dfrac{(x+2)(x+4)^2}{(x+4)}$

g) $\\dfrac{(x+1)(x+2)(x+3)}{(x+2)(x+3)(x+4)}$

h) $\\dfrac{x(x+3)^2}{x(x+1)(x+3)}$

Show Answers 

a) $x+6$

b) $x+1$

c) $\\dfrac{1}{x-4}$

d) $x+7$

e) $\\dfrac{x-3}{x+9}$

f) $(x+2)(x+4) = x^2+6x+8$

g) $\\dfrac{x+1}{x+4}$

h) $\\dfrac{x+3}{x+1}$

### Question 3

Simplify each of the following by factorising first.

a) $\\dfrac{x^2+5x+4}{x^2+4x+3}$

b) $\\dfrac{x^2+6x+9}{x^2-2x-15}$

c) $\\dfrac{x^2-2x}{x^2+2x-8}$

d) $\\dfrac{x^2-7x+10}{x^2+3x-10}$

e) $\\dfrac{x^2+8x+15}{x^2-x-12}$

f) $\\dfrac{x^2+13x+40}{x^2+14x+48}$

g) $\\dfrac{x^2-2x-8}{x^2+6x-40}$

h) $\\dfrac{x^2+10x+24}{x^2-36}$

i) $\\dfrac{x^2+4x-45}{x^2+10x+9}$

Show Answers 

a) $\\dfrac{x+4}{x+3}$

b) $\\dfrac{x+3}{x-5}$

c) $\\dfrac{x}{x+4}$

d) $\\dfrac{x-5}{x+5}$

e) $\\dfrac{x+5}{x-4}$

f) $\\dfrac{x+5}{x+6}$

g) $\\dfrac{x+2}{x+10}$

h) $\\dfrac{x+4}{x-6}$

i) $\\dfrac{x-5}{x+1}$

Multiplying and dividing algebraic fractions

### Question 4

Simplify each of the following.

a) $\\dfrac{2}{g} \\times \\dfrac{3}{h}$

b) $\\dfrac{3}{c} \\times \\dfrac{a}{4}$

c) $\\dfrac{w}{x} \\times \\dfrac{3}{a}$

d) $\\dfrac{3a}{7} \\times \\dfrac{2c}{g}$

e) $\\dfrac{a}{e} \\times \\dfrac{f}{b}$

f) $\\dfrac{e}{8} \\times \\dfrac{d}{8}$

g) $\\dfrac{x}{2} \\times \\dfrac{x}{5}$

h) $\\dfrac{7}{y} \\times \\dfrac{2}{y}$

i) $\\dfrac{3}{w} \\times \\dfrac{x}{4} \\times \\dfrac{y}{w}$

Show Answers 

a) $\\dfrac{6}{gh}$

b) $\\dfrac{3a}{4c}$

c) $\\dfrac{3w}{ax}$

d) $\\dfrac{6ac}{7g}$

e) $\\dfrac{af}{be}$

f) $\\dfrac{de}{64}$

g) $\\dfrac{x^2}{10}$

h) $\\dfrac{14}{y^2}$

i) $\\dfrac{3xy}{4w^2}$

### Question 5

Simplify each of the following.

d) $\\dfrac{1}{x+3} \\times \\dfrac{2}{x+1}$

e) $\\dfrac{3x+2}{3} \\times \\dfrac{x+1}{3}$

f) $\\dfrac{x+4}{x-4} \\times \\dfrac{x-2}{x+5}$

g) $\\dfrac{x+1}{x-7} \\times \\dfrac{x-5}{x+1}$

h) $\\dfrac{7}{2x+8} \\times \\dfrac{x+4}{14}$

i) $\\dfrac{4}{2x-1} \\times \\dfrac{6x-3}{x+7}$

Show Answers 

d) $\\dfrac{2}{(x+1)(x+3)}$

e) $\\dfrac{(3x+2)(x+1)}{9}$

f) $\\dfrac{(x+4)(x-2)}{(x-4)(x+5)}$

g) $\\dfrac{x-5}{x-7}$

h) $\\dfrac{1}{4}$

i) $\\dfrac{12}{x+7}$

### Question 6

Simplify each of the following.

a) $\\dfrac{x-4}{8} \\div \\dfrac{3x-12}{2}$

b) $\\dfrac{x+3}{x+2} \\div \\dfrac{x+1}{x+2}$

c) $\\dfrac{x+1}{2} \\div \\dfrac{2x+2}{3}$

d) $\\dfrac{3x+9}{2} \\div \\dfrac{x+3}{4}$

e) $\\dfrac{4}{x-2} \\div \\dfrac{3}{x^2-2x}$

f) $\\dfrac{11}{4x^2+2x} \\div \\dfrac{3}{2x+1}$

g) $\\dfrac{x+3}{x+1} \\div \\dfrac{x}{(x+1)^2}$

h) $\\dfrac{x^2}{7} \\div \\dfrac{6x^3+8x^2}{x^2-7x}$

i) $\\dfrac{x}{x+6} \\div \\dfrac{x+6}{x^2}$

Show Answers 

a) $\\dfrac{1}{12}$

b) $\\dfrac{x+3}{x+1}$

c) $\\dfrac{3}{4}$

d) $6$

e) $\\dfrac{4x}{3}$

f) $\\dfrac{11}{6x}$

g) $\\dfrac{(x+3)(x+1)}{x}$

h) $\\dfrac{x(x-7)}{14(3x+4)}$

i) $\\dfrac{x^3}{(x+6)^2}$

Adding algebraic fractions

### Question 7

Express as a single simplified fraction.

a) $\\dfrac{x}{3}+\\dfrac{x}{5}$

b) $\\dfrac{c}{2}+\\dfrac{c}{7}$

c) $\\dfrac{w}{3}+\\dfrac{w}{9}$

d) $\\dfrac{m}{3}+\\dfrac{2m}{7}$

e) $\\dfrac{3x}{5}+\\dfrac{x}{2}$

f) $\\dfrac{3c}{4}+\\dfrac{5c}{9}$

Show Answers 

a) $\\dfrac{8x}{15}$

b) $\\dfrac{9c}{14}$

c) $\\dfrac{4w}{9}$

d) $\\dfrac{13m}{21}$

e) $\\dfrac{11x}{10}$

f) $\\dfrac{47c}{36}$

Subtracting algebraic fractions

### Question 8

Express as a single simplified fraction.

a) $\\dfrac{x}{2}-\\dfrac{x}{3}$

b) $\\dfrac{a}{5}-\\dfrac{a}{9}$

c) $\\dfrac{m}{2}-\\dfrac{m}{8}$

d) $\\dfrac{m}{2}-\\dfrac{2m}{5}$

e) $\\dfrac{3n}{4}-\\dfrac{5n}{9}$

f) $\\dfrac{7h}{8}-\\dfrac{5h}{12}$

Show Answers 

a) $\\dfrac{x}{6}$

b) $\\dfrac{4a}{45}$

c) $\\dfrac{3m}{8}$

d) $\\dfrac{m}{10}$

e) $\\dfrac{7n}{36}$

f) $\\dfrac{11h}{24}$

### Question 9

Express as a single simplified fraction.

a) $\\dfrac{x+4}{3}+\\dfrac{x+1}{2}$

b) $\\dfrac{2x+1}{2}+\\dfrac{x+3}{5}$

c) $\\dfrac{5x-2}{3}+\\dfrac{2x+7}{4}$

d) $\\dfrac{3x+11}{2}+\\dfrac{x-20}{3}$

e) $\\dfrac{5x-4}{2}+\\dfrac{x+1}{3}$

f) $\\dfrac{x-4}{8}+\\dfrac{2x-3}{2}$

Show Answers 

a) $\\dfrac{5x+11}{6}$

b) $\\dfrac{12x+11}{10}$

c) $\\dfrac{26x+13}{12}$

d) $\\dfrac{11x-7}{6}$

e) $\\dfrac{17x-10}{6}$

f) $\\dfrac{9x-16}{8}$

### Question 10

Express as a single simplified fraction.

g) $\\dfrac{7x+4}{2}-\\dfrac{x+1}{3}$

h) $\\dfrac{9x+1}{5}-\\dfrac{x+2}{2}$

i) $\\dfrac{3x+11}{2}-\\dfrac{2x-3}{6}$

j) $\\dfrac{x-8}{3}-\\dfrac{x-5}{7}$

k) $\\dfrac{4x-7}{10}-\\dfrac{2x-9}{3}$

l) $\\dfrac{5-x}{6}-\\dfrac{8-3x}{10}$

Show Answers 

g) $\\dfrac{19x+10}{6}$

h) $\\dfrac{13x-8}{10}$

i) $\\dfrac{7x+36}{6}$

j) $\\dfrac{4x-41}{21}$

k) $\\dfrac{23-4x \\cdot \\frac{2}{3}}{...}$

l) $\\dfrac{2x+1}{30}$

Solving algebraic fractions

### Question 11

Solve each of the following.

a) $\\dfrac{3}{x+2}+\\dfrac{2}{x}=1$

b) $\\dfrac{2}{x+4}+\\dfrac{1}{x+2}=1$

c) $\\dfrac{4}{x-3}-\\dfrac{3}{x-2}=1$

d) $\\dfrac{1}{x-10}+\\dfrac{2}{x-10}=1$

e) $\\dfrac{1}{x}+\\dfrac{2}{x+1}=2$

f) $\\dfrac{1}{x+3}-\\dfrac{1}{x+1}=2$

g) $\\dfrac{3}{x+2}+\\dfrac{2}{x+4}=2$

h) $\\dfrac{2}{2x-1}+\\dfrac{1}{x-2}=1$

i) $\\dfrac{6}{x+1}-\\dfrac{1}{x+1}=3$

j) $1-\\dfrac{3}{x+3}=\\dfrac{1}{x-1}$

k) $\\dfrac{x}{x-3}+\\dfrac{4}{x+2}=2$

l) $1+\\dfrac{4}{3x-1}=\\dfrac{3}{2x-1}$

Show Answers 

a) $x=-1$ or $x=4$ (note: $x=-1$ makes $\\frac{2}{x}$ undefined, so $x=4$)

b) $x=-3$ or $x=0$ (note: $x=-3$ makes denominator zero, so $x=0$)

c) $x=1$ or $x=5$

d) $x=13$

e) $x=-\\dfrac{1}{2}$ or $x=1$

f) $x=-2$

g) $x=-\\dfrac{7}{2}$ or $x=0$

h) $x=1$ or $x=\\dfrac{7}{2}$

i) $x=\\dfrac{2}{3}$

j) $x=-1$ or $x=3$

k) $x=0$ or $x=8$

l) $x=0$ or $x=1$

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