GCSE Maths | 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) $\dfrac{12x^3}{20x^7}$
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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) $\dfrac{x(x+3)^2}{x(x+1)(x+3)}$
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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) $\dfrac{x^2+4x-45}{x^2+10x+9}$
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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) $\dfrac{x-5}{x+1}$
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) $\dfrac{3}{w} \times \dfrac{x}{4} \times \dfrac{y}{w}$
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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) $\dfrac{4}{2x-1} \times \dfrac{6x-3}{x+7}$
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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) $\dfrac{x}{x+6} \div \dfrac{x+6}{x^2}$
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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) $\dfrac{x^3}{(x+6)^2}$
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) $\dfrac{3c}{4}+\dfrac{5c}{9}$
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a) $\dfrac{8x}{15}$
b) $\dfrac{9c}{14}$
c) $\dfrac{4w}{9}$
d) $\dfrac{13m}{21}$
e) $\dfrac{11x}{10}$
f) $\dfrac{47c}{36}$
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) $\dfrac{7h}{8}-\dfrac{5h}{12}$
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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) $\dfrac{x-4}{8}+\dfrac{2x-3}{2}$
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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) $\dfrac{5-x}{6}-\dfrac{8-3x}{10}$
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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) $\dfrac{2x+1}{30}$
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) $1+\dfrac{4}{3x-1}=\dfrac{3}{2x-1}$
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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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