AQA Level 2 Further Maths · Level 2 Certificate · Chapter 2: Algebra II · Lesson 3

Algebraic fractions and equations with fractions

About 80 minutesNo calculator66 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $\frac{4}{9} \div \frac{10}{21}$ . Give your answer in its simplest form.
[2 marks]
Answer
A2
Needed today
Factorise $2 x^{2} - 7 x - 15$ .
[2 marks]
Answer
A3
Last lesson
Make $y$ the subject of $x = \frac{2 y - 1}{y + 3}$ .
[2 marks]
Answer
A4
From chapter 1
Rationalise the denominator of $\frac{6}{5 - 1}$ . Simplify your answer.
[2 marks]
Answer

Example, then your turn

Level 2

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

Example 1

Simplify fully $\frac{3 x + 12}{x ^{2} + x - 12}$ .

Hint Factorise the top and the bottom first. You can only cancel a whole bracket, never single terms.

B1
Simplify fully $\frac{x ^{2} - 25}{3 x ^{2} + 14 x - 5}$ .
[3 marks]
Answer

Example 2

Simplify fully $\frac{8 y - 12 x}{9 x ^{2} - 6 x y}$ .

Hint After taking out common factors, one bracket is the other one backwards. Use

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

B2
Simplify fully $\frac{10 b - 15 a}{6 a ^{2} - 4 ab}$ .
[2 marks]
Answer

Example 3

Simplify $\frac{x ^{2} - 4}{x + 3} \div \frac{x ^{2} + x - 6}{x ^{2} + 6 x + 9}$ .

Hint Turn the second fraction upside down and multiply. Factorise everything before you cancel.

B3
Simplify $\frac{x ^{2} + 5 x + 4}{2 x - 6} \times \frac{x ^{2} - 9}{x ^{2} + 4 x}$ .
[3 marks]
Answer

Example 4

Write $\frac{x + 2}{4} - \frac{2 x - 1}{6}$ as a single fraction in its simplest form.

Hint The lowest common denominator is

12

. Put the second numerator in a bracket so the minus sign reaches every term.

B4
Write $\frac{3}{x + 4} + \frac{2}{x - 3}$ as a single fraction in its simplest form.
[3 marks]
Answer

Example 5

Write $\frac{3}{x + 2} - \frac{12}{x ^{2} - 4}$ as a single fraction in its simplest form.

Hint

x^{2} - 4 = (x + 2) (x - 2)

already contains

x + 2

, so it is the common denominator. Do not multiply the two denominators together.

B5
Write $\frac{4}{x ^{2} - 2 x - 3} + \frac{1}{x + 1}$ as a single fraction in its simplest form.
[3 marks]
Answer

Practice

Level 2

C1
Simplify fully $\frac{6 x ^{2} y ^{3}}{9 x ^{4} y}$ .
[2 marks]
Answer
C2
Write $\frac{1}{x} + \frac{2}{x + 1} - \frac{3}{x + 2}$ as a single fraction in its simplest form.
[3 marks]
Answer

C3
Solve $\frac{x + 4}{3} - \frac{x - 2}{5} = 2$ .
[3 marks]
Answer
C4
Solve $\frac{3 x - 1}{4} + 2 = \frac{2 x + 5}{3}$ .
[3 marks]
Answer
C5
Solve $\frac{5}{x - 2} = \frac{3}{x + 4}$ .
[2 marks]
Answer
C6
Write $\frac{4 a ^{2} b}{15 c} \div \frac{6 a b ^{2}}{5 c ^{2}}$ as a single fraction in its simplest form.
[2 marks]
Answer

Exam-style questions

AQA exam style

D1
Mia says, " $\frac{2}{x + 1} + \frac{3}{x + 2} = \frac{5}{2 x + 3}$ , because you add the tops and add the bottoms."
Show that Mia is wrong. Write $\frac{2}{x + 1} + \frac{3}{x + 2}$ as a single fraction correctly.
[3 marks]
Answer

D2
Each part of this question uses the answer to the part before it.
(a)
Factorise $6 x^{2} + x - 15$ .
[2 marks]
Answer
(b)
Hence simplify fully $\frac{6 x ^{2} + x - 15}{4 x ^{2} - 9}$ .
[2 marks]
Answer
(c)
Hence solve $\frac{6 x ^{2} + x - 15}{4 x ^{2} - 9} = 4$ .
[2 marks]
Answer
Total for D2: 6 marks
D3
This question is about the expression $\frac{1}{x - 1} - \frac{1}{x + 1}$ .
(a)
Show that $\frac{1}{x - 1} - \frac{1}{x + 1} = \frac{2}{x ^{2} - 1}$ .
[2 marks]
(b)
Hence work out $\frac{1}{99} - \frac{1}{101}$ . Give your answer as a fraction.
[2 marks]
Answer
Total for D3: 4 marks
D4
Solve $\frac{6}{x + 1} - \frac{1}{x - 1} = 1$ . You must show your working.
[5 marks]
Answer
D5
A cyclist rides $30$ km at $x$ km/h, then a further $20$ km at $(x + 5)$ km/h.
(a)
Show that the total time, in hours, is $\frac{50 x + 150}{x ( x + 5 )}$ .
[2 marks]
(b)
The whole ride takes $3$ hours. Show that $3 x^{2} - 35 x - 150 = 0$ .
[2 marks]
(c)
Work out the speed for the first $30$ km.
[2 marks]
Answer $\dots\dots\dots$ km/h
Total for D5: 6 marks

Extension

Stretch

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

E1
Work out the exact value of $\frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \dots + \frac{1}{49 \times 51}$ .
[5 marks]
Hint 1 · What to try
Each term is $\frac{1}{( 2 n - 1 ) ( 2 n + 1 )}$ . Try writing it as a difference of two simpler fractions, the way $\frac{1}{x - 1} - \frac{1}{x + 1}$ worked in D3.
Hint 2 · The first line
$\frac{1}{2 n - 1} - \frac{1}{2 n + 1} = \frac{2}{( 2 n - 1 ) ( 2 n + 1 )}$ , so each term is $\frac{1}{2} (\frac{1}{2 n - 1} - \frac{1}{2 n + 1})$ .
Hint 3 · The full method
Add them up: $\frac{1}{2} (\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \dots + \frac{1}{49} - \frac{1}{51})$ . Everything cancels except $\frac{1}{1}$ and $- \frac{1}{51}$ , so the sum is $\frac{1}{2} \times \frac{50}{51} = \frac{25}{51}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify an algebraic fraction by factorising and cancelling.
I can multiply and divide algebraic fractions.
I can add and subtract algebraic fractions with a common denominator.
I can solve equations that have fractions in them.

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