KS3 Maths · Year 8 · Chapter 7: Fractions · Lesson 4

Algebraic fractions

Year 8About 70 minutesNo calculator48 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $\frac{4}{9} + \frac{5}{6}$ .
[1 mark]
Answer
A2
Needed today
Expand $3 (2 x - 5)$ .
[1 mark]
Answer
A3
Last lesson
Work out $\frac{3}{8} \div \frac{9}{20}$ .
[1 mark]
Answer
A4
Year 7
Simplify $4 a + 3 b - a + 2 b$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Simplify (a) $\frac{6 a}{9}$ (b) $\frac{8 x + 12}{4}$ .

Hint Divide the top and the bottom by the same number. In (b) every term on the top must be divided.

B1
Simplify (i) $\frac{10 t}{15}$ (ii) $\frac{12 y - 18}{6}$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 2

Write $\frac{x}{3} + \frac{3 x}{5}$ as a single fraction. Check your answer with $x = 15$ .

Hint Use the common denominator

15

, just as you would for

\frac{1}{3} + \frac{3}{5}

B2
Write $\frac{a}{4} + \frac{5 a}{6}$ as a single fraction.
[2 marks]
Answer

Example 3

Write $\frac{x + 4}{3} + \frac{x - 2}{4}$ as a single fraction.

Hint Over

12

, the first top is multiplied by

4

and the second by

3

. Keep the brackets.

B3
Write $\frac{a + 2}{5} + \frac{a - 3}{2}$ as a single fraction.
[3 marks]
Answer

Example 4

Write $\frac{2 x + 5}{6} - \frac{x - 1}{4}$ as a single fraction.

Hint The minus sign multiplies the whole second bracket:

- 3 (x - 1) = - 3 x + 3

B4
Write $\frac{3 a + 1}{4} - \frac{a - 2}{3}$ as a single fraction.
[3 marks]
Answer

Example 5

Simplify (a) $\frac{4 a}{9} \times \frac{3}{2 a}$ (b) $\frac{3 x}{5} \div \frac{x}{10}$ .

Hint Multiply tops and bottoms, then cancel. To divide, multiply by the reciprocal.

B5
Simplify (i) $\frac{5 y}{6} \times \frac{9}{10 y}$ (ii) $\frac{2 n}{3} \div \frac{4 n}{9}$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Practice

Year 8

C1
Write $\frac{5 x - 2}{7} + \frac{2 x + 9}{7}$ as simply as possible.
[2 marks]
Answer
C2
Write $3 + \frac{x}{4}$ as a single fraction.
[2 marks]
Answer

C3
Write $\frac{2 a}{3} - \frac{b}{5}$ as a single fraction.
[2 marks]
Answer
C4
Write $\frac{x}{2} + \frac{x}{3} + \frac{x}{6}$ as simply as possible.
[2 marks]
Answer
C5
Simplify $\frac{10 x - 15}{20 x + 5}$ by dividing the top and the bottom by a common factor.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Sam says: " $\frac{a}{3} + \frac{a}{5} = \frac{2 a}{8}$ ."
Sam is wrong.
(a)
Show that Sam is wrong by working out both sides when $a = 15$ .
[2 marks]
(b)
Write $\frac{a}{3} + \frac{a}{5}$ correctly as a single fraction.
[1 mark]
Answer
Total for D1: 3 marks

D2
The sides of a rectangle are $\frac{x + 3}{2}$ cm and $\frac{x - 1}{3}$ cm.
Not drawn accurately
(a)
Show that the perimeter is $\frac{5 x + 7}{3}$ cm.
[3 marks]
(b)
The perimeter is $9$ cm. Work out the value of $x$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
Show that $\frac{x + 1}{2} - \frac{x - 1}{3}$ can be written as $\frac{x + 5}{6}$ .
[3 marks]
D4
Solve $\frac{x + 5}{3} + \frac{x - 2}{4} = 7$ .
[4 marks]
Answer
D5
A cyclist rides $\frac{5 x}{4}$ km in $\frac{x}{6}$ hours. Show that her average speed is the same whatever the value of $x$ , and work it out.
[3 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Write $\frac{x}{2} + \frac{x}{6} + \frac{x}{12} + \frac{x}{20}$ as a single fraction, as simply as possible.
[4 marks]
Hint 1 · What to try
Find a common denominator. $60$ is a multiple of $2$ , $6$ , $12$ and $20$ .
Hint 2 · The first line
Over $60$ the tops are $30 x$ , $10 x$ , $5 x$ and $3 x$ .
Hint 3 · The full method
$\frac{48 x}{60}$ : divide the top and the bottom by $12$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify an algebraic fraction by dividing by a common factor.
I can add and subtract algebraic fractions using a common denominator.
I can multiply and divide algebraic fractions, cancelling first.
I can check a single fraction by substituting a value.

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