AQA Level 2 Further Maths · Level 2 Certificate · Chapter 1: Number and algebra I · Lesson 1

Ratio, percentages and simplifying expressions

About 80 minutesNo calculator77 marks

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Date

Do Now

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

A1
Needed today
Share £360 in the ratio $2 : 3 : 7$ . Work out the largest share.
[2 marks]
Answer
A2
Needed today
Expand and simplify $5 (2 x - 3) - 2 (x - 4)$ .
[2 marks]
Answer
A3
GCSE Higher
Increase £80 by 15%. Then decrease the new amount by 15%. Work out the final amount.
[2 marks]
Answer
A4
GCSE Higher
Simplify $\frac{12 a ^{5} b ^{2}}{4 a ^{2} b}$ .
[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

A tray holds apples, bananas and oranges. apples : bananas $= 2 : 3$ and bananas : oranges $= 5 : 4$ . Work out the smallest possible number of pieces of fruit on the tray.

Hint Make the bananas the same number of parts in both ratios: the LCM of

3

and

5

15

B1
$x : y = 5 : 6$ and $y : z = 4 : 9$ . Work out $x : y : z$ in its simplest form. Hence write down $x : z$ .
[3 marks]
Answer $x : y : z = \dots : \dots : \dots x : z = \dots : \dots$

Example 2

$a : b = 4 : 3$ . Write $a$ in terms of $b$ . Then work out $(a + 2 b) : (3 a - b)$ in its simplest form.

Hint Write

a = 4 k

and

b = 3 k

and put them into each bracket.

B2
$m : n = 2 : 5$ . Write $n$ in terms of $m$ . Then work out $(3 m + n) : (n - m)$ in its simplest form.
[3 marks]
Answer $n = \dots\dots \dots : \dots$

Example 3

In a sale every price is cut by 35%. A lamp costs £52 in the sale. Work out the price of the lamp before the sale.

Hint The sale price is

65%

of the old price. Find

1%

first.

B3
A train fare rises by $12.5%$ to £63. Work out the fare before the rise.
[2 marks]
Answer

Example 4

Simplify $3 (4 p - 5 q) - 2 (p - 3 q) + q$ . Then factorise your answer.

Hint The

- 2

multiplies both terms in its bracket, so

- 2 \times - 3 q = + 6 q

B4
Simplify $4 x (x + 3 y) - 2 y (5 x - 4 y) - 4 x^{2}$ . Then factorise your answer fully.
[3 marks]
Answer $\dots\dots\dots\dots$

Example 5

Simplify $\frac{8 a ^{3} b}{15 c ^{2}} \div \frac{4 a b ^{3}}{9 c}$ .

Hint Turn the second fraction upside down and multiply. Then cancel numbers and letters.

B5
Simplify fully $\frac{14 x ^{2} y ^{3}}{9 z} \div \frac{21 x y}{12 z ^{3}}$ .
[3 marks]
Answer $\dots\dots\dots\dots$

Practice

Level 2

C1
$A B C D$ is a straight line. $A B = 6$ cm, $A C = 15$ cm and $A D = 27$ cm.
Not drawn accurately
(a)
Write the ratio $4.5 km : 1350 m$ in its simplest form.
[2 marks]
Answer $\dots : \dots$
(b)
Work out $B C : C D$ in its simplest form.
[2 marks]
Answer $\dots : \dots$
Total for C1: 4 marks
C2
Work out
(a)
$37 \frac{1}{2} %$ of £96
[1 mark]
Answer
(b)
£850 decreased by $14%$
[2 marks]
Answer
Total for C2: 3 marks

C3
Work out each of these. Give your answers as fractions in their simplest form.
(a)
$\frac{2}{3} + \frac{5}{6} \div \frac{5}{4}$
[2 marks]
Answer
(b)
$3 \frac{3}{4} - 2 \frac{2}{5}$
[2 marks]
Answer
Total for C3: 4 marks
C4
Work out each of these without a calculator.
(a)
$7.2 - 0.46 \times 3.5$ , giving your answer to 2 significant figures.
[2 marks]
Answer
(b)
$0.24 \times 0.15 + 0.07$ , giving your answer to 1 significant figure.
[2 marks]
Answer
Total for C4: 4 marks
C5
Simplify
(a)
$4 a^{3} b^{2} \times 3 a b^{4}$
[1 mark]
Answer
(b)
$\frac{18 p ^{4} q ^{2} r}{24 p q ^{5}}$
[2 marks]
Answer
Total for C5: 3 marks
C6
Factorise or combine as asked.
(a)
Factorise fully $14 m^{3} n^{2} - 21 m^{2} n^{3} + 7 mn$ .
[2 marks]
Answer
(b)
Write $\frac{3 x}{4 t} - \frac{y}{6 t} + \frac{5 z}{3 t}$ as a single fraction.
[2 marks]
Answer
Total for C6: 4 marks

Exam-style questions

AQA exam style

D1
Priya writes $4 (2 a - 3 b) - 3 (a - 2 b) = 5 a - 18 b$ .
Explain the mistake Priya has made. Simplify $4 (2 a - 3 b) - 3 (a - 2 b)$ correctly.
[3 marks]
Answer $\dots\dots\dots\dots$

D2
$x : y = 7 : 3$
(a)
Circle the expression for $x$ in terms of $y$ . $\frac{3 y}{7} \frac{7 y}{3} 7 y - 3 \frac{7}{3 y}$
[1 mark]
(b)
Show that $(x + 2 y) : (3 x - 5 y) = 13 : 6$
[2 marks]
(c)
$y : z = 4 : 5$ . Work out $x : z$ in its simplest form.
[2 marks]
Answer $x : z = \dots : \dots$
Total for D2: 5 marks
D3
The ratio of Ali's age to Ben's age is $7 : 4$ . In $9$ years' time the ratio of their ages will be $10 : 7$ . Work out their ages now. Do not use trial and improvement. You must show your working.
[4 marks]
AnswerAli $\dots\dots$ Ben $\dots\dots$
D4
The angles of this shape are all $9 0^{\circ}$ or $27 0^{\circ}$ . Four of its side lengths, in cm, are given in terms of $x$ .
Not drawn accurately
(a)
Work out the perimeter of the shape in terms of $x$ . Give your answer in its simplest form.
[2 marks]
Answer $\dots\dots\dots$ cm
(b)
Work out the area of the shape in terms of $x$ . Give your answer in its simplest form.
[3 marks]
Answer $\dots\dots\dots$ cm²
Total for D4: 5 marks
D5
The price of a bike rises by $2%$ to £510.
(a)
Work out the price of the bike before the rise.
[2 marks]
Answer
(b)
Later, the price of £510 is reduced by $20%$ in a sale. Work out the overall percentage change from the price before the rise. State whether it is an increase or a decrease.
[3 marks]
Answer $\dots\dots %$ increase / decrease
Total for D5: 5 marks
D6
A rectangle of card is $(4 x + 6)$ cm long and $(2 x + 4)$ cm wide. A square of side $x$ cm is cut from each corner. The card that is left is the net of an open box.
Not drawn accurately
(a)
Work out the perimeter of the card that is left, in terms of $x$ .
[2 marks]
Answer $\dots\dots\dots$ cm
(b)
Work out the area of the card that is left, in terms of $x$ . Give your answer in its simplest form.
[2 marks]
Answer $\dots\dots\dots$ cm²
(c)
Show that the volume of the box is $8 x (x + 3)$ cm³.
[2 marks]
Total for D6: 6 marks

Extension

Stretch

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

E1
$x : y = 3 : 2$ and $x y - y = 20$ . Work out all the possible pairs of values of $x$ and $y$ .
[5 marks]
Hint 1 · What to try
Write $x$ and $y$ as multiples of one letter, so the second equation has one unknown.
Hint 2 · The first line
$x = 3 k$ and $y = 2 k$ give $6 k^{2} - 2 k = 20$ , so $3 k^{2} - k - 10 = 0$ .
Hint 3 · The full method
$(3 k + 5) (k - 2) = 0$ , so $k = 2$ or $k = - \frac{5}{3}$ . Then $x = 3 k$ and $y = 2 k$ .
Answer $x = \dots\dots, y = \dots\dots$ or $x = \dots\dots, y = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can combine two ratios and use a ratio given in letters.
I can work out a percentage change, and the amount before a change.
I can expand, simplify and factorise expressions, including with negative brackets.
I can multiply, divide and add algebraic fractions.

Answers and mark scheme

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