This PowerPoint helps students to consider the concept of infinity.
GCSE Linear Starters Higher
1. GCSE Linear Questions (H)
• Question set 1
• Question set 2
• Question set 3
• Question set 4
• Question set 5
• Question set 6
• Question set 7
• Question set 8
• Question set 9
• Question set 10
• Question set 11
• Question set 12
• Question set 13
• Question set 14
• Question set 15
• Question set 16
• Question set 17
• Question set 18
• Question set 19
• Question set 20
Odd numbers – Non
calculator
Even numbers -
Calculator
2. Number Algebra
Shape, Space, Measure Handling Data
Solve the following simultaneous
equations
Arrange these in ascending order
25
, 64
1
2, 2−1
, 80
, 16
1
4
5𝑥 + 2𝑦 = 16
3𝑥 − 𝑦 = 14
8 9 14 ?
The mean of the numbers of these 4
cards is 9. What is the number on the
fourth card?
4 + 3x
x + 6
The perimeter is equal to 32cm. What
is the value of x?
3. Number
Arrange these in ascending order
25, 64
1
2, 2−1, 80, 16
1
4
25 = 2 x 2 x 2 x 2 x 2 = 32
64
1
2 = 64 = 8
2−1 =
1
21
=
1
2
80 = 1
16
1
4 =
4
16 = 2
2−1
, 80
, 16
1
4, 64
1
2, 25
4. Algebra
Solve the following simultaneous
equations 5𝑥 + 2𝑦 = 16
3𝑥 − 𝑦 = 14
a
b
Multiply equation b by 2,
6𝑥 − 2𝑦 = 28 c
Add equations a and c
11𝑥 = 44
𝒙 = 𝟒
Substitute x back into one of the
original equations and solve
5. Shape, Space, Measure 4 + 3x
x + 6
The perimeter is equal to 32cm. What
is the value of x?
Perimeter = 32cm
Perimeter = 4 + 3𝑥 + 𝑥 + 6 + 4 + 3𝑥 + 𝑥 + 6
= 8𝑥 + 20
Therefore
8𝑥 + 20 = 32
8𝑥 = 12
𝑥 = 1.5
6. Handling Data
8 9 14 ?
The mean of the numbers of these 4
cards is 9. What is the number on the
fourth card?
Total of the four cards is the mean
multiplied by four
9 × 4 = 36
So find the missing card by subtraction
36 − 8 − 9 − 14 = 5
7. Number Algebra
Shape, Space, Measure Handling Data
Expand and Simplify the followingA bank account gains 6% compound
interest per annum. If Tom puts £700
into his account, how much could he
expect after 6 years?
4 𝑥 − 5 − 3(2𝑥 − 6)
(𝑥 + 2)(𝑥 − 7)
𝑡 + 5 2
Calculate the mean number of cars per
household
35°
Work out the missing length x
𝑥
15
No. of cars Frequency
0 4
1 8
2 7
3 2
8. Number
A bank account gains 6% compound
interest per annum. If Tom puts £700
into his account, how much could he
expect after 6 years?
To calculate the money after one year of interest, multiply by 1.06
𝑌𝑒𝑎𝑟 𝑜𝑛𝑒 − 700 × 1.06
𝑌𝑒𝑎𝑟 𝑡𝑤𝑜 − 700 × 1.06 × 1.06
𝑌𝑒𝑎𝑟 𝑡ℎ𝑟𝑒𝑒 − 700 × 1.06 × 1.06 × 1.06
𝑌𝑒𝑎𝑟 𝑠𝑖𝑥 − 700 × 1.06 × 1.06 × 1.06 × 1.06 × 1.06 × 1.06
10. Shape, Space, Measure
35°
Work out the missing length x
𝑥
15
35°
𝑥
15 Label your sides (in play)
h
a
Decide your triangle
h
a
c
Cover up what you are looking for and
write down your formula
ℎ =
𝑎
cos 𝜃
ℎ =
15
cos 35
ℎ = 18.31161883
ℎ = 18.3 (3. 𝑠. 𝑓)
11. Handling Data
Calculate the mean number of cars per
household
No. of cars Frequency
0 4
1 8
2 7
3 2
No. of cars Frequency Mean
0 4 4 x 0 = 0
1 8 8 x 1 =8
2 7 7 x 2 = 14
3 2 3 x 2 = 6
Total 21 28
𝑀𝑒𝑎𝑛 =
28
21
= 1. 3
12. Number Algebra
Shape, Space, Measure Handling Data
Write out the nth term for each
sequence. Hence work out what the
10th and 100th term will be.
Approximate the answer to
12.31 × 16.9
0.394 × 0.216
5, 8, 11, 14, …
10, 4, −2, −8, −14
4, 7, 12, 19, 28, …
6, 11, 18, 27, 38, …
Three cards are drawn from a deck and
replaced each time. What is the
probability of drawing 3 hearts?
Iron has a density of 8g/cm3 . What will
be the mass of the above cuboid?
4cm
2cm
5cm
13. Number
Approximate the answer to
12.31 × 16.9
0.394 × 0.216
Round all numbers to 1 significant figure
10 × 20
0.4 × 0.2
Calculate this sum
200
0.8
Use equivalent fractions to help divide by a decimal
200
0.8
=
2000
8
= 250
14. Algebra
Write out the nth term for each
sequence. Hence work out what the
10th and 100th term will be.
5, 8, 11, 14, …
10, 4, −2, −8, −14
4, 7, 12, 19, 28, …
6, 11, 18, 27, 38, …
Sequence nth term 10th term 100th term
5, 8, 11, 14, … 3𝑛 + 2 32 302
10, 4, −2, −8, −14, … 16 − 6𝑛 -44 -584
4, 7, 12, 19, 28, … 𝑛2 + 3 103 10003
6, 11, 18, 27, 38, … 𝑛2 + 2𝑛 + 3 123 10203
15. Shape, Space, Measure
Iron has a density of 8g/cm3 . What will
be the mass of the above cuboid?
4cm
2cm
5cm
𝐶𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 = 4𝑐𝑚 × 5𝑐𝑚
= 20𝑐𝑚2
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐶𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 × 𝑑𝑒𝑝𝑡ℎ
= 20 × 2
= 40𝑐𝑚3
The density is 8𝑔/𝑐𝑚3
this means, every 𝑐𝑚3
weighs 8g. So
the mass will be
𝑀𝑎𝑠𝑠 = 40 × 8
= 320𝑔
16. Handling Data
Three cards are drawn from a deck and
replaced each time. What is the
probability of drawing 3 hearts?
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝐻𝑒𝑎𝑟𝑡 𝐴𝑁𝐷 𝐻𝑒𝑎𝑟𝑡 𝐴𝑁𝐷 𝐻𝑒𝑎𝑟𝑡 =
1
4
×
1
4
×
1
4
𝑃 𝐻𝐻𝐻 =
1
64
17. Number Algebra
Shape, Space, Measure Handling Data
Solve the following quadratic equation,
leave your answers to 3 significant
figures.
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒
4.2 × 7.3
5.2 − 9.3
Write down your full calculator display
Round this number to 3 significant
figures
3𝑥2 − 5𝑥 = 18
A school of 800 pupils want to do a
survey on school dinners. They
decided to take a stratified sample of
30 pupils. How many of each year
group should they ask?
𝑥
Work out the size of angle x
18
13
Year 7 Year 8 Year 9 Year 10 Year 11
182 124 128 195 171
18. Number 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒
4.2 × 7.3
5.2 − 9.3
Write down your full calculator display
Round this number to 3 significant figures
For Casio fx-83GT plus
Type
4 . 2 x 7 .
3 5 . 2 -
9 . 3 = 𝑺 ≪=≫D
−7.478048
−7.48
19. Algebra Solve the following quadratic equation,
leave your answers to 3 significant
figures.
3𝑥2 − 5𝑥 = 18
Make it look like a usual quadratic
3𝑥2 − 5𝑥 = 18
3𝑥2
− 5𝑥 − 18 = 0
Difficult to factorise so use the formula.
𝐹𝑜𝑟 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 0
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑎 = 3
𝑏 = −5
𝑐 = −18
𝑥 =
− −5 ± −5 2 − 4 × 3 × −18
2 × 3
𝑥 =
5 ± 25 + 216
6
𝑥 =
5 + 241
6
𝑜𝑟 𝑥 =
5 − 241
6
𝑥 = 3.42 𝑜𝑟 𝑥 = −1.75
20. Shape, Space, Measure
𝑥
Work out the size of angle x
18
13
Label your sides (in play)
o
a
Decide your triangle
a
o
t
Cover up what you are looking for and
write down your formula
tan 𝑥 =
𝑜
𝑎
tan 𝑥 =
18
13
𝑥 = tan−1
18
13
𝑥 = 54.2°
𝑥
18
13
21. Handling Data
A school of 800 pupils want to do a
survey on school dinners. They
decided to take a stratified sample of
30 pupils. How many of each year
group should they ask?
Year 7 Year 8 Year 9 Year 10 Year 11
182 124 128 195 171
30 pupils out of 5 year groups –
must mean 6 from each year
group? Wrong.
We take a stratified sample – this
means we take a fair
representation of each year
group. There should be more year
7 than year 8 in the sample.
Year 7 =
182
800
× 30 = 6.825 = 7 𝑝𝑢𝑝𝑖𝑙𝑠
Year 8 =
124
800
× 30 = 4.65 = 5 𝑝𝑢𝑝𝑖𝑙𝑠
Year 9 =
128
800
× 30 = 4.8 = 5 𝑝𝑢𝑝𝑖𝑙𝑠
Year 10 =
195
800
× 30 = 7.3125 = 7 𝑝𝑢𝑝𝑖𝑙𝑠
Year 11 =
171
800
× 30 = 6.4125 = 6 𝑝𝑢𝑝𝑖𝑙𝑠
Check the student numbers add up
to your sample size.
7 + 5 + 5 + 7 + 6 = 30
Each year group gets a fair
representation.
22. Number Algebra
Shape, Space, Measure Handling Data
Factorise fullySimplify the following
48 =
3 × 12 =
2 5 × 3 10 =
300 − 75 =
5𝑥4 𝑦3 𝑧2 − 15𝑥6 𝑦5 𝑧
𝑥2
+ 11𝑥 − 12
𝑥2
− 64
9𝑥2 − 25𝑦2
4, 6, 2, 5, 7, 9, 9, 2, 1, 4, 8
Draw a box plot for the following data
Calculate the area of the rectangle
3 cm
5𝑥 − 3 cm
3𝑥 + 5 cm
23. Number Simplify the following
48 =
3 × 12 =
2 5 × 3 10 =
300 − 75 =
See Surds or the think SQUARE numbers
48 = 16 × 3
= 16 3
= 4 3
What is the largest square number
which is a factor of 48. 16
3 × 12 = 36
= 6
2 5 × 3 10 = 6 50
= 6 25 × 2
= 6 25 2
= 30 2
When adding fractions, the denominator needs to be the same number.
Similarly, when adding Surds, the Surds need to be the same number –
so simplify them
300 = 100 × 3
= 100 3
= 10 3
75 = 25 × 3
= 25 3
= 5 3
300 − 75 =
10 3 − 5 3 = 5 3
25. Shape, Space, Measure
Calculate the area of the rectangle
3 cm
5𝑥 − 3 cm
3𝑥 + 5 cm
Remember the features about a rectangle – two pairs of equal sides!
You can set up an equation using this information to work out the
length of the rectangle.
5𝑥 − 3 = 3𝑥 + 5
2𝑥 − 3 = 5
2𝑥 = 8
𝑥 = 4
Therefore the length is 3 x 4 + 5 = 17cm.
𝐴𝑟𝑒𝑎 = 17 × 3
= 51𝑐𝑚2
26. Handling Data
4, 6, 2, 5, 7, 9, 9, 2, 1, 4, 8
Draw a box plot for the following data
When working with Quartiles, Median and Range – it is always useful to
arrange your data in size order.
1, 2, 2, 4, 4, 5, 6, 7, 8, 9, 9
Need a few pieces of information for a box plot.
Highest value – 9
Lowest value – 1
The Median, since there are 11 numbers, the middle number will be the
11+1
2
= 6th number. So the 6th number is 5
Lower quartile will be the
11+1
4
= 3𝑟𝑑 number in our list. So LQ = 2
Upper quartile will be the
3 11+1
4
= 9𝑡ℎ number in our list. UQ = 8
27. Number Algebra
Shape, Space, Measure Handling Data
Using trial and improvement to find a
solution to 1 decimal place
The lengths of a room have been
calculated to the nearest metre.
Calculate the greatest and least area that
the room could be.
𝑥3 − 5𝑥 = 50
The probability of Man Utd winning a
match under David Moyes is 0.3 and
losing is 0.2.
Man Utd play 3 matches, what is the
probability that out of these, 2 are
won and one is drawn.
Calculate the volume and surface area.
Leave your answer to 3 significant
figures.
8 𝑐𝑚
5 𝑐𝑚
9
4
28. Number The lengths of a room have been
calculated to the nearest metre.
Calculate the greatest and least area that
the room could be.
9
4
Lower bound
3.5
8.5
𝐴𝑟𝑒𝑎 = 3.5 × 8.5
= 29.75 𝑚2 Upper bound
9.5
4.5
𝐴𝑟𝑒𝑎 = 4.5 × 9.5
= 42.75 𝑚2
29. Algebra Using trial and improvement to find a
solution to 1 decimal place
𝑥3 − 5𝑥 = 50
𝒙 𝒙 𝟑
− 𝟓𝒙 Comment
4 43
− 5 × 4 = 44 Low
5 53
− 5 × 5 = 100 High
4.3 4.33
− 5 × 4.3 = 58.007 High
4.2 4.23
− 5 × 4.2 = 53.088 High
4.1 4.13
− 5 × 4.1 = 48.421 Low
Draw a table – it helps!
4.153 − 5 × 4.15 = 50.723375
Since 4.15 is too high, everything about it must be too high as well.
Therefore the solution to 1 decimal place is 4.1
31. Handling Data
The probability of Man Utd winning a
match under David Moyes is 0.3 and
losing is 0.2.
Man Utd play 3 matches, what is the
probability that out of these, 2 are
won and one is drawn.
Probability of drawing a match is
1 − 0.3 − 0.2 = 0.5
List all the possible combinations
Win Win Draw
Win Draw Win
Draw Win Win
𝑃 𝑊 𝐴𝑁𝐷 𝑊 𝐴𝑁𝐷 𝐷 = 0.3 × 0.3 × 0.5 = 0.045
𝑃 𝑊 𝐴𝑁𝐷 𝐷 𝐴𝑁𝐷 𝑊 = 0.3 × 0.5 × 0.3 = 0.045
𝑃 𝐷 𝐴𝑁𝐷 𝑊 𝐴𝑁𝐷 𝑊 = 0.5 × 0.3 × 0.3 = 0.045
Therefore, the probability of winning two and drawing one
match is 0.045 + 0.045 + 0.045 = 0.135
32. Number Algebra
Shape, Space, Measure Handling Data
Solve the following
Leave all answers as mixed numbers
1
3
+
4
5
=
3
5
7
− 1
2
3
=
4
9
×
3
8
=
4𝑥 + 6 = 𝑥 − 12
7𝑥 − 8 = 20 − 3𝑥
5𝑥 + 4
6
=
3𝑥 + 8
5
Draw a cumulative frequency graph for the
following data. Construct a box plot from this
information.
Calculate the distance between the
coordinates (-1, 4) and (11, 9)
Marks Frequency
0 – 10 2
11 – 20 7
21 – 30 13
31 – 40 8
36. Handling Data
Draw a cumulative frequency graph for the
following data. Construct a box plot from this
information.
Marks Frequency
0 – 10 2
11 – 20 7
21 – 30 13
31 – 40 8Marks Frequency Cumulative
Frequency
0 – 10 2 2
11 – 20 7 9
21 – 30 13 22
31 – 40 8 30
Plot your graph using the end
points and cumulative
frequency.
37. Number Algebra
Shape, Space, Measure Handling Data
Rearrange the formula to make x the
subject
Tins of paint are on offer, buy 5 get 1
free. John the painter needs 27 tins of
paint. If a tin of paint costs £3.43, how
much will John have to pay?
𝑦 = 4𝑥 − 2
j𝑥 + 𝑡 = 𝑘𝑥 − 𝑚
Calculate an estimate for the mean
Table on time of goal scored
Calculate the area of the sector and
the arc length. Leave all answers to 1
decimal place.
132° 3 𝑐𝑚
Time of match Frequency
0 < 𝑥 ≤ 15 8
15 < 𝑥 ≤ 30 4
30 < 𝑥 ≤ 45 5
45 < 𝑥 ≤ 60 7
60 < 𝑥 ≤ 75 7
75 < 𝑥 ≤ 90 10
38. Number Tins of paint are on offer, buy 5 get 1 free. John
the painter needs 27 tins of paint. If a tin of
paint costs £3.43, how much will John have to
pay?
If John buys 5 tins he gets 6. So using this, if he buys 20
tins, he will actually get 24. Therefore he only needs to
purchase another 3 tins to have 27.
John needs to buy 23 tins.
23 × 3.43 = £78.89
40. Shape, Space, Measure
Calculate the area of the sector and
the arc length. Leave all answers to 1
decimal place.
132° 3 𝑐𝑚
3 𝑐𝑚
Calculate the area and circumference of
the full circle.
𝐴𝑟𝑒𝑎 = 𝜋 × 𝑟𝑎𝑑𝑖𝑢𝑠2
𝐴𝑟𝑒𝑎 = 𝜋 × 32
𝐴𝑟𝑒𝑎 = 9𝜋
𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝜋 × 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝜋 × 6
𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 6𝜋
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟 =
9𝜋
360
× 132
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟 = 10.36725576
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟 = 10.4 𝑐𝑚2
𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ =
6𝜋
360
× 132
𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 6.911503838
𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 6.9 𝑐𝑚
41. Handling Data Calculate an estimate for the mean
Table on time of goal scored
Time of
match
Frequency Midpoint Midpoint x
Frequency
0 < 𝑥 ≤ 15 8 7.5 60
15 < 𝑥 ≤ 30 4 22.5 90
30 < 𝑥 ≤ 45 5 37.5 187.5
45 < 𝑥 ≤ 60 7 52.5 367.5
60 < 𝑥 ≤ 75 7 67.5 472.5
75 < 𝑥 ≤ 90 10 82.5 825
Total 41 2002.5
Time of match Frequency
0 < 𝑥 ≤ 15 8
15 < 𝑥 ≤ 30 4
30 < 𝑥 ≤ 45 5
45 < 𝑥 ≤ 60 7
60 < 𝑥 ≤ 75 7
75 < 𝑥 ≤ 90 10
Not sure what time the goal was
scored – so we use the mid point
as an estimate.
𝑀𝑒𝑎𝑛 =
2002.5
41
𝑀𝑒𝑎𝑛 = 48.8
A goal was scored on average
at 48.8 minutes.
42. Number Algebra
Shape, Space, Measure Handling Data
Evaluate
80 =
64
1
2 =
7−2 =
4
81
1
2
=
𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒
𝑥2
+ 7𝑥 + 12
𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒
3𝑥2 + 14𝑥 + 15
𝐻𝑒𝑛𝑐𝑒 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦
𝑥2 + 7𝑥 + 12
3𝑥2 + 14𝑥 + 15
Draw a histogram for the following data.
Work out the value of x in this regular
pentagon
Marks Frequency
0 < x ≤ 5 3
5 < x ≤ 15 14
15 < x ≤ 30 18
30 < x ≤ 40 85x - 12
45. Shape, Space, Measure Work out the value of x in this regular
pentagon
5x - 12
Interior angles of a pentagon add up to 540˚.
Regular pentagon has equal angles, 540 ÷ 5 = 108°
Therefore, 5𝑥 − 12 = 108
5𝑥 = 120
𝑥 = 24°
46. Handling Data
Draw a histogram for the following data.
Marks Frequency
0 < x ≤ 5 3
5 < x ≤ 15 14
15 < x ≤ 30 18
30 < x ≤ 40 8
Marks Frequency Frequency Density
0 < x ≤ 5 3 3 ÷ 5 = 0.6
5 < x ≤ 15 14 14 ÷ 10 = 1.4
15 < x ≤ 30 18 18 ÷ 15 = 1.2
30 < x ≤ 40 8 8 ÷ 10 = 0.8
Calculate the frequency density by
frequency ÷ class width.
Area of the bar is the frequency
47. Number Algebra
Shape, Space, Measure Handling Data
Fill in the table of values and complete
the quadratic graph, 𝑦 = 𝑥2 − 5𝑥 + 6
for -4 ≤ x ≤ 4
1) Out of a class of 28, 19 of the pupils
support Barnsley. What percentage
of pupils do not support Barnsley.
2) An antique was bought for £210, it
was later sold for £400. What
percentage of the price it was sold
for, was profit?
There are 7 green balls and 3 red balls
in a bag. A ball is chosen at random
and not replaced.
What is the probability of picking 3
balls and
a) Them being all the same colour
b) 2 green balls and a redCalculate the area of the Isosceles
triangle
x -4 -3 -2 -1 0 1 2 3 4
y 42 2
42°
6 𝑐𝑚
48. Number 1) Out of a class of 28, 19 of the pupils
support Barnsley. What percentage
of pupils do not support Barnsley.
2) An antique was bought for £210, it
was later sold for £400. What
percentage of the price it was sold
for, was profit?
19 pupils support Barnsley,
this means 9 pupils don’t
support Barnsley.
9
28
= 9 ÷ 28
= 0.32142857
= 32%
£190 was profit. So
190
400
=190 ÷ 400
= 0.475
= 47.5%
49. Algebra Fill in the table of values and complete
the quadratic graph, 𝑦 = 𝑥2 − 5𝑥 + 6
for -4 ≤ x ≤ 4
x -4 -3 -2 -1 0 1 2 3 4
y 42 2
x -4 -3 -2 -1 0 1 2 3 4
y 42 30 20 12 6 2 0 0 2
50. Shape, Space, Measure
Calculate the area of the Isosceles
triangle
42°
6 𝑐𝑚
𝐴𝑟𝑒𝑎 =
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
2
21°
3 𝑐𝑚
Need to calculate the height of
the triangle. Use Trigonometry
Label your sides (in play)
o
a
Decide your triangle
a
o
t
Cover up what you are looking for and
write down your formula
𝑜 = tan 𝑥 × 𝑎
𝑜 = tan 42 × 3
𝑜 = 2.701
𝐴𝑟𝑒𝑎 =
6 × 2.701
2
𝐴𝑟𝑒𝑎 = 8.1 𝑐𝑚2 (3. 𝑠. 𝑓)
51. Handling Data There are 7 green balls and 3 red balls
in a bag. A ball is chosen at random and
not replaced.
What is the probability of picking 3 balls
and
a) Them being all the same colour
b) 2 green balls and a red
a) List the combinations –
All three green or all three red
𝑃 𝑅 𝐴𝑁𝐷 𝑅 𝐴𝑁𝐷 𝑅 =
3
10
×
2
9
×
1
8
=
6
720
𝑃 𝐺 𝐴𝑁𝐷 𝐺 𝐴𝑁𝐷 𝐺 =
7
10
×
6
9
×
5
8
=
210
720
𝑃 𝐴𝑙𝑙 𝑔𝑟𝑒𝑒𝑛 𝒐𝒓 𝐴𝑙𝑙 𝑟𝑒𝑑 =
6
720
+
210
720
=
𝟐𝟏𝟔
𝟕𝟐𝟎
=
𝟑
𝟏𝟎
b) List the combinations –
GGR or GRG of RGG
𝑃 𝐺𝐺𝑅 =
7
10
×
6
9
×
3
8
=
126
720
𝑃 𝐺𝑅𝐺 =
7
10
×
3
9
×
6
8
=
126
720
𝑃 𝑅𝐺𝐺 =
3
10
×
7
9
×
6
8
=
126
720
𝑃 𝐺𝐺𝑅 𝒐𝒓 𝐺𝑅𝐺 𝒐𝒓 𝑅𝐺𝐺 =
126
720
+
126
720
+
126
720
=
𝟑𝟕𝟖
𝟕𝟐𝟎
=
𝟐𝟏
𝟒𝟎
52. Number Algebra
Shape, Space, Measure Handling Data
Estimate
387 − 43
0.18
𝐸𝑥𝑝𝑎𝑛𝑑 𝑎𝑛𝑑 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦
3 2𝑥 − 5 − 7 6 − 𝑥
3𝑥3 𝑦2 𝑧5 3
(3𝑥 − 2𝑦)(2𝑥 + 5𝑦)
Below is a table to show how James spends his
day. Draw a pie chart to represent this data
Calculate the perimeter of this
isosceles triangle
5x - 711 - x
4x + 6
Activity Hours
Sleeping 9 hours
Work 8 hours
Eating/Cleaning/Cooking 3 hours
Reading 2 hours
Commuting 2 hours
53. Number
Round all numbers to 1 significant figure
400 − 40
0.2
Calculate this sum
360
0.2
Use equivalent fractions to help divide by a decimal
360
0.2
=
3600
2
= 1800
Estimate
387 − 43
0.18
56. Handling Data Below is a table to show how James spends his
day. Draw a pie chart to represent this data
Activity Hours
Sleeping 9 hours
Work 8 hours
Eating/Cleaning/Cooking 3 hours
Reading 2 hours
Commuting 2 hours
Activity Hours Degrees
Sleeping 9 hours 135
Work 8 hours 120
Eating/Cleaning/Cooking 3 hours 45
Reading 2 hours 30
Commuting 2 hours 30
24 hours in a day. Find out what
each hour is worth
360 ÷ 24 = 15°
Each hour is worth 15˚ on our pie
chart. So sleeping is 9 x 15˚ = 135˚
57. Number Algebra
Shape, Space, Measure Handling Data
Solve the following Simultaneous
Equations
5𝑥 − 𝑦 = 9
15𝑥 − 2𝑦 = 24
2𝑥 + 3𝑦 = 58
1) A gardens perimeter is 34m
(rounded to the nearest m), garden
fence panels are 130cm (rounded to
the nearest 10cm).
a) What is the most number of
panels that may be needed?
b) What is the least numbers of panels
that may be need?
Probability of Eric passing his Maths
exam is 0.7, the probability of passing
his English exam is independent of
this, and is 0.6.
What is the probability of Eric passing
at least one of his exams?
What is the straight line distance
between the coordinates (4, -5)
and ( 10, -3)
58. Number 1) A gardens perimeter is 34m (rounded to
the nearest m), garden fence panels are
130cm (rounded to the nearest 10cm).
a) What is the most number of panels
that may be needed?
b) What is the least numbers of panels that
may be need?
Change measurements into the same
units. 34m = 3400cm.
a) The most number of fence panels
needed are when you have a
large perimeter and small fence
panels.
Upper bound for perimeter = 3450cm
Lower bound for fence panel = 125cm
How many ‘small’ fence panels will you need for a ‘large’ perimeter.
𝟑𝟒𝟓𝟎 ÷ 𝟏𝟐𝟓 = 𝟐𝟕. 𝟔 You would need 28 panels!
b) The least number of fence panels needed
are when you have a small perimeter and large
fence panels
Lower bound for perimeter = 3350cm
Upper bound for fence panel = 135cm
How many ‘large’ fence panels will you need for a ‘small’ perimeter.
𝟑𝟑𝟓𝟎 ÷ 𝟏𝟑𝟓 = 𝟐𝟒. 𝟖𝟏 𝟒 You would need 25 panels!
60. Shape, Space, Measure What is the straight
line distance between
the coordinates (4, -5)
and ( 10, -3)
(4,-5)
(10,-3)
−3 − −5 = 2
10 − 4 = 6
𝑈𝑠𝑒 𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑎𝑠 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒
𝑎2
+ 𝑏2
= 𝑐2
62 + 22 = 𝑐2
36 + 4 = 𝑐2
40 = 𝑐2
6.32 (3.s.f)= 𝒄
61. Handling Data Probability of Eric passing his Maths
exam is 0.7, the probability of passing
his English exam is independent of
this, and is 0.6.
What is the probability of Eric passing
at least one of his exams?
List the combinations –
Pass Maths (0.7), Fail English (0.4)
Pass English (0.6), Fail Maths (0.3)
Pass English (0.6), Pass Maths (0.7)
𝑃 𝑃𝑎𝑠𝑠𝑀 𝐴𝑁𝐷 𝐹𝑎𝑖𝑙𝐸 = 0.7 × 0.4
= 0.28
𝑃 𝑃𝑎𝑠𝑠𝐸 𝑎𝑛𝑑 𝐹𝑎𝑖𝑙𝑀 = 0.6 × 0.3
= 0.18
𝑃 𝑃𝑎𝑠𝑠𝐸 𝑎𝑛𝑑 𝑃𝑎𝑠𝑠𝑀 = 0.6 × 0.7
= 0.42
𝑃 𝑃𝑎𝑠𝑠 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 = 0.28 + 0.18 + 0.42
= 𝟎. 𝟖𝟖
Alternative Solution.
Probability of passing at least once = 1 – probability of not passing either exam
𝑃 𝐹𝑎𝑖𝑙𝐸 𝑎𝑛𝑑 𝐹𝑎𝑖𝑙𝑀 = 0.4 × 0.3
= 0.12
𝟏 − 𝟎. 𝟏𝟐 = 𝟎. 𝟖𝟖
62. Number Algebra
Shape, Space, Measure Handling Data
Put the following, in ascending order
𝑟0, 2−5, 81
1
2,
1
4
−2
,
3
2
2
𝑆𝑜𝑙𝑣𝑒
4𝑥 − 1 = 2 − 𝑥
10𝑥 + 5 = 8𝑥 − 9
𝑥 − 1
2
+
4𝑥 − 3
3
= 4
Construct a frequency polygon for the
following
The area of the rectangle is 60𝑐𝑚2
calculate the perimeter
𝑥 − 7 Cost x (£) Frequency
0 ≤ x < 10 8
10 ≤ x < 20 11
20 ≤ x < 30 5
30 ≤ x < 40 14
40 ≤ x < 50 6
𝑥
63. Number Put the following, in ascending order
𝑟0, 2−5, 81
1
2,
1
4
−2
,
3
2
2𝑟0 = 1 (anything to the power 0 is
equal to 1.
2−5
=
1
25 =
1
32
81
1
2 = 81 = 9
1
4
−2
=
4
1
2
=
42
12
=
16
1
= 16
3
2
2
=
32
22 =
9
4
Therefore,
𝟐−𝟓, 𝒓 𝟎,
𝟑
𝟐
𝟐
, 𝟖𝟏
𝟏
𝟐,
𝟏
𝟒
−𝟐
65. Shape, Space, Measure The area of the rectangle is 60𝑐𝑚2
calculate the perimeter
𝑥 − 7
𝑥
𝐴𝑟𝑒𝑎 = 60𝑐𝑚2
𝐴𝑟𝑒𝑎 = 𝑥 𝑥 − 7
= 𝑥2 − 7𝑥
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑥2
− 7𝑥 = 60
𝑀𝑎𝑘𝑒 𝑖𝑡 𝑙𝑜𝑜𝑘 𝑙𝑖𝑘𝑒 𝑎 𝑢𝑠𝑢𝑎𝑙 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
−60 𝑥2
− 7𝑥 − 60 = 0 −60
𝑁𝑜𝑤 𝑠𝑜𝑙𝑣𝑒. 𝐼𝑡 𝑤𝑖𝑙𝑙 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒, 𝑏𝑢𝑡 𝑦𝑜𝑢 𝑐𝑎𝑛 𝑢𝑠𝑒 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎
𝑥 − 12 𝑥 + 5 = 0
𝑥 − 12 = 0 𝑜𝑟 𝑥 + 5 = 0
𝑥 = 12 𝑜𝑟 𝑥 = −5
We are working with lengths, so we will ignore the -5, 𝑥 = 12.
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 𝑥 − 7 + 𝑥 + 𝑥 − 7 + 𝑥
= 4𝑥 − 14
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑖𝑛 𝑥 = 12
= 4 12 − 14
= 𝟑𝟒𝒄𝒎
66. Handling Data Construct a frequency polygon for the
following
Cost x (£) Frequency
0 ≤ x < 10 8
10 ≤ x < 20 11
20 ≤ x < 30 5
30 ≤ x < 40 14
40 ≤ x < 50 6
See polygon – think straight lines.
Is this 8 things at £0 or £9.99? We
can’t be sure so use the midpoint.
67. Number Algebra
Shape, Space, Measure Handling Data
Calculate the 𝑛𝑡ℎ term for the
following sequences
3, 7, 11, 15, 19, …
8, 11, 16, 23, 32, …
2, 7, 14, 23, 34, …
Barnsley Football club is organising travel for
an away game. 1300 adults and 500 juniors
want to go. Each coach holds 48 people and
costs £320 to hire. Tickets to the match are
£18 for adults and £10 for juniors.
The club is charging adults £26 and juniors
£14 for travel and a ticket. How much profit
does the club make out the trip?
Calculate an estimate for the meanThese two shapes are Mathematical Similar.
Calculate the missing volume and surface
area
4cm
12cm
Surface area =
Volume = 72𝑐𝑚3 Surface area = 729𝑐𝑚2
Volume =
Cost x (£) Frequency
0 ≤ x < 10 8
10 ≤ x < 20 11
20 ≤ x < 30 5
30 ≤ x < 40 14
40 ≤ x < 50 6
68. Number Barnsley Football club is organising travel for
an away game. 1300 adults and 500 juniors
want to go. Each coach holds 48 people and
costs £320 to hire. Tickets to the match are
£18 for adults and £10 for juniors.
The club is charging adults £26 and juniors
£14 for travel and a ticket. How much profit
does the club make out the trip?
Cost to supporter
1300 × £26 = £33,800
500 × £14 = £7,000
𝑇𝑜𝑡𝑎𝑙 = £40,800
Cost to the club
Ticket costs
1300 × £18 = £23,400
500 × £10 = £5,000
𝑇𝑜𝑡𝑎𝑙 = £28,400
Coach costs
Coaches needed: (1300 +
Profit to the club
£40,800 − £40,560 = £240
70. Shape, Space, Measure These two shapes are Mathematical Similar.
Calculate the missing volume and surface
area
4cm
12cm
Surface area =
Volume = 72𝑐𝑚3 Surface area = 729𝑐𝑚2
Volume =
Mathematical similar – they are
in scale.
𝑆𝑐𝑎𝑙𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 =
12
4
= 3
Shape B has been enlarged by
the linear scale factor of 3.
The surface area will therefore
be 32
as big.
729
32
= 81𝑐𝑚2
The volume will be 33
as big.
72 × 33
= 1944𝑐𝑚3
A
B
𝟖𝟏𝒄𝒎 𝟐
𝟏𝟗𝟒𝟒𝒄𝒎 𝟑
Top Tip:
Area is measured in 𝒄𝒎 𝟐
, so when
enlarging a shape – don’t forget to
square the scale factor when working
out the enlarged area.
Volume is measured in 𝒄𝒎 𝟑
, so when
enlarging a shape – don’t forget to
cube the scale factor when working
out the enlarged volume.
71. Calculate an estimate for the mean
Time of
match
Frequency Midpoint Midpoint x
Frequency
0 ≤ x < 10 8 5 40
10 ≤ x < 20 11 15 165
20 ≤ x < 30 5 25 125
30 ≤ x < 40 14 35 490
40 ≤ x < 50 6 45 270
Total 44 1090
Not sure what time the cost is –
so we use the mid point as an
estimate.
𝑀𝑒𝑎𝑛 =
1090
44
𝑀𝑒𝑎𝑛 = £24.78
Handling Data
Cost x (£) Frequency
0 ≤ x < 10 8
10 ≤ x < 20 11
20 ≤ x < 30 5
30 ≤ x < 40 14
40 ≤ x < 50 6
72. Number Algebra
Shape, Space, Measure Handling Data
𝐹𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑒
4𝑥2
− 18𝑥
𝑥2
− 12𝑥 + 32
6𝑥2 + 13𝑥 − 5
25𝑥2 − 9𝑦2
Give two criticisms of the following
questionnaire
A rectangles width to length ratio is
2:5, if the perimeter is 84cm, what is
the area?
2.1 x 0.48 =
0.021 x 4.8 =
162.9 ÷ 0.09 =
Do you agree that Barnsley FC are by far the
greatest team the world has ever seen?
Strongly agree Agree Not sure
73. Number 2.1 x 0.48 =
0.021 x 4.8 =
162.9 ÷ 0.09 =
Ignore the decimal places and do your calculations when multiplying
decimals.
𝟐𝟏 × 𝟒𝟖 = 𝟏𝟎𝟎𝟖
Now add your decimal point into your answer. The question had three
decimal places – so your answer needs three decimal places.
1 008.
Ignore the decimal places and do your calculations when multiplying
decimals.
𝟐𝟏 × 𝟒𝟖 = 𝟏𝟎𝟎𝟖
Now add your decimal point into your answer. The question had four
decimal places – so your answer needs four decimal places.
1 008
.
Use equivalent fractions when dividing with decimals.
162.9
0.09
=
16290
9
= 1810
75. Shape, Space, Measure
A rectangles width to length ratio is
2:5, if the perimeter is 84cm, what is
the area?
5𝑥
2𝑥
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 84𝑐𝑚
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 5𝑥 + 2𝑥 + 5𝑥 + 2𝑥
= 14𝑥
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 14𝑥 = 84
÷ 14 𝑥 = 6 ÷ 14
𝐴𝑟𝑒𝑎 = 2𝑥 × 5𝑥
= 10𝑥2
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑥 = 6
𝐴𝑟𝑒𝑎 = 10 62
= 𝟑𝟔𝟎𝒄𝒎 𝟐
76. Handling Data Give two criticisms of the following
questionnaire
Do you agree that Barnsley FC are by far the
greatest team the world has ever seen?
Strongly agree Agree Not sure
1. A leading question, do you agree, puts pressure on
the person answering the question.
2. No response boxes for if you disagree (but why
would you?)
77. Number Algebra
Shape, Space, Measure Handling Data
Fill in the table of values and complete
the cubic graph, 𝑦 = 𝑥3 − 2𝑥 +1 for
-3 ≤ x ≤ 3
2
3
7
− 4
1
4
=
5
8
×
4
15
=
4
5
÷
6
7
=
A bag contains 3 blue balls, 5 red balls
and 4 green balls. When a ball is taken
out, it is not replaced. What is the
probability of picking three balls that
are all the same colour.
Calculate the area of the following
quadrilateral
x -3 -2 -1 0 1 2 3
y -3 1 22
79. Algebra Fill in the table of values and complete
the cubic graph, 𝑦 = 𝑥3 − 2𝑥 +1 for
-3 ≤ x ≤ 3
x -3 -2 -1 0 1 2 3
y -3 1 22
x -3 -2 -1 0 1 2 3
y -20 -3 2 1 0 5 22
80. Shape, Space, Measure Calculate the area of the following
quadrilateralWill need these
(found at the front
of your exam)
Find the area of the two
triangles and add these
together...
1
2
Sin B
b
=
Sin A
a
𝑆𝑖𝑛 B
3
=
Sin 83
12
𝑆𝑖𝑛 𝐵 =
3𝑆𝑖𝑛 83
12
𝐵 = 𝑆𝑖𝑛−1
3 sin 83
12
𝐵 = 14.367°
1
B
𝑇ℎ𝑒 𝑙𝑎𝑠𝑡 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 1
180 − 83 − 14.367 = 82.633°
2
𝐹𝑖𝑛𝑑 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 2 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑐𝑜𝑠𝑖𝑛𝑒 𝑟𝑢𝑙𝑒
𝑎𝑛𝑦 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑎𝑝𝑝𝑙𝑖𝑐𝑎𝑏𝑙𝑒
𝑎2
= 𝑏2
+ 𝑐2
− 2𝑏𝑐 𝐶𝑜𝑠 𝐴
122
= 102
+ 42
− 2 × 10 × 4 𝐶𝑜𝑠 𝐴
144 = 116 − 80𝐶𝑜𝑠 𝐴
28 = −80 𝐶𝑜𝑠 𝐴
−
28
80
= 𝐶𝑜𝑠 𝐴
𝐶𝑜𝑠−1
−
28
80
= 𝐴
𝐴 = 110.487
81. Shape, Space, Measure Calculate the area of the following
quadrilateralWill need these
(found at the front
of your exam)
Find the area of the two
triangles and add these
together...
1
2
𝑈𝑠𝑒 𝐴𝑟𝑒𝑎 =
1
2
𝑎𝑏𝑆𝑖𝑛 𝐶
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 1 =
1
2
× 12 × 3 × 𝑆𝑖𝑛 82.633
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 1 = 17.851𝑐𝑚2
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 2 =
1
2
× 10 × 4 × 𝑆𝑖𝑛 110.487
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 2 = 18.735𝑐𝑚2
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 = 17.851 + 18.735
= 𝟑𝟔. 𝟓𝟖𝟔
82. Handling Data A bag contains 3 blue balls, 5 red balls
and 4 green balls. When a ball is taken
out, it is not replaced. What is the
probability of picking three balls that
are all the same colour.
List the combinations –
Blue, Blue, Blue,
Red, Red, Red,
Green, Green, Green
𝑃 𝐵 𝐴𝑁𝐷 𝐵 𝐴𝑁𝐷 𝐵 =
3
12
×
2
11
×
1
10
=
6
1320
𝑃 𝑅 𝐴𝑁𝐷 𝑅 𝐴𝑁𝐷 𝑅 =
5
12
×
4
11
×
3
10
𝑃 𝐺 𝐴𝑁𝐷 𝐺 𝐴𝑁𝐷 𝐺 =
4
12
×
3
11
×
2
10
𝑃 𝐵𝐵𝐵 𝑂𝑅 𝑅𝑅𝑅 𝑂𝑅 𝐺𝐺𝐺 =
6
1320
+
60
1320
+
24
1320
=
90
1320
=
3
44
83. Number Algebra
Shape, Space, Measure Handling Data
𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦
ℎ0
3𝑟2
𝑡4
𝑧 3
10𝑥𝑦2
𝑧4
2𝑥3 𝑦2 𝑧
𝑥2
+ 7𝑥 + 12
𝑥 + 4
The mean weight of ten footballers is 73.5kg. A
new player comes along and the mean weight
goes down to 72kg. How much does the new
player weigh?
Calculate the area of the isosceles
triangle
Calculate the area and perimeter of
the rectangle. Leave your answer in
Surd form if applicable
18
2
10cm
13cm
84. Number Calculate the area and perimeter of
the rectangle. Leave your answer in
Surd form if applicable
18
2
𝐴𝑟𝑒𝑎 = 18 × 2
= 36
= 𝟔 𝒖𝒏𝒊𝒕𝒔 𝟐
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 18 + 2 + 18 + 2
= 9 × 2 + 2 + 9 × 2 + 2
= 9 2 + 2 + 9 2 + 2
= 3 2 + 2 + 3 2 + 2
= 𝟖 𝟐 𝒖𝒏𝒊𝒕𝒔
86. Shape, Space, Measure Calculate the area of the
isosceles triangle
10cm
13cm
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
2
Need to calculate the perpendicular
height of the triangle.
13cm
5cm
h 𝑈𝑠𝑒 𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑎𝑠 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 ℎ𝑒𝑖𝑔ℎ𝑡
𝑎2
+ 𝑏2
= 𝑐2
52
+ 𝑏2
= 132
25 + 𝑏2 = 169
𝑏2 = 144
𝑏 = 12
𝐻𝑒𝑖𝑔ℎ𝑡 𝑖𝑠 12𝑐𝑚. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒,
𝐴𝑟𝑒𝑎 =
12 × 5
2
𝐴𝑟𝑒𝑎 = 𝟑𝟎𝒄𝒎 𝟐
87. Handling Data The mean weight of ten footballers is
73.5kg. A new player comes along and the
mean weight goes down to 72kg. How much
does the new player weigh?
10 footballers in total weigh
10 × 73.5𝑘𝑔 = 735𝑘𝑔.
11 footballers in total weigh
11 × 72 = 792𝑘𝑔
The eleventh footballer must have weight
792 − 735 = 𝟓𝟕𝒌𝒈
88. Number Algebra
Shape, Space, Measure Handling Data
The formula to convert Celsius to
Fahrenheit is
9
5
𝐶 + 32 = 𝐹
Use your calculator to work out the value of
3.92 − 1.42
Write down the full display.
Round your answer to 2 significant figures
Complete a histogram for the following
data
Height, h (cm) Frequency
151 ≤ h < 153 64
153 ≤ h < 154 43
154 ≤ h < 155 47
155 ≤ h < 159 96
159 ≤ h < 160 12
What is the temperature in Fahrenheit,
when it 18˚ Celsius?
What is the temperature in Celsius
when it is 53.6˚ Fahrenheit?
An exterior angle of a
regular polygon is 30˚.
Work out the number of
sides of the polygon.
89. Number Use your calculator to work out the value of
3.92 − 1.42
Write down the full display.
Round your answer to 2 significant figures
3
For Casio fx-83GT plus
Type
. 9 - 1 . 4 𝒙 𝟐 =
3.640054945
3.6
𝒙 𝟐
90. Algebra The formula to convert Celsius to
Fahrenheit is
9
5
𝐶 + 32 = 𝐹
What is the temperature in Fahrenheit,
when it 18˚ Celsius?
What is the temperature in Celsius
when it is 53.6˚ Fahrenheit?
9 × 18
5
+ 32 = ℉
64.4 = ℉
𝟏𝟖℃ = 𝟔𝟒. 𝟒℉
9
5
𝐶 + 32 = 53.6
−32
9
5
𝐶 = 21.6 −32
× 5 9𝐶 = 108 × 5
÷ 9 𝐶 = 12 ÷ 9
𝟏𝟐℃ = 𝟓𝟑. 𝟔℉
91. Shape, Space, Measure
An exterior angle of a
regular polygon is 30˚.
Work out the number of
sides of the polygon.
Exterior angles add up to 360°, therefore
360 ÷ 30 = 12
The polygon has 12 sides (Dodecagon)
30°
92. Handling Data
Complete a histogram for
the following data
Height, h (cm) Frequency
151 ≤ h < 153 64
153 ≤ h < 154 43
154 ≤ h < 155 47
155 ≤ h < 159 96
159 ≤ h < 160 12
Marks Frequency Frequency Density
151 ≤ h < 153 64 64 ÷ 2 = 32
153 ≤ h < 154 43 43 ÷ 1 = 43
154 ≤ h < 155 47 47 ÷ 1 = 47
155 ≤ h < 159 96 96 ÷ 4 = 24
159 ≤ h < 160 12 12 ÷ 1 = 12
Calculate the frequency density by
frequency ÷ class width.
Area of the bar is the frequency
93. Number Algebra
Shape, Space, Measure Handling Data
𝐸𝑥𝑝𝑎𝑛𝑑 𝑎𝑛𝑑 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦
3𝑦 4𝑦 − 2
4 𝑥 − 2 − 5 𝑥 + 3
𝑥 + 4 𝑥 − 7
(3𝑎 + 2𝑏)(2𝑎 − 3𝑏)
What is the probability of rolling a dice
three times and getting two square
numbers and a prime number.
Convert 8𝑐𝑚3 to 𝑚𝑚3
Convert 40,000𝑐𝑚2 to 𝑚2
Four pumps usually empty water
from a tank in 1 hour 36 minutes.
One of the pumps breaks down.
How long will three pumps, working
at the same rate, take to empty the
same tank.
94. Number
Four pumps usually empty water
from a tank in 1 hour 36 minutes.
One of the pumps breaks down.
How long will three pumps, working
at the same rate, take to empty the
same tank.
1 hour 36 minutes = 96 minutes
If four pumps take 96 minutes to
empty a tank, that must mean one
pump would take four times as long
4 × 96 = 384 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
If three pumps were to do this, they
would do it in a third of the time
384 ÷ 3 = 128 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
128 minutes = 2 hours 8 minutes.
97. Handling Data What is the probability of rolling a dice
three times and getting two square
numbers and a prime number.
List the combinations –
Square, Square, Prime
Square, Prime, Square
Prime, Square, Square
𝑃 𝑆 𝐴𝑁𝐷 𝑆 𝐴𝑁𝐷 𝑃 =
1
3
×
1
3
×
1
2
=
1
18
𝑃 𝑆 𝐴𝑁𝐷 𝑃 𝐴𝑁𝐷 𝑆 =
1
3
×
1
2
×
1
3
=
1
18
𝑃 𝑃 𝐴𝑁𝐷 𝑆 𝐴𝑁𝐷 𝑆 =
1
2
×
1
3
×
1
3
=
1
18
Square numbers 1, 4
𝑃 𝑆𝑞𝑢𝑎𝑟𝑒 =
1
3
Prime numbers 2, 3, 5
𝑃 𝑃𝑟𝑖𝑚𝑒 =
1
2 𝑃 𝑆𝑆𝑃 𝑂𝑅 𝑆𝑃𝑆 𝑂𝑅 𝑃𝑆𝑆 =
1
18
+
1
18
+
1
18
=
3
18
=
1
6
98. Number Algebra
Shape, Space, Measure Handling Data
Write down all the integers n that
satisfy
5 ≤ 𝑛 < 9
−3 < 2𝑛 ≤ 1
−2 ≤ 4𝑛 + 2 ≤ 0
£1 = $1.67
A laptop costs £320 in the UK, and $530 in
the USA. Where is the laptop cheaper, and by
how much?
The table shows van rentals for the
company VansRCool. Calculate a 3
point moving average for the months
below
Hayley can run 100m in 13
seconds. What is her average
speed in miles per hour?
99. Number
£1 = $1.67
A laptop costs £320 in the UK, and $530 in
the USA. Where is the laptop cheaper, and by
how much?
UK price £320
USA price $530 ÷ 1.67 = £317.37
USA cheaper by £2.63
UK price £320 × 1.67 = $534.40
USA price $530
USA cheaper by $4.40
OR
101. Shape, Space, Measure
Key Facts:
60 seconds in a minute
60 minutes in a hour
1600m in a mile.
Hayley can run 100m in 13
seconds. What is her average
speed in miles per hour?
100 𝑚𝑒𝑡𝑟𝑒𝑠 𝑖𝑛 13 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
100
13
𝑚𝑒𝑡𝑟𝑒𝑠 𝑖𝑛 1 𝑠𝑒𝑐𝑜𝑛𝑑
60 × 100
13
𝑚𝑒𝑡𝑟𝑒𝑠 𝑖𝑛 1 𝑚𝑖𝑛𝑢𝑡𝑒
60 × 60 × 100
13
𝑚𝑒𝑡𝑟𝑒𝑠 𝑖𝑛 1 ℎ𝑜𝑢𝑟
360000
13
𝑚𝑒𝑡𝑟𝑒𝑠 𝑖𝑛 1 ℎ𝑜𝑢𝑟
360000
13 × 1600
𝑚𝑖𝑙𝑒𝑠 𝑖𝑛 1 ℎ𝑜𝑢𝑟.
17.3 miles per hour
102. Handling Data
3 point moving average – work
out the mean in groups of 3
The table shows van rentals for the
company VansRCool. Calculate a 3
point moving average for the months
below
9 + 22 + 37
3
= 22. 6
22 + 37 + 14
3
= 24. 3
37 + 14 + 18
3
= 23
14 + 18 + 24
3
= 18. 6
𝑇ℎ𝑒 3 𝑝𝑜𝑖𝑛𝑡 𝑚𝑜𝑣𝑖𝑛𝑔 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑠 22.6, 24.3, 23, 18.6
