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

Linear equations, and algebra with number

About 75 minutesNo calculator69 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back the last lesson.

A1
Needed today
Solve $7 x - 4 = 3 x + 18$ .
[2 marks]
Answer
A2
Needed today
Write $\frac{a}{6} + \frac{b}{8}$ as a single fraction.
[2 marks]
Answer
A3
Last lesson
$x : y = 2 : 5$ and $y : z = 3 : 4$ . Work out $x : z$ in its simplest form.
[2 marks]
Answer
A4
Last lesson
A price rises by $8%$ to £54. Work out the price before the rise.
[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

Solve $5 (3 x - 4) = 2 (4 x + 1) - 3 x$ .

Hint Multiply out both sides first. Then collect the

x

terms on the side with more of them.

B1
Solve $3 (2 y + 5) - 4 (y - 3) = 7 (y + 1)$ .
[3 marks]
Answer $y = \dots\dots$

Example 2

Solve $\frac{x + 4}{3} - \frac{2 x - 1}{5} = 1$ .

Hint Multiply EVERY term by

15

, the LCM of

3

and

5

. Put each top in a bracket first.

B2
Solve $\frac{3 t - 2}{4} - \frac{t + 5}{6} = 1$ .
[3 marks]
Answer $t = \dots\dots$

Example 3

A rectangle is $(3 x + 1)$ cm long and $5$ cm wide. A square has sides of $(x + 6)$ cm. They have the same perimeter. Work out the area of the square.

Not drawn accurately

Hint Write each perimeter in terms of

x

and set them equal.

B3
The sum of four consecutive odd numbers is $136$ . The smallest of them is $n$ . Write an equation in $n$ and solve it. Write down the largest of the four numbers.
[3 marks]
Answerlargest $= \dots\dots$

Example 4

$35%$ of $a$ is equal to $14%$ of $b$ . Work out $a$ as a percentage of $b$ .

Hint Write the percentages as decimals, then make

a

the subject.

B4
$p$ increased by $20%$ is equal to $q$ decreased by $10%$ . Work out $p$ as a percentage of $q$ .
[3 marks]
Answer $\dots\dots %$

Example 5

$a$ is $60%$ of $b$ , and $b : c = 5 : 4$ . Show that $4 a = 3 c$ .

Hint Write

a

in terms of

b

and

b

in terms of

c

. Then put one into the other.

B5
$x$ is $125%$ of $y$ , and $y : z = 4 : 7$ . Show that $7 x = 5 z$ .
[3 marks]

Practice

Level 2

C1
Write an expression for each of these in its simplest form.
(a)
$c %$ of $d$
[1 mark]
Answer
(b)
$k$ decreased by $35%$
[1 mark]
Answer
(c)
$m$ increased by $2.5%$
[1 mark]
Answer
Total for C1: 3 marks
C2
Maya is five times as old as her niece Ella. In $18$ years' time Maya will be twice as old as Ella. Ella is $e$ years old now.
(a)
Write an expression for Maya's age in $18$ years' time.
[1 mark]
Answer
(b)
Write an equation in $e$ and solve it. How old is Maya now?
[2 marks]
AnswerMaya is $\dots\dots$
Total for C2: 3 marks

C3
A test has $30$ questions. A correct answer scores $3$ marks and $1$ mark is taken off for each wrong answer. A question not tried scores nothing. Jo tries $a$ questions and gets $c$ of them correct.
(a)
Write an expression for Jo's total mark. Simplify it.
[2 marks]
Answer
(b)
Jo tries $24$ questions and scores $52$ . Work out how many she gets correct.
[2 marks]
Answer
Total for C3: 4 marks
C4
$m : n = 5 : 8$ and $r$ is $40%$ of $n$ . Work out $m : r$ in its simplest form.
[2 marks]
Answer $m : r = \dots : \dots$
C5
$y$ is $30%$ greater than $x$ . $w$ is $30%$ less than $y$ . Work out $w : x$ in its simplest form.
[2 marks]
Answer $w : x = \dots : \dots$
C6
In a triangle, the largest angle is four times the smallest angle, $s$ . The third angle is $3 0^{\circ}$ more than the smallest. Write an equation in $s$ and work out the three angles.
[3 marks]
Not drawn accurately
Answer $\dots^{\circ}, \dots^{\circ}, \dots^{\circ}$

Exam-style questions

AQA exam style

D1
Tom solves $\frac{x + 5}{2} - \frac{x - 1}{3} = 4$ . He writes $3 (x + 5) - 2 (x - 1) = 4$ , so $x + 17 = 4$ and $x = - 13$ .
Explain Tom's mistake. Solve the equation correctly.
[3 marks]
Answer $x = \dots\dots$

D2
The ratio of boys to girls in a club is $5 : 4$ . Then $10$ boys leave and $8$ girls join. The ratio of boys to girls is now $3 : 4$ . Work out the number of boys and the number of girls in the club now. Do not use trial and improvement. You must show your working.
[4 marks]
Answerboys $\dots\dots$ girls $\dots\dots$
D3
$p$ is $20%$ greater than $q$ . $p$ is also $20%$ less than $r$ .
(a)
Show that $r = 1.5 q$
[2 marks]
(b)
Work out $q : p : r$ in its simplest form.
[2 marks]
Answer $\dots : \dots : \dots$
Total for D3: 4 marks
D4
The diagram shows a square and an isosceles triangle. All lengths are in centimetres. The square and the triangle have the same perimeter.
Not drawn accurately
(a)
Work out the value of $a$ .
[3 marks]
Answer $a = \dots\dots$
(b)
Work out the area of the square.
[2 marks]
Answer $\dots\dots$ cm²
Total for D4: 5 marks
D5
Here is an equation. $4 (3 x - 2) - 2 (x + 1) = 5 (2 x - 3)$
(a)
Show that the equation has no solution.
[2 marks]
(b)
The $3$ on the right side is changed to $k$ . Then $4 (3 x - 2) - 2 (x + 1) \equiv 5 (2 x - k)$ is true for every value of $x$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
Total for D5: 4 marks
D6
Ali, Bev and Cal share some money. Bev gets £10 more than twice what Ali gets. Cal gets $\frac{3}{4}$ of what Bev gets. Altogether they get £152.50. Work out how much Cal gets. You must show your working.
[4 marks]
Answer£ $\dots\dots$

Extension

Stretch

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

E1
$p$ is $m %$ greater than $q$ . $p$ is also $m %$ less than $r$ . Write $r : q$ in terms of $m$ . Then work out the value of $m$ for which $r : q = 3 : 2$ .
[5 marks]
Hint 1 · What to try
Write $p$ in two ways, once using $q$ and once using $r$ .
Hint 2 · The first line
$p = q (1 + \frac{m}{100})$ and $p = r (1 - \frac{m}{100})$ .
Hint 3 · The full method
So $\frac{r}{q} = \frac{100 + m}{100 - m}$ . Set this equal to $\frac{3}{2}$ : $200 + 2 m = 300 - 3 m$ , so $m = 20$ .
Answer $r : q = \dots\dots m = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve linear equations with brackets and with fractions.
I can form an equation from a problem and solve it.
I can write percentage changes in algebra and use them with ratios.
I can tell when an equation has no solution or is true for every value.

Answers and mark scheme

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