KS3 Maths · Key Stage 3 · Unit 4: Fractions, Percentages and Brackets

Fractions, decimals and percentages, brackets, and angles in polygons

Years 7–8About 85 minutesNo calculator84 marks

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NameClassDate

Before you start

Recall

You will need these in every question below.

A1
Write down the lowest common multiple of 6 and 8.
[1 mark]
Answer

A2
Simplify $5 x + 3 - 2 x + 4$ .
[1 mark]
Answer
A3
Work out $\frac{3}{4}$ of 36.
[1 mark]
Answer
A4
Two angles on a straight line add up to $18 0^{\circ}$ . One of them is $7 4^{\circ}$ . Work out the other.
[1 mark]
Answer
A5
Write 0.7 as a percentage, and as a fraction.
[2 marks]
Answer $\dots\dots %$ and $\dots\dots$

Examples, then your turn

Core

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

Example 1

Work out $\frac{3}{4} + \frac{2}{5}$ . Give your answer as a mixed number.

Hint Change both fractions to the same denominator. 20 is a multiple of 4 and of 5. Then add the tops only.

B1
Work out $\frac{5}{6} - \frac{3}{8}$ .
[2 marks]
Answer

Example 2

Each exterior angle of a regular polygon is $2 4^{\circ}$ . How many sides does the polygon have? Work out the size of each interior angle.

Hint The exterior angles of any polygon add up to

36 0^{\circ}

. An interior angle and its exterior angle add up to

18 0^{\circ}

B2
Each exterior angle of a regular polygon is $4 0^{\circ}$ . How many sides does the polygon have? Work out the size of each interior angle.
[3 marks]
Answer $\dots\dots$ sides interior angle $\dots \dots^{\circ}$

Example 3

Write $\frac{3}{8}$ as a decimal and as a percentage.

Hint Divide 3 by 8 using short division, adding zeros after the decimal point. To change a decimal to a percentage, multiply by 100.

B3
Write $\frac{7}{40}$ as a decimal and as a percentage.
[2 marks]
Answer $\dots\dots$ and $\dots\dots %$

Example 4

Expand and simplify $3 (2 x - 5) - 2 (x - 4)$ .

Hint Multiply every term inside each bracket by the number in front, with its sign.

- 2 \times - 4 = + 8

. Then collect the

x

terms and the numbers.

B4
Expand and simplify $5 (x + 3) - 3 (2 x - 1)$ .
[3 marks]
Answer

Practice

Core

Each question asks for something different. Show your working.

C1
Work out $\frac{7}{9} - \frac{1}{6}$ . Give your answer in its simplest form.
[2 marks]
Answer

C2
Work out $4 \frac{1}{6} - 2 \frac{3}{4}$ . Give your answer as a mixed number.
[3 marks]
Answer
C3
Write these numbers in order of size. Start with the smallest.
$\frac{5}{8} 0.6 63% \frac{2}{3}$
[2 marks]
Answer
C4
Answer both parts.
(a)
Write 0.45 as a fraction in its simplest form.
[1 mark]
Answer
(b)
Write $\frac{11}{12}$ as a percentage. Give your answer to 1 decimal place.
[2 marks]
Answer
Total for C4: 3 marks
C5
Answer both parts.
(a)
Expand $- 4 (3 x - 5)$ .
[1 mark]
Answer
(b)
Factorise $20 x - 28$ fully.
[2 marks]
Answer
Total for C5: 3 marks
C6
A rectangle is 6 cm wide and $(2 x + 5)$ cm long. Its area is 78 cm². Work out the value of $x$ .
[3 marks]
Answer $x = \dots\dots$
C7
The interior angles of a hexagon are $10 0^{\circ}$ , $11 5^{\circ}$ , $13 0^{\circ}$ , $12 5^{\circ}$ , $14 0^{\circ}$ and $x^{\circ}$ . Work out the value of $x$ .
[2 marks]
Answer $x = \dots\dots$
C8
The diagram shows two parallel lines cut by a straight line. Work out the size of angle $x$ . Give a reason for your answer.
[2 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$
C9
Four of the exterior angles of a pentagon are $6 2^{\circ}$ , $7 5^{\circ}$ , $8 0^{\circ}$ and $5 8^{\circ}$ . Work out the fifth exterior angle, and the interior angle at that corner.
[2 marks]
Answerexterior $\dots \dots^{\circ}$ interior $\dots \dots^{\circ}$

Test-style questions

Stretch

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Ella says, " $\frac{3}{4} + \frac{2}{7} = \frac{5}{11}$ ."
Explain why Ella is wrong, and work out the correct answer.
[3 marks]
Answer

D2
Two shops sell the same game. The normal price in both shops is £120.
Shop A takes $\frac{3}{8}$ off the price. Shop B takes $35%$ off the price.
Which shop sells the game for less, and by how much? You must show your working.
[4 marks]
AnswerShop $\dots$ by £ $\dots\dots$
D3
The diagram shows a pentagon. Its interior angles are $x^{\circ}$ , $2 x^{\circ}$ , $(x + 30)^{\circ}$ , $(2 x - 20)^{\circ}$ and $11 0^{\circ}$ .
Not drawn accurately
(a)
Show that $6 x + 120 = 540$ .
[2 marks]
(b)
Work out the value of $x$ .
[2 marks]
Answer $x = \dots\dots$
(c)
Work out the size of the largest exterior angle of the pentagon.
[2 marks]
Answer
Total for D3: 6 marks
D4
Show that $4 (3 x + 5) - 2 (3 x - 2)$ is a multiple of 6 for every whole number $x$ .
[3 marks]
D5
The diagram shows two parallel lines cut by a straight line. Work out the value of $x$ , and the size of each of the marked angles.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ angle $= \dots \dots^{\circ}$
D6
Jo spends $\frac{1}{3}$ of her pocket money on a game and $\frac{1}{4}$ of it on food. She saves the rest. She saves £10.
How much pocket money did Jo have?
[3 marks]
Answer
D7
There are 25 students in a class. $\frac{2}{5}$ of them walk to school. $0.28$ of them come by car. $12%$ of them cycle. The rest come by bus.
(a)
What percentage of the class comes by bus?
[2 marks]
Answer
(b)
How many students come by bus?
[1 mark]
Answer
Total for D7: 3 marks
D8
Show that a regular polygon cannot have interior angles of $14 5^{\circ}$ .
[2 marks]
D9
The perimeter of a rectangle is $(10 x + 16)$ cm. The length of the rectangle is $(3 x + 5)$ cm. Find an expression for its width. Give your answer in its simplest form.
[3 marks]
Answer
D10
A square, a regular hexagon and a third regular polygon fit together at a point with no gaps.
How many sides does the third polygon have?
[3 marks]
Answer

Challenge

UKMT level

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Ana, Ben and Cal share a bag of sweets. Ana takes $\frac{1}{3}$ of the sweets. Ben takes $\frac{1}{4}$ of the sweets that are left. Cal takes the other 18 sweets.
How many sweets were in the bag?
[3 marks]
Hint 1 · What to try
Work out what fraction of the whole bag each person takes.
Hint 2 · The first line
After Ana, $\frac{2}{3}$ of the bag is left. Ben takes $\frac{1}{4}$ of $\frac{2}{3}$ , which is $\frac{1}{6}$ of the bag.
Hint 3 · The full method
Cal takes $1 - \frac{1}{3} - \frac{1}{6} = \frac{1}{2}$ of the bag. Half the bag is 18 sweets, so the bag held 36 sweets.
Answer

E2
Two whole numbers add up to 96. $40%$ of the larger number is equal to $\frac{2}{3}$ of the smaller number. What is the larger number?
[3 marks]
Hint 1 · What to try
Write $40%$ as a fraction, then compare the two numbers.
Hint 2 · The first line
$\frac{2}{5}$ of the larger equals $\frac{2}{3}$ of the smaller, so the larger is $\frac{5}{3}$ of the smaller. The numbers are in the ratio $5 : 3$ .
Hint 3 · The full method
$5 + 3 = 8$ parts make 96, so one part is 12. The larger number is $5 \times 12 = 60$ . Check: $40%$ of 60 is 24, and $\frac{2}{3}$ of 36 is 24.
Answer
E3
$a$ and $b$ are numbers. $a (3 x + 2) - 2 (x - b) = 10 x + 22$ for every value of $x$ . Find $a$ and $b$ .
[3 marks]
Hint 1 · What to try
Expand the left-hand side. The number of $x$ s on each side must match, and so must the numbers on their own.
Hint 2 · The first line
$3 a x + 2 a - 2 x + 2 b$ . The $x$ terms give $3 a - 2 = 10$ .
Hint 3 · The full method
$3 a - 2 = 10$ gives $a = 4$ . The numbers give $2 a + 2 b = 22$ , so $8 + 2 b = 22$ and $b = 7$ .
Answer $a = \dots\dots$ $b = \dots\dots$
E4
Regular polygon $A$ has 3 more sides than regular polygon $B$ . Each exterior angle of $A$ is $1 0^{\circ}$ smaller than each exterior angle of $B$ .
How many sides does polygon $A$ have?
[4 marks]
Hint 1 · What to try
An exterior angle of a regular polygon with $n$ sides is $360 \div n$ .
Hint 2 · The first line
Try polygons $B$ with 3, 4, 5, … sides. For each one, work out the exterior angle of $B$ and of the polygon with 3 more sides.
Hint 3 · The full method
$B$ with 9 sides has exterior angles of $4 0^{\circ}$ . A polygon with 12 sides has exterior angles of $3 0^{\circ}$ . The difference is $1 0^{\circ}$ , so $A$ has 12 sides.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can add and subtract fractions and mixed numbers by finding a common denominator.
I can change between fractions, decimals and percentages, and put them in order of size.
I can expand a single bracket, including with a negative number in front, and factorise fully.
I can find angles in polygons and on parallel lines, and form an equation from them.

Answers and mark scheme

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