Sec 2 Maths Algebraic fractions and formulae worksheet with answers

A free Sec 2 Maths worksheet on algebraic fractions and formulae: 7 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 2 Maths worksheet: Algebraic fractions and formulae

Total: 30 marksSuggested time: 45 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

Express

\dfrac{1}{x - 3} - \dfrac{5}{x + 3}

as a single fraction in its simplest form.

Answer[2]

Simplify

\dfrac{2}{11p} \times \dfrac{7q^2}{10p}

, giving your answer in its simplest form.

Answer[2]

Simplify

\dfrac{7a^2}{6b^2} \times \dfrac{10}{3a^2}

, giving your answer in its simplest form.

Answer[2]

(a)Make

x

the subject of

y = \sqrt{5x - 5}

Answer (a)[2]

(b)Make

x

the subject of

y = \dfrac{4x - 5}{3x + 2}

Answer (b)[2]

Section B

(a)Make

x

the subject of

y = \dfrac{9}{x} + 2

Answer (a)[2]

(b)Express

\dfrac{3}{x - 5} - \dfrac{4}{x + 4}

as a single fraction in its simplest form.

Answer (b)[2]

(a)Simplify

\dfrac{x^{2} + x - 6}{x^{2} - 2x - 3} \times \dfrac{x - 3}{x + 3}

Answer (a)[2]

(b)Make

x

the subject of

y = \dfrac{2x - 6}{5}

Answer (b)[2]

(c)(i)Simplify

\dfrac{10}{7ab^2} \times \dfrac{9}{5a}

, giving your answer in its simplest form.

Answer (c)(i)[2]

(c)(ii)Simplify

\dfrac{4y^2}{5} \div \dfrac{8xy^2}{9}

, giving your answer in its simplest form.

Answer (c)(ii)[2]

(a)Simplify

\dfrac{p}{5} \times \dfrac{5pq}{6}

, giving your answer in its simplest form.

Answer (a)[2]

(b)Express

\dfrac{3}{x - 4} + \dfrac{1}{x - 3}

as a single fraction in its simplest form.

Answer (b)[2]

(c)(i)Make

x

the subject of

y = \dfrac{2}{x} + 5

Answer (c)(i)[2]

(c)(ii)Make

x

the subject of

y = 4\sqrt{x} + 5

Answer (c)(ii)[2]

END OF PAPER

Answers and worked solutions

Question 1

Answer:

= \dfrac{-4x + 18}{(x - 3)(x + 3)}

[2]

\dfrac{1}{x - 3} - \dfrac{5}{x + 3} = \dfrac{1(x + 3) - 5(x - 3)}{(x - 3)(x + 3)}

= \dfrac{x + 3 - (5x - 15)}{(x - 3)(x + 3)}

= \dfrac{-4x + 18}{(x - 3)(x + 3)}

Question 2

Answer:

= \dfrac{7q^2}{55p^2}

[2]

\dfrac{2}{11p} \times \dfrac{7q^2}{10p} = \dfrac{14 \times q \times q}{110 \times p \times p}

The HCF of

14

and

110

2

\dfrac{14}{110} = \dfrac{7}{55}

= \dfrac{7q^2}{55p^2}

Question 3

Answer:

= \dfrac{35}{9b^2}

[2]

\dfrac{7a^2}{6b^2} \times \dfrac{10}{3a^2} = \dfrac{70 \times a \times a}{18 \times a \times a \times b \times b}

The HCF of

70

and

18

2

\dfrac{70}{18} = \dfrac{35}{9}

= \dfrac{35}{9b^2}

Question 4

(a)Answer:

x = \dfrac{y^{2} + 5}{5}

[2]

y^{2} = 5x - 5

y^{2} + 5 = 5x

x = \dfrac{y^{2} + 5}{5}

(b)Answer:

x = \dfrac{-5 - 2y}{3y - 4}

[2]

y(3x + 2) = 4x - 5

3xy + 2y = 4x - 5

3xy - 4x = -5 - 2y

x(3y - 4) = -5 - 2y

x = \dfrac{-5 - 2y}{3y - 4}

Question 5

(a)Answer:

x = \dfrac{9}{y - 2}

[2]

y - 2 = \dfrac{9}{x}

x(y - 2) = 9

x = \dfrac{9}{y - 2}

(b)Answer:

= \dfrac{-x + 32}{(x - 5)(x + 4)}

[2]

\dfrac{3}{x - 5} - \dfrac{4}{x + 4} = \dfrac{3(x + 4) - 4(x - 5)}{(x - 5)(x + 4)}

= \dfrac{3x + 12 - (4x - 20)}{(x - 5)(x + 4)}

= \dfrac{-x + 32}{(x - 5)(x + 4)}

Question 6

(a)Answer:

= \dfrac{x - 2}{x + 1}

[2]

\dfrac{x^{2} + x - 6}{x^{2} - 2x - 3} \times \dfrac{x - 3}{x + 3}

= \dfrac{x^{2} + x - 6}{x^{2} - 2x - 3} \times \dfrac{x - 3}{x + 3}

= \dfrac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \times \dfrac{x - 3}{x + 3}

= \dfrac{x - 2}{x + 1}

(b)Answer:

x = \dfrac{5y + 6}{2}

[2]

5y = 2x - 6

5y + 6 = 2x

x = \dfrac{5y + 6}{2}

(c)(i)Answer:

= \dfrac{18}{7a^2b^2}

[2]

\dfrac{10}{7ab^2} \times \dfrac{9}{5a} = \dfrac{90}{35 \times a \times a \times b \times b}

The HCF of

90

and

35

5

\dfrac{90}{35} = \dfrac{18}{7}

= \dfrac{18}{7a^2b^2}

(c)(ii)Answer:

= \dfrac{9}{10x}

[2]

\dfrac{4y^2}{5} \div \dfrac{8xy^2}{9} = \dfrac{4y^2}{5} \times \dfrac{9}{8xy^2}

= \dfrac{36 \times y \times y}{40 \times x \times y \times y}

The HCF of

36

and

40

4

\dfrac{36}{40} = \dfrac{9}{10}

= \dfrac{9}{10x}

Question 7

(a)Answer:

= \dfrac{p^2q}{6}

[2]

\dfrac{p}{5} \times \dfrac{5pq}{6} = \dfrac{5 \times p \times p \times q}{30}

The HCF of

5

and

30

5

\dfrac{5}{30} = \dfrac{1}{6}

= \dfrac{p^2q}{6}

(b)Answer:

= \dfrac{4x - 13}{(x - 4)(x - 3)}

[2]

\dfrac{3}{x - 4} + \dfrac{1}{x - 3} = \dfrac{3(x - 3) + 1(x - 4)}{(x - 4)(x - 3)}

= \dfrac{3x - 9 + (x - 4)}{(x - 4)(x - 3)}

= \dfrac{4x - 13}{(x - 4)(x - 3)}

(c)(i)Answer:

x = \dfrac{2}{y - 5}

[2]

y - 5 = \dfrac{2}{x}

x(y - 5) = 2

x = \dfrac{2}{y - 5}

(c)(ii)Answer:

x = \left(\dfrac{y - 5}{4}\right)^{2}

[2]

y - 5 = 4\sqrt{x}

\sqrt{x} = \dfrac{y - 5}{4}

x = \left(\dfrac{y - 5}{4}\right)^{2}

Practice questions by skill

Sec 2 Maths Algebraic fractions and formulae worksheet with answers

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Algebraic fractions and formulae notes. Or practise it online with step-by-step help when you get one wrong.

Answers and worked solutions

Question 1

Answer:

= \dfrac{-4x + 18}{(x - 3)(x + 3)}

[2]

\dfrac{1}{x - 3} - \dfrac{5}{x + 3} = \dfrac{1(x + 3) - 5(x - 3)}{(x - 3)(x + 3)}

= \dfrac{x + 3 - (5x - 15)}{(x - 3)(x + 3)}

= \dfrac{-4x + 18}{(x - 3)(x + 3)}

Question 2

Answer:

= \dfrac{7q^2}{55p^2}

[2]

\dfrac{2}{11p} \times \dfrac{7q^2}{10p} = \dfrac{14 \times q \times q}{110 \times p \times p}

The HCF of

14

and

110

2

\dfrac{14}{110} = \dfrac{7}{55}

= \dfrac{7q^2}{55p^2}

Question 3

Answer:

= \dfrac{35}{9b^2}

[2]

\dfrac{7a^2}{6b^2} \times \dfrac{10}{3a^2} = \dfrac{70 \times a \times a}{18 \times a \times a \times b \times b}

The HCF of

70

and

18

2

\dfrac{70}{18} = \dfrac{35}{9}

= \dfrac{35}{9b^2}

Question 4

(a)Answer:

x = \dfrac{y^{2} + 5}{5}

[2]

y^{2} = 5x - 5

y^{2} + 5 = 5x

x = \dfrac{y^{2} + 5}{5}

(b)Answer:

x = \dfrac{-5 - 2y}{3y - 4}

[2]

y(3x + 2) = 4x - 5

3xy + 2y = 4x - 5

3xy - 4x = -5 - 2y

x(3y - 4) = -5 - 2y

x = \dfrac{-5 - 2y}{3y - 4}

Question 5

(a)Answer:

x = \dfrac{9}{y - 2}

[2]

y - 2 = \dfrac{9}{x}

x(y - 2) = 9

x = \dfrac{9}{y - 2}

(b)Answer:

= \dfrac{-x + 32}{(x - 5)(x + 4)}

[2]

\dfrac{3}{x - 5} - \dfrac{4}{x + 4} = \dfrac{3(x + 4) - 4(x - 5)}{(x - 5)(x + 4)}

= \dfrac{3x + 12 - (4x - 20)}{(x - 5)(x + 4)}

= \dfrac{-x + 32}{(x - 5)(x + 4)}

Question 6

(a)Answer:

= \dfrac{x - 2}{x + 1}

[2]

\dfrac{x^{2} + x - 6}{x^{2} - 2x - 3} \times \dfrac{x - 3}{x + 3}

= \dfrac{x^{2} + x - 6}{x^{2} - 2x - 3} \times \dfrac{x - 3}{x + 3}

= \dfrac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \times \dfrac{x - 3}{x + 3}

= \dfrac{x - 2}{x + 1}

(b)Answer:

x = \dfrac{5y + 6}{2}

[2]

5y = 2x - 6

5y + 6 = 2x

x = \dfrac{5y + 6}{2}

(c)(i)Answer:

= \dfrac{18}{7a^2b^2}

[2]

\dfrac{10}{7ab^2} \times \dfrac{9}{5a} = \dfrac{90}{35 \times a \times a \times b \times b}

The HCF of

90

and

35

5

\dfrac{90}{35} = \dfrac{18}{7}

= \dfrac{18}{7a^2b^2}

(c)(ii)Answer:

= \dfrac{9}{10x}

[2]

\dfrac{4y^2}{5} \div \dfrac{8xy^2}{9} = \dfrac{4y^2}{5} \times \dfrac{9}{8xy^2}

= \dfrac{36 \times y \times y}{40 \times x \times y \times y}

The HCF of

36

and

40

4

\dfrac{36}{40} = \dfrac{9}{10}

= \dfrac{9}{10x}

Question 7

(a)Answer:

= \dfrac{p^2q}{6}

[2]

\dfrac{p}{5} \times \dfrac{5pq}{6} = \dfrac{5 \times p \times p \times q}{30}

The HCF of

5

and

30

5

\dfrac{5}{30} = \dfrac{1}{6}

= \dfrac{p^2q}{6}

(b)Answer:

= \dfrac{4x - 13}{(x - 4)(x - 3)}

[2]

\dfrac{3}{x - 4} + \dfrac{1}{x - 3} = \dfrac{3(x - 3) + 1(x - 4)}{(x - 4)(x - 3)}

= \dfrac{3x - 9 + (x - 4)}{(x - 4)(x - 3)}

= \dfrac{4x - 13}{(x - 4)(x - 3)}

(c)(i)Answer:

x = \dfrac{2}{y - 5}

[2]

y - 5 = \dfrac{2}{x}

x(y - 5) = 2

x = \dfrac{2}{y - 5}

(c)(ii)Answer:

x = \left(\dfrac{y - 5}{4}\right)^{2}

[2]

y - 5 = 4\sqrt{x}

\sqrt{x} = \dfrac{y - 5}{4}

x = \left(\dfrac{y - 5}{4}\right)^{2}