YEAR 9

MATHS

Name: ______________________________

1
Contents
Number:

Laws of indices

Algebra:
nth term
Expanding and factorising (single brackets)
Solving linear equations
Solving linear inequalities
Substituting

Shape, Space and Measure:

Angles in parallel lines
Area and circumference of circles
Area problems

Data Handling:
Averages problems

Ratio and Proportion:

Dividing into a ratio

Fractions, decimals and percentages

2
 Laws of Indices
Things to remember:
 The exam question will use the word “simplify”
 When multiplying, add the indices
 When dividing, subtract the indices
 With brackets, multiply the indices
 If the exam question has the words “work out the value of”, or “evaluate” it means the
answer is a number.
 Anything to the power zero is 1
 Anything to the power one is itself
 Anything to a negative power becomes a reciprocal

Questions:
1. (a) Write down the reciprocal of 5
...........................................................
(1)
(b) Evaluate 3−2
...........................................................
(1)
(Total for Question is 2 marks)

2. (a) Write down the value of

...........................................................
(1)
(b) Work out the value of 52 + 2 3
...........................................................
(2)
(Total for Question is 3 marks)

3. Write these numbers in order of size. Start with the smallest number.

5-1 0.5 -5 50

..............................................................................................................................................
(Total for Question is 2 marks)

4. (a) Solve 3x2 = 147

...........................................................
(2)
(b) Work out the value of 2–3
...........................................................
(1)
(c) Simplify (3x2)3
...........................................................
(2)
(Total for question = 5 marks)
3
5. (a) Simplify a4 × a5
...........................................................
(1)
45𝑒 6 𝑓8
(b) Simplify
5𝑒𝑓2

...........................................................
(2)
(c) Write down the value of 9½
...........................................................
(1)
(Total for Question is 4 marks)

6. (a) Simplify 54 × 56
...........................................................
(1)
(b) Simplify 75 ÷ 72
...........................................................
(1)
(Total for Question is 2 marks)

7. Write down the value of

(i) 7°
...........................................................
(ii) 5−1
...........................................................
(iii) 9½
...........................................................
(Total for Question is 3 marks)

8. (a) Work out 34

...........................................................
(1)
(b) Write down the cube root of 64
...........................................................
(1)
(Total for Question is 2 marks)

4
nth term
Things to remember:
 The gap between the numbers is the number that goes in front of n e.g. 4n
 Then add on the zero term.
 If you’re asked to write down terms of a sequence – use n=1, n=2, n=3 etc.

Questions:
1. Here are some patterns made from sticks.

In the space below, draw Pattern number 4

(1)
(b) Complete the table.

(1)
(c) How many sticks make Pattern number 15?

…........................................................
(1)
(Total for Question is 3 marks)

2. Here are the first four terms of a number sequence.

6 10 14 18
(a) Write down the next term in this sequence.

…........................................................
(1)
(b) Find the 10th term in this sequence.

…........................................................
(1)
(c) The number 101 is not a term in this sequence. Explain why.

…..................................................................................................................................

…..................................................................................................................................
(1)
(Total for Question is 3 marks)

5
3. Here are the first four terms of a number sequence.
3 7 11 15
(a) Write down the next term of this sequence.
…........................................................
(1)
The 50th term of this number sequence is 199
(b) Write down the 51st term of this sequence.

…........................................................
(1)
The number 372 is not a term of this sequence.
(c) Explain why.

…..................................................................................................................................

…..................................................................................................................................
(1)
(Total for Question is 3 marks)
4. Here are the first 5 terms of an arithmetic sequence.
6, 11, 16, 21, 26
Find an expression, in terms of n, for the nth term of the sequence.

...........................................................
(Total 2 marks)

5. Here are the first five terms of a number sequence.

3 7 11 15 19
(a) Work out the 8th term of the number sequence.

...........................................................
(1)
(b) Write down an expression, in terms of n, for the nth term of the number sequence.

...........................................................
(2)
(Total 3 marks)

6. The first five terms of an arithmetic sequence are

2 9 16 23 30
Find, in terms of n, an expression for the nth term of this sequence.

...........................................................
(Total 2 marks)

6
7. Here are the first four terms of a number sequence.
2 7 12 17
(a) Write down the 6th term of this number sequence.

...........................................................
(1)
The nth term of a different number sequence is 4n + 5
(b) Work out the first three terms of this number sequence.

............ ............ ............

(2)
(Total 3 marks)
8. The nth term of a number sequence is given by 3n + 1
(a) Work out the first two terms of the number sequence.

...........................................................
(1)
Here are the first four terms of another number sequence.
1 5 9 13
(b) Find, in terms of n, an expression for the nth term of this number sequence.

...........................................................
(2)
(Total 3 marks

7
Expanding and Factorising (Single Brackets)
Things to remember:
 Expand brackets means to multiply what is outside the bracket with everything inside the
bracket.
 Factorising is the opposite of expanding – put the HCF outside the brackets to factorise
fully.

Questions:
1. (a) Expand 5(m + 2)

...........................................................
(1)
(b) Factorise y2 + 3y

...........................................................
(1)
(c) Simplify a5 × a4

...........................................................
(1)
(Total for Question is 3 marks)

2. (a) Expand 2m(m + 3)

...........................................................
(1)
(b) Factorise fully 3xy2 − 6xy

...........................................................
(2)
(Total for Question is 3 marks)

3. (a) Expand 3(x + 4)

...........................................................
(1)
(b) Expand x(x2 + 2)

...........................................................
(2)
(c) Factorise x2 − 6x

...........................................................
(1)
(Total for Question is 4 marks)

8
4. (a) Expand and simplify 5(x + 7) + 3(x – 2)

...........................................................
(2)
(b) Factorise completely 3a2b + 6ab2

...........................................................
(2)
(Total for Question is 4 marks)

5. (a) Expand 3(2y – 5)

...........................................................
(1)
(b) Factorise completely 8x2 + 4xy

...........................................................
(2)
(Total for Question is 3 marks)

6. (a) Factorise 3x + 6

...........................................................
(1)

(b) Expand and simplify 5(y − 2) + 2(y − 3)

...........................................................
(2)
(Total for Question is 3 marks)

7. (a) Factorise 4x + 10y

...........................................................
(1)
(b) Factorise x2 + 7x

...........................................................
(1)
(Total for Question is 2 marks)

9
Solving Equations
Things to remember:
 “Solve” means to find the value of the variable (what number the letter represents).
 The inverse of + is – and the inverse of x is ÷
 Work one step at a time, keeping you = signs in line on each new row of working.

Questions:
1. Solve 4x + 3 = 19

x =………………………
(Total 2 marks)

2. (a) Solve 6x – 7 = 38

x = .................................
(2)
(b) Solve 4(5y – 2) = 40

y = .................................
(3)
(Total 5 marks)

3. Solve 5(2y + 3) = 20

y = .................................
(Total 3 marks)

10
4. (a) Solve 7x + 18 = 74

x = ………………………
(2)
(b) Solve 4(2y – 5) = 32

y = ………………………
(2)
(c) Solve 5p + 7 = 3(4 – p)

p = ………………………
(3)
(Total 7 marks)

5. (a) Solve 7p + 2 = 5p + 8

p = ............................
(2)
(b) Solve 7r + 2 = 5(r – 4)

r = ...........................
(2)
(Total 4 marks)

11
6. Solve
4y + 1 = 2y + 8

y =………………………
(Total 2 marks)

7. Solve 4y + 3 = 2y + 8

y = ..............................
(Total 2 marks)

12
Inequalities
Things to remember:
 < means less than
 > means greater than
 ≤ means less than or equal to
 ≥ means greater than or equal to
 An integer is a whole number
 On a number line, use a full circle to show a value can be equal, and an empty circle to
show it cannot.

Questions:
1. (i) Solve the inequality
5x – 7 < 2x – 1

...........................................................
(ii) On the number line, represent the solution set to part (i).

–5 –4 –3 –2 –1 0 1 2 3 4 5

(Total 3 marks)
2. (a) List all the possible integer values of n such that
–2 ≤ n < 3
...........................................................
(2)
(b) Solve the inequality
4p – 8 < 7 – p

(2)
(Total 4 marks)

3. (a) –3 ≤ n < 2
n is an integer.
Write down all the possible values of n.
...........................................................
(2)
(b) Solve the inequality
5x < 2x – 6

...........................................................
(2)
(Total 4 marks)
13
4. (a) Solve the inequality
3t + 1 < t + 12

...........................................................
(2)
(b) t is a whole number.
Write down the largest value of t that satisfies
3t + 1 < t + 12
...........................................................
(1)
(Total 3 marks)

5. Solve 4 < x – 2 ≤ 7

...........................................................
(Total 3 marks)

6. Solve 5x + 3 > 19

...........................................................
(Total 2 marks)

14
Substitution
Things to remember:
 There is always 1 mark just for writing down the numbers you have had to put into the
expression.
 Your answer must be a number – don’t forget to finish the sum
 The question will always use the words “Work out the value of”

Questions:
1. (a) Work out the value of 3x – 4y when x = 3 and y = 2

...........................................................
(2)
p(q – 3)
(b) Work out the value of 4 when p = 2 and q = –7

...........................................................
(3)
(Total 5 marks)

2. Find the value of

t² – 4t when t = -3

...........................................................
(Total 2 marks)

3. P = x² - 7x
Work out the value of P when x = -5

P = ...........................................................
(Total 2 marks)

15
4. T, x and y are connected by the formula
T = 5x + 2y
x = -3 and y = 4
(a) Work out the value of T.

T = ...........................................................
(2)
T = 16 and x = 7
(b) Work out the value of y.

y = ...........................................................
(3)
(Total 5 marks)

5. P = 4k – 10
P = 50
(a) Work out the value of k.

...........................................................
(2)
y = 4n – 3d
n=2
d=5
(b) Work out the value of y.

...........................................................
(2)
(Total 4 marks)

16
6. h = 5t2 + 2
(i) Work out the value of h when t = –2

...........................................................
(ii) Work out a value of t when h = 47

...........................................................
(Total 3 marks)

17
Angle Rules
Things to remember:
 Angles in a triangle sum to 180°
 Angles on a straight line sum to 180°
 Angles around a point sum to 360°
 Vertically opposite angles are equal
 Alternate angles are equal
 Corresponding angles are equal
 Supplementary angles sum to 180°

Questions:
*1. ABC is parallel to EFGH.
GB = GF
Angle ABF = 65°

Work out the size of the angle marked x.

Give reasons for your answer.

(Total for Question is 4 marks)

18
*2. ABCD and EFG are parallel lines.
BC = CF
Angle BFE = 70°

Work out the size of the angle marked x.

Give reasons for each stage of your working.

(Total for question = 4 marks)

19
3. AFB and CHD are parallel lines.
EFD is a straight line.

Diagram NOT accurately drawn

Work out the size of the angle marked x.

x = ...........................................................
(Total for Question is 3 marks)

20
*4. ABC is a straight line.
DEFG is a straight line.
AC is parallel to DG.
EF = BF.
Angle BEF = 50°.

Work out the size of the angle marked x.

Give reasons for your answer.

...........................................................°
(Total for Question is 4 marks)

21
5.

(i) Find the size of the angle marked x.

........................................................... °

(ii) Give a reason for your answer.

.......................................................................................................................................

.......................................................................................................................................
(Total for Question is 2 marks)

22
6. ABC and DEF are parallel lines.
BEG is a straight line.
Angle GEF = 47°.

Diagram NOT accurately drawn

Work out the size of the angle marked x.
Give reasons for your answer.

........................................................... °
(Total for Question is 3 marks)

23
Circles
Things to remember:
 πr² sounds like area to me, when I need the circumference I’ll just use πD.
 Read the question carefully and check if you are being asked to find circumference or area
and whether they have given you the radius or the diameter.
 Remember the diameter is twice the radius.

Questions:
1. The diameter of a wheel on Harry’s bicycle is 0.65 m.
Calculate the circumference of the wheel.
Give your answer correct to 2 decimal places.
Diagram NOT accurately drawn

................................. m
(Total 2 marks)

2. Diagram NOT accurately drawn

The radius of this circle is 8 cm.
Work out the circumference of the circle.
Give your answer correct to 2 decimal places. 8 cm

.............................. cm
(Total 2 marks)

3. Diagram NOT accurately drawn

The radius of the circle is 9.7 cm. 9.7 cm
Work out the area of the circle.
Give your answer to 3 significant figures.

…………………………… cm²
(Total 2 marks)

4. A circle has a radius of 6.1 cm.

Work out the area of the circle.

6.1 cm

..........................................
(Total 3 marks)

24
5. The top of a table is a circle.
The radius of the top of the table is 50 cm.
(a) Work out the area of the top of the table.

………………………cm²
(2)
The base of the table is a circle.
The diameter of the base of the table is 40 cm.
(b) Work out the circumference of the base of the table.

………………………cm
(2)
(Total 4 marks)
6. The diagram shows two small circles inside a large circle.
The large circle has a radius of 8 cm.
Each of the two small circles has a diameter of 4 cm.
(a) Write down the radius of each of the small circles. 4 cm

8 cm
4 cm

............................. cm
(1)
(b) Work out the area of the region shown shaded in
the diagram.
Give your answer correct to one decimal place.

...................................... cm²
(4)
(Total 5 marks)

25
Area Problems
Things to remember:
 Area of a rectangle = base x height
 Area of a triangle = ½ x base x height
 Area of a parallelogram = base x height
 Area of a trapezium = ½ (a + b) h, where a and b are the parallel sides and h is the height
 The perimeter is the distance around the edge of the shape

Questions:
*1. The diagram shows the floor plan of Mary's conservatory.

Mary is going to cover the floor with tiles.

The tiles are sold in packs.
One pack of tiles will cover 2m2
A pack of tiles normally costs £24.80
Mary gets a discount of 25% off the cost of the tiles.
Mary has £100
Does Mary have enough money to buy all the tiles she needs?
You must show all your working.

(Total for question = 5 marks)

26
*2. Mr Weaver's garden is in the shape of a rectangle.
In the garden there is a patio in the shape of a rectangle and two ponds in the shape of
circles with diameter 3.8 m.
The rest of the garden is grass.

Diagram NOT accurately drawn

Mr Weaver is going to spread fertiliser over all the grass.

One box of fertiliser will cover 25 m2 of grass.
How many boxes of fertiliser does Mr Weaver need?
You must show your working.

(Total for Question is 5 marks)

27
*3. The diagram shows the plan of Mrs Phillips' living room.

Mrs Phillips is going to cover the floor with floor boards.

One pack of floor boards will cover 2.5 m2.
How many packs of floor boards does she need?
You must show your working.

(Total for Question is 4 marks)

28
4. A piece of card is in the shape of a trapezium.

Diagram NOT accurately drawn

A hole is cut in the card.

The hole is in the shape of a trapezium.
Work out the area of the shaded region.

. . . . . . . . . . . . . . . . . . . . . . cm2
(Total for Question is 3 marks)

29
5. Mrs Kunal's garden is in the shape of a rectangle.
Part of the garden is a patio in the shape of a triangle.
The rest of the garden is grass.

Mrs Kunal wants to spread fertiliser over all her grass.

One box of fertiliser is enough for 32 m2 of grass.
How many boxes of fertiliser will she need?
You must show your working.

………………………………………………
(Total for Question is 4 marks)

30
*6. The diagram shows a flower bed in the shape of a circle.

The flower bed has a diameter of 2.4 m.

Sue is going to put a plastic strip around the edge of the flower bed.
The plastic strip is sold in 2 metre rolls.
How many rolls of plastic strip does Sue need to buy?
You must show all your working.

(Total for Question is 4 marks)

31
Averages
Things to remember:
 Mode is most – the number that occurs the most frequently.
 Median is middle – put the numbers in order then identify the middle number.
 Mean is mean to work out – add all the numbers together and divide by the quantity in the
list.
 Range is the difference from the biggest to the smallest.

Questions:
1. Mrs Smith asked each student in her class to record the numbers of times they used their
mobile phone last Saturday.
Here are the results for the boys.
Boys 8 10 8 9 7 9 8 13 14
(a) Work out the median.

...........................................................
(2)
Here are the results for the girls.
Girls 6 8 9 9 10 14 14
*(b) Compare the numbers of times the boys used their mobile phones with the numbers
of times the girls used their mobile phones.

(4)
(Total for question = 6 marks)

2. There are 18 packets of sweets and 12 boxes of sweets in a carton.

The mean number of sweets in all the 30 packets and boxes is 14
The mean number of sweets in the 18 packets is 10
Work out the mean number of sweets in the boxes.

...........................................................
(Total for question = 3 marks)
32
3. 25 students in class A did a science exam.
30 students in class B did the same science exam.
The mean mark for the 25 students in class A is 67.8
The mean mark for all the 55 students is 72.0
Work out the mean mark for the students in class B.

...........................................................
(Total for Question is 3 marks)

4. There are 10 boys and 20 girls in Mrs Brook's class.

Mrs Brook gave all the class a test.
The mean mark for all the class is 60
The mean mark for the girls is 56
Work out the mean mark for the boys.

...........................................................
(Total for Question is 3 marks)

5. Here are four number cards.

One of the cards is turned over so you cannot see the number on it.

The mean of the four numbers is 6

Work out the number you cannot see.

...........................................................
(Total for Question 10 is 3 marks)

33
*6. There are two trays of plants in a greenhouse.
The first tray of plants was given fertiliser.
The second tray of plants was not given fertiliser.
On Monday the heights of the plants were measured in centimetres.
The boxes show some information about the heights of the plants.

Compare the distribution of the heights of the plants given fertiliser to the distribution of the
heights of the plants not given fertiliser.

(Total for Question is 4 marks)

7. 23 girls have a mean height of 153 cm.

17 boys have a mean height of 165 cm.
Work out the mean height of all 40 children.

........................................................... cm
(Total for Question is 3 marks)

34
8. Hertford Juniors is a basketball team.
At the end of 10 games, their mean score is 35 points per game.
At the end of 11 games, their mean score has gone down to 33 points per game.
How many points did the team score in the 11th game?

...........................................................
(Total for Question is 3 marks)

9. Mr Brown gives his class a test.

The 10 girls in the class get a mean mark of 70%
The 15 boys in the class get a mean mark of 80%

Nick says that because the mean of 70 and 80 is 75 then the mean mark for the whole
class in the test is 75%
Nick is not correct.

Is the correct mean mark less than or greater than 75%?

You must justify your answer.

.............................................................................................................................................

.............................................................................................................................................

.............................................................................................................................................
(Total for question = 2 marks)

10. Walkden Reds is a basketball team.

At the end of 11 games, their mean score was 33 points per game.
At the end of 10 games, their mean score was 2 points higher.

Jordan says,
"Walkden Reds must have scored 13 points in their 11th game."

Is Jordan right?
You must show how you get your answer.

...........................................................
(Total for question is 3 mark

35
Dividing into a Ratio
Things to remember:
 Start by dividing the quantity by the total number of parts, then multiply by each share.
 Don’t forget to include units throughout your working.

Questions:
1. Keith and Graham share £105 in the ratio 4:3
Work out how much Keith gets.

...........................................................
(Total for Question is 2 marks)

*2. Talil is going to make some concrete mix.

He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight.
Talil wants to make 180 kg of concrete mix.
Talil has
15 kg of cement
85 kg of sand
100 kg of gravel
Does Talil have enough cement, sand and gravel to make the concrete mix?

(Total for Question is 4 marks)

3. Liam, Sarah and Emily shared some money in the ratio 2 : 3 : 7

Emily got £80 more than Liam.
How much money did Sarah get?

...........................................................
(Total for question = 3 marks)

36
4. A pile of sand has a weight of 60 kg.
The sand is put into a small bag, a medium bag and a large bag in the ratio 2 : 3 : 7
Work out the weight of sand in each bag.

small bag ........................................................... kg

medium bag ........................................................... kg

large bag ........................................................... kg

(Total for Question is 3 marks)

5. A shop sells freezers and cookers.

The ratio of the number of freezers sold to the number of cookers sold is 5 : 2
The shop sells a total of 140 freezers and cookers in one week.
*(a) Work out the number of freezers and the number of cookers sold that week.

(3)
Jake buys this freezer in a sale.
The price of the freezer is reduced by 20%.
(b) Work out how much Jake saves.

£ ...........................................................
(2)
(Total for Question is 5 marks)
6. Graham and Michael share £35 in the ratio 5 : 2
Work out the amount of money that Graham gets.

£...........................................................
(Total for Question is 2 marks)

37
7. 5 schools sent some students to a conference.
One of the schools sent both boys and girls.
This school sent 16 boys.
The ratio of the number of boys it sent to the number of girls it sent was 1 : 2
The other 4 schools sent only girls.
Each of the 5 schools sent the same number of students.
Work out the total number of students sent to the conference by these 5 schools.

...........................................................
(Total for Question is 4 marks)

38
Fractions, Decimals and Percentages
Things to remember:

Questions:
1. (a) Write 0.1 as a fraction.
...........................................................
(1)
(b) Write ¼ a decimal.
...........................................................
(1)
(Total for Question is 2 marks)

2. (a) Write as a decimal.

...........................................................
(1)
(b) Write 0.3 as a fraction.
...........................................................
(1)
(Total for Question is 2 marks)

3. (a) Write as a decimal.

...........................................................
(1)
(b) Write 0.15 as a fraction.
...........................................................
(1)
(c) Write 17 out of 40 as a fraction.
...........................................................
(1)
(Total for question = 3 marks)

39
4. (a) Write 7⁄10 as a decimal.
...........................................................

(1)
(b) Write 0.45 as a percentage.
...........................................................

(1)
(c) Write 30% as a fraction.
Give your fraction in its simplest form.
...........................................................
(2)
(Total for Question is 4 marks)

5. (a) Write 0.7 as a fraction.

...........................................................
(1)
(b) Write 0.3 as a percentage.
...........................................................
(1)
(c) Write 8⁄12 in its simplest form.

...........................................................
(1)
(Total for Question is 3 marks)

6. Write these numbers in order of size. Start with the smallest number.

…………………………………………………………………………………………………..
(Total for question = 2 marks)

7. Write these numbers in order of size. Start with the smallest number.

…………………………………………………………………………………………………..
(Total for question = 2 marks)
40
8. Celina and Zoe both sing in a band.
One evening the band plays for 80 minutes.
Celina sings for 65% of the 80 minutes.

Zoe sings for of the 80 minutes.

Celina sings for more minutes than Zoe sings.
Work out for how many more minutes.
You must show all your working.

........................................................... minutes
(Total for question = 4 marks)

41