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Level 3/4 Mathematics

All the basics you need to All the basics you need to know for the Level 3/4 know for the Level 3/4

course.......course.......

I can ....I can ....11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414

1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828

2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242

4343 4444 4545 4646 4747 4848 4949 5050 5151 5252 5353 5454 5555 5656

5757 5858 5959 6060 6161 6262 6363 6464 6565 6666 6767 6868 6969 7070

7171 7272 7373 7474 7575 7676 7777 7878 7979 8080 8181 8282 8383 8484

8585 8686 8787 8888 8989 9090 9191 9292 9393 9494 9595 9696 9797

Created by Mr. Lafferty

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Question 1Question 1

multiply small decimal multiply small decimal numbers together ?numbers together ?

0.04 x 0.7answer

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Answer to Question 1Answer to Question 1

multiply numbers together

then count number of decimal places.

0.0280.028

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Question 2Question 2

divide small decimal divide small decimal numbersnumbers

15.3 ÷ 0.0315.3 ÷ 0.03

answer

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Answer to Question 2Answer to Question 2

Do not divide by a decimal scale it up to a whole number

Scale the number to be divided by the same amount

1530 ÷ 3 = 510

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Question 3

tell you the meaning of the words

(a) Profit(b) Loss(c) Salary(d) Overtime

answer

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Answer to Question 3

Profit:Profit: When you sell something for When you sell something for MOREMORE than than you bought it.you bought it.

Loss : Loss : When you sell something for When you sell something for LESS than you than you bought bought it.it.

Salary : Salary : How much a person is paid to do a job.How much a person is paid to do a job.

Can be measured weekly, monthly or Can be measured weekly, monthly or annually.annually.

Overtime : Extra work outside normal work time.Usually at night or weekends.

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answer

Question 4

tell you the meaning of the words

(a)Hire Purchase(b)Foreign Exchange

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Answer to Question 4

Hire Purchase : Hire Purchase : When you pay for goods byWhen you pay for goods by

putting down a deposit and then putting down a deposit and then

paying the rest up in small fixed paying the rest up in small fixed

payments over a period of time.payments over a period of time.

Foreign Exchange :Foreign Exchange : Converting one currency to another.Converting one currency to another.

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answer

Question 5

explain the meaning of

Significant Figures

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Answer to Question 5

A measure of both:A measure of both:

(i)(i) QuantityQuantity

(ii)(ii)AccuracyAccuracy

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Question 6

explain the term double negative.

(-4) – (-6)

answer

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Answer to Question 6

Double negative changes to a positive

(-4) – (-6) = (-4) + 6 = 2

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Question 7

tell you the instrument that helps us to add and subtract integers.

answer

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Answer to Question 7Answer to Question 7

thermometerthermometer

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answer

Question 8

explain the rules of multiplication and

division of integers.

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Answer to Question 8

Different signs = negative answer

Same signs = positive answer

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Question 9

explain BODMAS

3 – ( 2 x (-4))2

answer

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Answer to Question 9

B - BracketsO – OtherM - MultiplicationA - AdditionS- Subtraction

3 – ( 2 x (-4))2

3 – (-8)2

3 – 64

– 61

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Question 10

explain what an integer is.

answer

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Answer to Question 10

A positive or negative number including zero.

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Question 11

explain the term scientific notation / standard form

and convert to it.

(a)5680000

(b)0.00623answer

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Answer to Question 11

a x 10n

a between 1 and 10

n is an integer

(a) 5.68 x 106

(b) 6.23 x 10-3

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Question 12

answer

write numbers in normal form

(a) 5.93 x 104

(b) 7.43 x 10-5

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Answer to Question 12

Normal Form

(a) 59300

(b) 0.0000743

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Question 13

write these numbers to 3 significant figures

(a)92418400(b)0.04057090100

answer

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Answer to Question 13

3 significant figures

(a)92400000(b)0.0406

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Question 14

round decimals to any significant figures

answer

(a) 93.0108 ( to 2 s.f.)(b) 0.0706194 ( to 4 s.f.)

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Answer to Question 14

(a) 93 ( to 2 s.f.)(b) 0.07062 ( to 4

s.f.)

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Question 15

write numbers to any decimal place

(a) 29.0864 ( to 2 dp)(b) 0.0109964 ( to 4 dp)

answer

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Answer to Question 15

Higher Mathematics: ©HappySmile Higher Mathematics: ©HappySmile ProductionsProductions

(a) 29.09 ( to 2 dp)(b) 0.0110 ( to 4 dp)

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answer

Question 16

explain how to get triangular numbers

and write down the first 6 triangular

numbers.

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Answer to Question 16

1, 3, 6, 10, 15, 21

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answer

Question 17

explain how to get square numbers

and write down the first 10

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Answer to Question 17

1, 4, 9, 16, 25, 3649, 64, 81, 100

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answer

Question 18

decompose a number into product of primes

e.g. 18

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Answer to Question 18

18

3

3 6

23

18

= 3 x 6

= 3 x 3 x 2All Prime !

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answer

Question 19

explain what the LCM is between a set of

numbers

e.g. LCM for 3, 4 and 5

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Answer to Question 19

the lowest number that 3, 4, 5 all divide into

without leaving a remainder is 60

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answer

Question 20

explain what the HCF is between a set of

numbers

e.g. HCF for 25, 50 and 100

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Answer to Question 20

the highest number that divides into 25, 50, 100

without leaving a remainder is 25

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answer

Question 21

know the basic angle properties at Level E

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Answer to Question 21

120o 95

o

Angles round a pointAdd up to 360

o

115o

Two angles making astraight line add to 180

o

angles opposite each otherat a cross are equal.

34o 34

o

3 angles in a triangle ALWAYSadd up to 180

o.

50o

40o

65o

90o

146o

146o

145o

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answer

Question 22

explain the term alternate angles

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Answer to Question 22

two angles that form a Z - shape

= equal angles

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answer

Question 23

explain the term corresponding angles

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Answer to Question 23

two angles that form a F - shape

= equal angles

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answer

Question 24

explain the term interior angle of a shape

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Answer to Question 24

Pentagon

72o

Hexagon

60o

Octagon

45o

Interior AnglesInterior Angle

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Higher Mathematics: ©HappySmile Higher Mathematics: ©HappySmile ProductionsProductions

answer

Question 25

explain the term exterior angle of a shape

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Pentagon

This is called the“Exterior angle”

B

A

E

O

Q

Answer to Question 25

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answer

Question 26

find the exterior angle of a shape

Pentagon Find Exterior angle

B

A

E

O

Q

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Answer to Question 26

Exterior angle = 180o – interior angle

Pentagon

This is called the“Exterior angle”

B

A

E

CD

O

Q

Exterior angle = 180o – 108o = 72o

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answer

Question 27

find the interior angle of a shape

Octagon

45o

Find the interior angles

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Answer to Question 27

Octagon

45o

Interior angle = (180o – 45o) = 135o

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answer

Question 28

use angle properties to calculate missing angles

Find all missing angles ao , bo and co

d = 115o

co

ao

bo

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Answer to Question 28

ao = co = 115o

bo = 65o

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answer

Question 29

remove a single bracket

(a) -2 ( 8y – 3)(b) 4b ( – 5b + a)

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(a) -16y + 6

(b) -20b2 + 4ab

Answer to Question 29

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remove a single bracketand simplify

(a) 2 (x + 5) - 7(b) 3(a + 4) + 2(a – 1)

Question 30

answer

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Answer to Question 30

(a) 2x + 10 -7 =2x + 3

(b) 3a + 12 + 2a – 2 =5a + 10

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answer

Question 31

evaluate algebraic expressions

22 3 30a b c

a = 3 ; b = 4 and c = -1

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Answer to Question 31

22 3 30a b c

(2 3 3) (3 4) (30 ( 1)) (2 ) (3 ) (30 )a a b c

18 12 30 0

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answer

Question 32

factorise algebraic expressions

(a) 4y + 80(b) 9x – 6x2

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Answer to Question 32

(a) 4y + 80 = 4(y + 20)

(b) 9x – 6x2 = 3x(3 – 2x)

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answer

Question 33

simplify algebraic expressions

(a) -6x + 7y – 16x + 11y(b) 2b – 3b2 – 5b + 5b2

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Answer to Question 33

(a) -22x + 18y

(b) -3b + 2b2

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answer

Question 34

find a percentage of quantity (with a calculator)

What is 17.5 % of £450

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Answer to Question 34

17.5

450100

= £76.50

17.5 % of £450

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answer

Question 35

explain the term appreciation with reference to percentages

The average house prices in Glasgow have appreciated by 50% over the past 10 years. If you bought the house for £80 000 ten years ago.

How much would the house be worth now?

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Answer to Question 35

when an item has increased in value by a certain percentage.

Appreciation = 50% x £ 80 000= 0.50 x £80 000

= £ 40 000

New value = Old Value + Appreciation

= £ 80 000 + £40 000= £ 120 000

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answer

Question 36

explain the term depreciation with reference to percentages

A Mini Cooper cost £15 000 in 2005.At the end 2006, it will have depreciated by 20%What will the mini cooper worth at end

2006?

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Answer to Question 36

when an item has decreased in value by a certain percentage.

End 2006 Depreciation = 20% x £15 000

= 0.2 x £15 000= £3 000

New value = Old value - Depreciation= £15 000 - £3000

= £12 000

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answer

Question 37

express a value as a percentage of another value.

Frances scored 13 out of 20 in a Maths test.What was her percentage score?

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Answer to Question 37

13= 100

20=65%

scored 13 out of 20

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answer

Question 38

work backwards to find initial value.

After a 10% increase the price of a house is £88 000. What was the price before the increase.

Tricky

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Answer to Question 38

Price before is 100% :

£800 x 100 = £80 000

£88 000 110 = £8001 % :

110 % = £88 000We have :

100 % + 10 % = £88 000

Deduce from question :

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Question 39

work backwards to find initial value.

Tricky

The value of a car depreciated by 15%. It is now valued at £2550. What was it’s original price.

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Answer to Question 39

Price before is 100% :

£30 x 100 = £3 000

£2 550 85 = £301 % :

85 % = £2 550We have :

100 % - 15 % = £2 550

Deduce from question :

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answer

Question 40

find a percentage of quantity (with out a calculator)

What is 17.5 % of £600

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Answer to Question 40

10% 600 ÷ 10 = £ 60 5% half of 10% = £ 30

2.5% half of 5% = £ 15

17.5% £105

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answer

Question 41

identify the main parts of a circle.Name the main parts of the circle

O

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Answer to Question 41

O

radius

Circumference Diameter

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answer

Question 42

explain the connections between the radius, diameter and the

circumference of a circle.

O

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Answer to Question 42

radius

O

Circumference Diameter2D r

12

r D

C D

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answer

Question 43

explain the difference between the circumference and the area

of a circle.

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Circumference is the name given to the perimeter of the circle

(the length round the outside of the circle)

The area is the inside of the circle

Answer to Question 43

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answer

Question 44

calculate the perimeter and area for a circle.

Find the area and perimeter of the circle ?

4cm

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Answer to Question 44

2A r

24A250.26A cm

4cm

C D

8C

225.13C cm

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answer

Question 45

explain how to calculate the area of a

composite shape.

20cm

5 cm

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Answer to Question 45

20cm

5 cm

Area = rectangle + semicircle21

2

A l b r

2120 5 10

2( )A

100 157 A 2257 cmA

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answer

Question 46

calculate the area of any triangle.

Find the area below.

10cm

4cm

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Answer to Question 46

14 10

2Area

10cm

4cm

12

Area b h

Altitude h outside triangle this time.

220Area cm

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answer

Question 47

calculate the area of a rhombus.

Find the area below.

5cm

2cm

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Answer to Question 47

1Rhombus Area ( )

2D d

1Area = (5 2)

2

2Area = 5cm5cm

2cm

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answer

Question 48

calculate the area of a kite and inverted kite.

Find the area below.

9cm

4cm 7cm4cm

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9cm

4cm 7cm4cm

Answer to Question 48

1Kite Area ( )

2D d

1Area = (9 4)

2

2Area = 18cm

1Kite Area ( )

2D d

1Area = (7 4)

2

2Area = 14cm

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answer

Question 49

calculate the area of a parallelogram.

Find the area below.

9cm

3cm

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Answer to Question 49

Parallelogram Area b h

Area = 9 32Area = 27cm

9cm

3cm

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answer

Question 50

calculate the area of a trapezium.

Find the area below.

5cm

6cm

4cm

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Answer to Question 50

5cm

6cm

4cm1

Trapezium Area = (5 6) 42

1Trapezium Area = ( )

2a hb

2Trapezium Area = 22cm

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answer

Question 51

calculate the area of composite shapes.

Find the area shaded grey below.

10cm

11cm

8cm

4cm

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Answer to Question 51

10cm

11cm

8cm

4cm

1Trapezium Area = ( )

2a b h

Trapezium Area - Triangle Area

1Triangle Area =

2bh

21= (10 8) 11 99

2cm

21 = 4 11 22

2cm

2Shaded Area = 99 - 22 77cm

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answer

Question 52

write down the area formula for(a) Any triangle(b) Rhombus(c) Parallelogram(d) Kite(e) Trapezium

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Answer to Question 52

The area formula for(a) Any triangle(b) Rhombus(c) Parallelogram(d) Kite(e) Trapezium

1Area = ( )

2a b h

Area b h

1Area ( )

2D d

1Area ( )

2D d

12

Area b h

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answer

Question 53

write down in words Pythagoras Theorem

for right-angled triangles.

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Answer to Question 53

a

b c2 2 2a b c

Two shorter sides squared and added together

equal to the longest side

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answer

Question 54

calculate the longest side given the two shorter sides.

8

12

c

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Answer to Question 54

8

12

c2 2 2c =a +b2 2 2c =12 +8

2c =208

c = 208 =14.42km

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answer

Question 55

calculate a shorter sidegiven the longest side

and the other shorter side.

20cm 12cm

a cm

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2 2 2c =a +b

2 2 2a =20 -122a =256

a= 256 =16cm

2 2 2a =c - b

Check answer ! Always smaller

than hypotenuse

Answer to Question 55

20cm 12cm

a cm

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answer

Question 56

write down the three different versions of Pythagoras

Theorem

a

cb

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2 2 2c =a +b

2 2 2a =c - b

2 2 2b =c - a

Finding shorter side a

Findingshorter side b

a

cb

Answer to Question 56

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answer

Question 57

write down the rule for a linear pattern.

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Answer to Question 57

b = 3s + 8

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answer

Question 58

write down the steps for finding the rule for a

linear pattern.

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Answer to Question 58

1. Find the difference

2. Write down part of formula

3. Find the correction factor.

4. Write down the full formula

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answer

Question 59

convert hour and minutes to decimal hours.

48 minutes to decimal is

2hr 15 minutes to decimal is

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Answer to Question 59

48

60= 0.8 hr

152

60 = 2.25 hr

To change minutes to a decimal divide by 60

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answer

Question 60

convert decimal hours to hours and minutes.

0.7 hrs to minutes is

3.4 hrs to hours and minutes is

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Answer to Question 60

0.7 hrs to minutes is 0.7 60= 42 mins

3.4 hrs to hours and minutes is

3+0.4 60= 3 hrs 24 mins

To change decimal time to minutes ‘multiply by 60’

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answer

Question 61

calculate the speed of an object.

Daniel drove from his house to the Blackpool, a distance of 135 miles.

It took him 2hrs 15mins.

What was his average speed?

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Answer to Question 61

DS =

T135

=2.25

135 miles=

12 hrs

4

S =60mph

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answer

Question 62

calculate the time taken for an object.

How long did the bus journey take if it travelled a total distance of 60 km at an average speed of 40 km/hr.

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Answer to Question 62

DT =

S=1.5 hrs

60 km=

40 km/ hr

T =1 hr 30 mins

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answer

Question 63

calculate the distance travelled for an object.

A racing car travelled at 50 km/hr. What is the distance covered in 6 hours ?

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Answer to Question 63

D =S T =300 km=50 6

D =300 km

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answer

Question 64

write down the three formulae linking distance, speed and time.

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Answer to Question 64

D

S T DT =

SD

S =T

Simple way to remember the 3 formulae !

D = S T

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answer

Question 65

simplify ratios

6 : 812 : 45

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Answer to Question 65

3 : 4

4 : 15

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answer

Question 66

express ratios in unity form

1 : 6

2

0.1 : 4

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Answer to Question 66

1 : 12

1 : 40

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answer

Question 67

solve questions involving ratio

The ratio of boys to girls is 4:5. If there are 16 boys, how many girls are there.

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Answer to Question 67

The ratio of boys to girls is 4:5. If there are 16 boys, how many girls are there.

boys girls4 516 20

x 4 x 4

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Question 68

work out portions using ratios.

Ryan and Kerry share a raffle win of £400 in the ratio 3:5.

How much does each get ?

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Answer to Question 68

Ryan and Kerry share a raffle win of £400 in the ratio 3:5. How much does each get ?

Step 1 : Since the ratio is 3:5, there are :

3+5 = 8 shares

Step 2 : Each share is worth : 50

8 400

Step 3 : Ryan gets 3 x 50 = £150

Kerry gets 5 x 50 = £250Check !

150 + 250 = 400

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answer

Question 69

draw a graph of direct proportionand

state the key points of the graph.

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Answer to Question 69

Two quantities which are in DIRECT PROPORTION

always lie on a straight linepassing through the origin.

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answer

Question 70

solve problems involvingdirect proportion

The cost of 6 cakes is £4.20. Find the cost of 5 cakes.

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Answer to Question 70

19 Apr 202319 Apr 2023

Easier methodCakes Pence6 420

5

Are we expecting more

or less

5

6420

(less)

350 £3.50

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answer

Question 71

It takes 10 worker 12 months to build a house.How long should it take 8 men.

solve problems involvingindirect (inverse) proportion

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Answer to Question 71

10 12

1 12 x 10 = 120 Easier methodWorkers months

10 12 8

10

812

(more)

15 months

Are we expecting more

or less

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answer

Question 72

add two simple fractions

5 36 4

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6 4

18+

3824

2420

14124

÷2

÷2

14124

7112

Answer to Question 72

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answer

Question 73

5 16 5

subtract two simple fractions

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25

Answer to Question 73

6-

5 16 5

30

1930

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answer

Question 74

multiply two simple fractions

3 34 5

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Answer to Question 74

3 34 5

3 34 5

9

20

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answer

Question 75

divide two simple fractions

5 28 3

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Answer to Question 75

5 28 3

358 2

1516

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answer

Question 76

add two mixed fractions

1 22 3

2 3

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1 22 3

2 3

1 22 3

2 3

1 25

2 3 4+

76

6

3

116

15 1

6

16

6

Answer to Question 76

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answer

Question 77

7 27 4

8 3

subtract two mixed fractions

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Answer to Question 77

7 27 4

8 3

7 27 4

8 3

7 23

8 3 16-

524

24

21

53

24

53

24

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answer

Question 78

3 14 1

4 5

multiply two mixed fractions

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Answer to Question 78

3 14 1

4 5

19 64 5

19 64 5

11420

145

20

75

10

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answer

Question 79

3 23 1

4 3

divide two mixed fractions

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Answer to Question 79

3 23 1

4 3

15 54 3

3154 5

4520

52

20 1

24

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answer

Question 80

solve simple equations

2 12x 5 11x

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5 11x

Answer to Question 80

2 12x

16x

11 5x

6x

12 2x

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answer

Question 81

solve simple equations

4 7 17n

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Answer to Question 81

4 17 7n 4 7 17n

6n 24 4n

4 24n

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answer

Question 82

solve more complex equations

6 1 2 21x x

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Answer to Question 82

6 1 2 21x x

5x

6 2 21 1x x 4 20x

20 4x

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answer

Question 83

2(2 3) 24x x

solve equations with brackets

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Answer to Question 83

2(2 3) 24x x

4 6 24x x

3 18x

18 3 6x

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answer

Question 84

5(3 2) 2(4 3) 2 36x x x

solve equations with double brackets

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Answer to Question 84

5(3 2) 2(4 3) 2 36x x x

15 10 8 6 2 36x x x 7 16 2 36x x

20 5 4x

5 16 36x 5 20x

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answer

Question 85

13 7

2x

solve equations with fractions

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Answer to Question 85

13 7

2x

6 14x 14 6 8x

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answer

Question 86

3 2(2 1) 1

4 3x x

solve equations with brackets & fractions

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Answer to Question 86

3 2(2 1) 1

4 3x x

9(2 1) 8 12x x 18 9 8 12x x

26 21x 21

26x

Multiply EVERY

term by 12

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answer

Question 87

solve simple inequations (inequalities)

2 1 7x

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Answer to Question 87

2 1 7x 2 7 1x

2 8x 4x

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answer

Question 88

solve harder inequations (inequalities)

2(2 3) 9x x

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Answer to Question 88

2(2 3) 9x x 4 6 9x x 3 6 9x

3 15x15 3x 5x

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answer

Question 89

explain the meaning of the terms

MeanMedianModeRange

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Answer to Question 89

1. The Mean

Sum of all the data ÷ by the number of data values

2. The Median (put the data in order then find the MIDDLE value)

3. The Mode (the number that appears the most)

4. Range

Highest value - the lowest value

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Question 90

construct a frequency table12 23 41 55 7715 32 40 51 6921 12 16 43 5671 32 75 34 4255 76 21 73 2256 41 19 20 4778 17

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Answer to Question 90

Class Class IntervalsIntervals

TallyTally FrequencFrequencyy

1

6

3

5

6

6

5

10-19

20-29

30-39

40-49

50-59

60-69

70-79

Choose suitable

Class interval

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answer

Question 91

add a third column to a frequency table

and find the mean from it.

No of No of SiblingSiblings (s (SS))

Freq.Freq.

(f)(f)

9

6

1

30

1

13

0

1

2

3

5

Totals

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Answer to Question 91

No of No of SiblingSiblings (s (SS))

Freq.Freq.

(f)(f)

9

6

1

30

1

13

0

1

2

3

5

Totals

0 x 9 =0

2 x 6 = 12

5 x 1 = 5

3 x 1 = 3

1 x 13 = 13

33

S x S x ff

Mean 33= =1.1

30

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answer

Question 92DayDay Freq.Freq.

(f)(f)

2

1

5

6

3

1

2

3

4

5

2

6

17

12

5

Cum. Freq.Cum. Freq.Total so farTotal so far

7

6 8

4

25

29

find the medianfrom a

cumulative frequency graph.

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Answer to Question 92

median=29 2 =14.5

DayDay Freq.Freq.

(f)(f)

2

1

5

6

3

1

2

3

4

5

2

6

17

12

5

Cum. Freq.Cum. Freq.Total so farTotal so far

7

6 8

4

25

29

it occurred on day 5

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Question 93

construct a Pie-chart

Rugby

Football

Cricket

Ice Hockey

75

90

45

60

Favourite Sport

Squash 30

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Answer to Question 93

Rugby

Football

Cricket

Ice Hockey

75

90

45

60

Favourite Sport

Squash 30

Total 30075

Rugby angle = 360 0300

9 ox

90Football angle = 360 8

300

01 ox

45Cricket angle = 360 4

3005 ox

60Ice Hockey angle = 360 2

3007 ox

30Squash angle = 360 6

3003 ox

RugbyFo

otba

l

l

Cricket Ice Hockey

Squash

90o108o

54o72o

36o

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Question 94

construct a stem-leaf diagram

1212 4040 5757 5454 55551313 5555 1515 3232 55553232 1515 4141 2121 40402323 4141 2929 5151 1212

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Answer to Question 94

1 2

2345

stem leaves

n = 20 Key : 2 3 means 23

2

1

0

33 9

2 20 1 1

1 4 5 5

5 5

5 7

1212 4040 5757 5454 55551313 5555 1515 3232 55553232 1515 4141 2121 40402323 4141 2929 5151 1212

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Question 95

construct a scattergraphand describe the main features

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Answer to Question 95

xxx x

x x

Strong positive correlation

xx

x xx

x

Strong negative correlation

Best fit lineBest fit line

When two quantities are strongly connected we say there is a strong correlation between them.

xx

xxx

x

x

No correlation

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Question 96

explain the term probabilityin words and numbers

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Answer to Question 96

Probability can be thought of as a fraction or decimal.

It always lies between 0 and 1.

0 meaning impossible ( could not happen)

1 meaning certain ( will definitely happen)

The chance of an event happening

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Question 97

evaluate a probability

There are 3 red and 4 green balls in a bag. What is the probability a green ball is picked.

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Answer to Question 97

There are 3 red and 4 green balls in a bag. What is the probability a green ball is picked.

4 4

3 4 7P( )green

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Online Revision Exercises

Integers

Angles

Type of numbers

Scientific Notation 1 2

Algebra

Decimals

Statistics

Percentages

Area of Quadrilaterals

Circle

Fractions

Ratio & Proportion

Equations & Inequalities

DST

Linear Patterns 1 2

Pythagoras 1 2

Level F exam type question