CHAPTER 1 IMPORTANT POINTS - Number
There are 9 different types of numbers you may come
across. Write them down.
Know your Squares, Square roots (up to the first 15), Cubes &
cube roots (up to the first 12) and triangle numbers (first 10)
Remember how to use Factor Trees to find the HCF & LCM
Remember to how to work with directed numbers e.g.
−14 − 4 =
What are the tests for divisibility?
Order of Operations
Remember how to round numbers, when the answer is not exact
or the an exact answer is not asked for.
Money rounds to _____ dp, angles round to _____ dp & all else if
you are not told to round to ______ sf.
These are the types of numbers you will come across
The Universal set of all the numbers we use is the set of Real Numbers (ℝ)
Inside the Universal set there are these numbers –
Natural Numbers (ℕ) 1, 2, 3, 4, 5, …, ∞
Integers (ℤ) are 0, 1, 2, 3, 4, 5, …, ∞ and are -1, -2, -3, -4, -5, …, −∞ but to distinguish them we use (ℤ− ) for Negative Integers
Rational Numbers (ℚ) can be expresses as a fraction
𝑎
𝑏
Odd Number
Even Numbers
Prime numbers
Square Numbers
Cube numbers
Irrational Numbers cannot be expressed as a fraction
Triangle numbers
Here are 36 numbers. Your task is to use the put the numbers
in a table under the correct headings.
3
-6, -5, -4, -3, -2 -1, 0, 0. 6, 1, 2, 3, 𝜋, 4, 5, 192, 6, 7, 8, 9,
3
111, 11, 12, 13, 14, 15.098, 17, 19, 23, 552, 25, 17576,
27, 36, 49, 64, 125
64
3
19, 23, 552, 25, 17576, 36
3
27, 36, 49, 64, 125
0
Irrational
𝜋
Triangle
11, 12, 13, 14, 15.098, 17,
Cube
0. 6
Square
111
Prime
192, 6, 7, 8, 9, 111,
Even
3
17576
Odd
3
Rational
-6
Negative
integer
0. 6, 1, 2, 3, 𝜋, 4, 5,
Positive
integer
Natural
Here are 36 numbers. Your task is to use the put
the numbers in a table under the correct
headings.
-6, -5, -4, -3, -2 -1, 0,
64
36
0
Irrational
𝜋
Triangle
0. 6
Rational
111
Negative
integer
3
17576
3
Positive
integer
Natural
-6
Square Cube
Prime
Even
Odd
CHAPTER 2 IMPORTANT POINTS – Making Sense of Algebra
Remember, by convention, when writing an expression write the
terms in order of powers, highest power first.
All variables a in alphabetical order.
The expression is simplified by adding & subtracting like terms or
simplifying between numerator and denominator.
Write out and understand the the 6 index laws
12×𝑎×𝑏
3𝑎2 𝑏3
Write the terms simply e.g. 16 + 4𝑎𝑏2 =
Rearrange this expression to suit the above convention.
𝑥
𝑥 2 − 5𝑥 4 + 𝑥 − 5𝑦𝑥𝑝 + 16 − 12𝑝𝑥𝑦 + 17𝑥 5 + 6 − 𝑑𝑓𝑤𝑝𝑘 − 4
Forming Equations – Exs 2.1
Substitution – Exs 2.2
Simplifying Expressions – Exs 2.3
Multiplying and Dividing Expressions – Exs 2.4
Expanding Brackets – Exs 2.5 to Exs 2.6
Expand & simplify challenge
1
5𝑥 2 −4𝑥 + 5𝑦 − 2𝑦 3𝑦 2 − 2
𝑥
−1
2
5𝑦 2 𝑥 4
0
2
2
−
× 7𝑥𝑦 − 5𝑥 𝑦 + 5 −𝑥 8 3 𝑥
𝑥𝑦 2
Then calculate a solution if 𝑥 = 2 & 𝑦 = 3
38
Expand & simplify
1
2
2
5𝑥 −4𝑥 + 5𝑦 − 2𝑦 3𝑦 − 2
𝑥
−1
2
5𝑦 2 𝑥 4
0
2
2
3𝑥
−
×
7𝑥𝑦
−
5𝑥
𝑦
+
5
−𝑥
8
𝑥𝑦 2
Then calculate a solution if 𝑥 = 2 & 𝑦 = −3
−6𝑦 3 + 2𝑥 2 𝑦 − 25𝑥 2 if 𝑥 = 2 & 𝑦 = −3 ⇒ −6𝑦 3 + 2𝑥 2 𝑦 − 25𝑥 2 = 38
CHAPTER 3 IMPORTANT POINTS -angles Complementary
Know your angle terms and sum of angles. Give examples of
each –
Acute
Supplementary
Right-angle
Angles in isosceles triangles
Obtuse
Angles in equilateral triangles
Straight angle
Angles in quadrilaterals
Reflex
Angles in regular polygons
Revolution (angles about a point)
Sum of interior angles
Vertically opposite
Sum of exterior angles
Alternate
Exterior angle of a triangle = _________________________
Co-interior
Know how to find an exterior angle of a regular polygon given
the number of sides and vice versa.
Corresponding
Transversal
Angles in triangles
Know how to find an interior angle of a regular polygon given
the number of sides & vice versa.
In the diagram 𝐴𝐷, 𝐸𝐹, 𝐺𝐻 are parallel and
𝐵𝐶, 𝐼𝐷, 𝐽𝐹are parallel. 𝐹𝐽𝐾 is a straight line.
1. In terms of 𝑥 write down the size of angle
𝑞. What property did you use?
2. In the diagram the lines 𝐴𝐵𝐶 represent one
vertex of a regular polygon. Calculate the
exterior angle of the polygon (hint calculate
𝑥 - use the properties of angles around a
point).
3. Calculate the number of sides in the
polygon.
4. Calculate the sum of the interior angles –
use two methods.
5. What is the name of angles 𝑗 & 𝑞?
6. Calculate the size of angles 𝑦 & 𝑧.
7. What is the name for angles 𝑘 & 𝑝? What
do they add to?
8. What is the name for angles 𝑗 & 𝑝?
9. Name two supplementary angles.
10. What three angles form a right angle?
𝑝
𝑗
𝑘
𝑞
14𝑥
4𝑥
3𝑥
3𝑥
2𝑥
𝑧
𝑦
CHAPTER 4 IMPORTANT POINTS – Representing Data
Remember all the vocabulary in your booklets – especially
the types of data. (Exs 4.2)
Understand how to use and interpret frequency tables
discrete and grouped data. (Exs 4.3)
Understand and interpret Stem & Leaf Plots. Remember the
Leaf is the largest digit and the should be in numerical
order. The rest of the digit(s) are the stem. Only do a back to
back stem & leaf plot if you are comparing 2 sets of data.
Always have a key/legend and title. (Exs 4.4)
Two-Way Tables are for reading data in two directions.
Always check the totals add in both directions. (Exs 4.5 &
4.6)
Other ways to display data are through Pictograms (ensure
you have a legend saying what each picture represents) (Exs
4.7); bar charts with gaps for discrete data; compound bar
charts where to compare two or more sets of data for each
discrete variables. With bar charts or any graphs have a title
and label the axes. (Exs 4.8)
Pie charts – remember to find the angle to be measured in the pie
𝑑𝑎𝑡𝑎 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
chart = 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 × 360°. Know how to go backwards from
the pie chart to find the frequency. (Exs 4.9)
If data changes with time you can use a line graph to represent
the data.
Know which chart to choose (Exs 4.10)
10B1 Grades
19, 35, 48, 55, 59, 63, 63, 64, 65, 66, 67, 70, 71, 74, 78, 78, 79, 80, 81, 82, 83, 88, 90, 91, 92, 93, 93, 97, 98, 100
10B2 Grades
22, 34, 40, 45, 58, 59, 60, 62, 64, 65, 66, 68, 69, 77, 78, 79, 79, 80, 80, 82, 84, 85, 86, 87, 90, 90, 91, 98, 100, 100
Above is some data for the final exam for two classes from another school.
1. Calculate the mean, median, mode and range and compare the two schools.
2. Represent the data in a back to back stem and leaf plot – discuss the what the Stem & Leaf plot shows
3. Fill in the grouped frequency table
0 ≤ 𝑔 < 20
20 ≤ 𝑔 < 40
40 ≤ 𝑔 < 60
60 ≤ 𝑔 < 80
80 ≤ 𝑔 < 100
10B1
1
1
3
12
13
10B2
0
2
4
11
13
Exam Grade
1. Calculate an estimate of the mean, and range.
2. In which interval does the median and mode lie.
3. Compare the two classes.
4. Draw a compound bar chart
5. Draw a cumulative frequency graph and calculate 𝑄1 , 𝑄2 , 𝑄3 , 𝐼𝑄𝑅.
6. Write down the 5 – figure summary
7. Draw a box and whisker plot for each class and write a least three statements about what you notice.
8. Draw a pie chart for each class.
9. Draw a pictogram for each class.
10. Combine the two intervals 0 ≤ 𝑔 < 20 and 20 ≤ 𝑔 < 40 into 0 ≤ 𝑔 < 40 and draw a histogram.
CHAPTER 5 IMPORTANT POINTS – Fractions & Standard Form
FRACTIONS
Remember how to –
Cancel down fractions (Exs 5.1)
Add and subtract fractions by finding a common
denominator (Exs 5.3)
Multiply fractions by multiplying the numerators together
and multiplying the denominators together. Don’t forget to
simplify whenever you can. Remember of means
multiplication(Exs 5.2)
Dividing fractions – the first fraction stays the same, the
division becomes multiplication and the fraction after that
you write down it’s reciprocal. Now the division of fractions
is a multiplication of fractions. (Exs 5.4)
When dividing where there is a decimal in the numerator
and/or the denominator, multiply the numerator by a
power of 10 so the decimal is a whole number. Then carry
out the division. (Exs 5.5)
Know how to find fractions of a quantity. (Exs 5.6)
PERCENTAGES
Remember percent means per _____
Percentages an be expressed as fractions and decimals and
fractions and decimals can be expressed as percentages. Know
how to do this (Exs 5.7)
Know how to express two quantities as a percentage (Exs 5.8)
If a quantity increases or a quantity decreases, know how to
calculate by what percentage has this quantity increased or
decreased by. (Exs 5.9)
Know how to increase a quantity by a given percentage and how
to decrease a quantity by a given percentage (Exs 5.10)
Know how to calculate the multiplier. E.g. if an item is increased
by 13% then the multiplier is 1.13 (100%+13%=113%=1.13).
If an item is decreased by 13%, then the multiplier is 0.87 – by
doing what?
Reverse percentages – to find the original amount, divide by the
multiplier. (Exs 5.11)
Try this “probably shouldn’t exist” fraction, percentage and decimal question (without a calculator of course)–
3
2 1 5
1
15
3
×4 + − ×2 ÷
+ 𝑜𝑓10 − 10% + 0.64 ÷ 8% 𝑜𝑓 2 − (1 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑏𝑦 90%)
8
3 3 7
3 0.75 5
Try this “probably shouldn’t exist” fraction, percentage and decimal question (without a
calculator of course)–
3
2 1 5
1
15
3
×4 + − ×2 ÷
+ 𝑜𝑓10 − 10% + 0.64 ÷ 8% 𝑜𝑓 2 − (1 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑏𝑦 90%)
8
3 3 7
3 0.75 5
Georgiou buys a Porsche for £38,000 plus VAT of 17.5%. Calculate the total cost of the car.
As soon as the Porsche leaves the salesroom it loses value. If he was to sell the car he loses the VAT and other
government charges for imported cars, which amount to 5.5% of the list price. Calculate the value of the car.
Unfortunately on the first day a stone flicks up off the road and damages the car. The car’s value immediately goes
down by £132. Calculate the % loss on the car’s value.
Georgiou buys a spoiler for the back of the car for £672, which has been discount by 7%. Calculate the original price
of the spoiler.
Georgiou's car is now valued at?
Each year the Porsche depreciates by 3.6%. What is the car’s value after 10 years (based on the above value).
Express the value of the car as a % of the original price Georgiou bough the car for.
In what year will the value of the car be £2000? (I might be able to afford it then).
Georgiou keeps the car in excellent for 50 years and it becomes a collectors item. He sells it for £50000. Calculate the
% profit on the original price of the car.
However due to inflation £50000 is only worth £31000. Calculate the average inflation rate over the last 50 years?
Georgiou buys a Porsche for £38,000 plus VAT of 17.5%. Calculate the total cost of the car.
As soon as the Porsche leaves the salesroom it loses value. If he was to sell the car he loses the VAT and other
government charges for imported cars, which amount to 5.5% of the list price. Calculate the value of the car.
Unfortunately on the first day a stone flicks up off the road and damages the car. The car’s value immediately goes
down by £132. Calculate the % loss on the car’s value.
Georgiou buys a spoiler for the back of the car for £672, which has been discount by 7%. Calculate the original price
of the spoiler.
Georgiou's car is now valued at?
Each year the Porsche depreciates by 3.6%. What is the car’s value after 10 years (based on the above value).
Express the value of the car as a % of the original price Georgiou bough the car for.
In what year will the value of the car be £2000? (I might be able to afford it then).
Georgiou keeps the car in excellent for 50 years and it becomes a collectors item. He sells it for £50000. Calculate the
% profit on the original price of the car.
However due to inflation £50000 is only worth £31000. Calculate the average inflation rate over the last 50 years?
CHAPTER 5 IMPORTANT POINTS – Fractions & Standard Form continued
STANDARD FORM
Any number whether very small or very large and
everything in between can be expressed in standard form.
Convert the ordinary number into standard form 0.005082 = 5.083 × 10−3 (Exs 5.13)
50820000 = 5.082 × 107 (Exs 5.12)
Understand when the power should be negative or positive.
Note there is only on digit to the left of the decimal point
and the number is multiplied by a power of 10. The number
left of the digit can only be the numbers 1 to 9, never 0. This
is called standard form (or scientific notation).
Also know how to add, subtract, multiply and divide
numbers in Standard form. (Exs 5.12 & Exs 5.13)
Know how to type in and read answers in standard form on
your calculator.
ESTIMATION
You estimate an answer to ascertain if what you have calculated
makes sense.
To estimate a calculation round all numbers in the calculation to
one significant figure and then mentally work out your estimation.
The idea is to make the calculation easy to calculate, but
sometimes there maybe no need to round to one significant
figure e.g. 0.24 ÷ 0.835 Rounding to 1sf is 0.2 ÷ 0.8 = which is
harder to do in your head than say 0.24 ÷ 0.8 = 0.03
(Exs 5.14)
CHAPTER 5 IMPORTANT POINTS – Fractions & Standard Form continued
STANDARD FORM
Distance of planets from the sun –
Mercury 57.9 million km
Jupiter 778.5 million km
Venus
108.2 million km
Saturn 1.434 billion km
Earth
149.6 million km
Uranus 2.871 billion km
Mars
227.9 million km
Neptune 4.495 billion km
A space traveller wants to visit all the known planets from
Mercury to Neptune, starting from Earth. Using Standard
form 1. Calculate the total distance from Earth to Mercury and
back.
2. If the space traveller completed a rectangular trip from
Earth to Mercury where the width of the rectangle is 1.25 ×
104 𝑘𝑚 wide, calculate the area the inside this rectangle.
3. Calculate how many times further Neptune is from the
sun than Earth?
4. Calculate the distance from Venus to Saturn
5. Calculate the total distance travelled from Earth to
Mercury to Neptune and back to Earth.
6. The distance across a human cells is 100𝜇𝑚. Calculate
how may times larger the distance is from Earth to Neptune
than across a cell.
ESTIMATION
Use your calculator to determine the answer to the expression
below.
Then use estimation (by rounding to 1sf or use a more suitable
rounding) to check your calculator rounding. Your answers should
be quite close. If not find out where you went wrong.
3
15.8 × 3.894 − 1 9.5042 × 8.02
30.194 − 20.739
−
+
5.0872
8.763 − 4.092
1÷9
CHAPTER 6 IMPORTANT POINTS – Equations & Rearranging Formula
Know how to, understand how to and remember how to
expand brackets with negatives outside of them. (Exs 6.1)
Remember when solving linear equations rearrange the
terms so all the unknowns are on one side and all the
numbers are on the other side of the equals sign.
What does linear mean? What is the highest power in a
linear function?
When moving from one side of the equals sign to the other
side remember to do the inverse operations. What is the
3
2
3
inverse operation of +, −, ×, ÷,
,
,
,
?
(Exs 6.2)
Factoring expressions – remember to find the highest
common factor of the numbers and the letters. (Exs 6.3)
Rearranging Formula - Remember all the terms with the
variable you are trying to isolate go on one side of the
equation and all other terms go on the other side of the
equation until you have isolated the variable/letter you are
required to make the subject. (Exs 6.4)
1. Rearrange this crazy expression to make 𝑘 the subject
𝑝𝑞 =
1
15 − 8 𝑑𝑘 3 + 5
3
𝑘 𝑎𝑏 2 − 6
2. I think of a number, I double it, I add 𝑝, I halve it, and finally I
subtract the number I thought of and the answer is 5. Solve to
find the value of 𝑝.
3. Solve for 𝑥
4. Solve for 𝑥
1
× 3𝑥 + 3 = 2𝑥 − 6 3𝑥 − 4
2
216
1
𝑥−3
3
= 1296− 𝑥−8
5. Factorise 3𝑎4 𝑏2 − 3𝑎3 𝑏 + 6𝑎2 𝑏 − 9𝑎𝑏 3 + 9𝑎3 𝑏2 − 12𝑎2 𝑏
6. Factorise and simplify
𝑥 2 +𝑥−12
𝑥 2 +4𝑥
6𝑥 2 −17𝑥+12
𝑥 2 +3𝑥
× 3𝑥 3−4𝑥 2 × 𝑥 2−9
CHAPTER 6 IMPORTANT POINTS – Equations & Rearranging Formula
Know how to, understand how to and remember how to
expand brackets with negatives outside of them. (Exs 6.1)
Remember when solving linear equations rearrange the
terms so all the unknowns are on one side and all the
numbers are on the other side of the equals sign.
What does linear mean? What is the highest power in a
linear function?
When moving from one side of the equals sign to the other
side remember to do the inverse operations. What is the
3
2
3
inverse operation of +, −, ×, ÷,
,
,
,
?
(Exs 6.2)
Factoring expressions – remember to find the highest
common factor of the numbers and the letters. (Exs 6.3)
Rearranging Formula - Remember all the terms with the
variable you are trying to isolate go on one side of the
equation and all other terms go on the other side of the
equation until you have isolated the variable/letter you are
required to make the subject. (Exs 6.4)
1. Rearrange this crazy expression to make 𝑘 the subject
𝑝𝑞 =
𝑘=
−25+6𝑝𝑞 3
or
𝑎𝑏 2 𝑝𝑞+8𝑑
1
15 − 8 𝑑𝑘 3 + 5
3
𝑘=
𝑘 𝑎𝑏 2 − 6
3
25−6𝑝𝑞
𝑎𝑏 2 𝑝𝑞−8𝑑
2. I think of a number, I double it, I add 𝑝, I halve it, and finally I
subtract the number I thought of and the answer is 5. Solve to
find the value of 𝑝. 𝑝 = 10
3. Solve for 𝑥
4. Solve for 𝑥
1
× 3𝑥 + 5 = 2𝑥 − 6 3𝑥 − 4
2
216
1
𝑥−3
3
= 1296− 𝑥−8
38
𝑥 = 35
𝑥=7
5. Factorise 3𝑎4 𝑏2 − 3𝑎3 𝑏 + 6𝑎2 𝑏2 − 9𝑎3 𝑏3 + 9𝑎3 𝑏2 − 3𝑎2 𝑏
3𝑎2 𝑏 𝑎2 𝑏 − 𝑎 + 2𝑏 − 3𝑎𝑏2 + 3𝑎𝑏 − 1
6. Factorise and simplify
2𝑥−3
2𝑥 2
𝑥 2 +𝑥−12
𝑥 2 +4𝑥
6𝑥 2 −17𝑥+12
𝑥 2 +3𝑥
× 3𝑥 3−4𝑥 2 × 2𝑥 2−18
CHAPTER 7 IMPORTANT POINTS – Perimeter, Area & Volume
Perimeter is –
The metric units are –
Area is –
The metric units are –
Volume is –
The metric units we use are –
Capacity is –
The metric units are – 𝑚𝑙, 𝑙𝑖𝑡𝑟𝑒𝑠,
1𝑐𝑚3 = 1𝑚𝑙 and 1000𝑐𝑚3 = 1𝑙𝑖𝑡𝑟𝑒 and 1𝑚3 = 1000 𝑙
A circle is – a locus of points equidistant from a single point
Circumference is – the distance around the circle
An arc is – part of a circumference
A sector is –
A segment is A vertex is –
An edge or side is –
A face is –
Surface area is –
Curved surface is –
A compound shapes is –
A net is –
Know how to draw accurate nets of solid (3D) shapes and
use this to find the surface area of said shapes
Write down, understand and remember the area formula for –
Rectangle
Square
Triangles
Parallelograms
Rhombus
Trapezium
Circles
Sectors
Compound Shapes
Write down, understand and remember how to find perimeters
for –
All the above shapes
Remember the perimeter of a circle is called a circumference.
Write down, understand and remember the formulas for the
curved surface area, the total surface area and volume of –
Spheres
Cones
Pyramids
Cylinders
Remember these important points about the relationship between sectors and cones –
A sector can be folded into a shape of a cone.
The arc of the sector becomes the circumference of the base of the cone.
The area of the sector is the area of the curved surface of the cone.
The radius of the sector becomes the slant length of the cone.
Which shape has the least surface area – the cone, sphere or square based pyramid? Show all your working.
Use exact values or long calculator values for all calculations. Round at the end.
A sector, made from tin, has a radius of 6cm and an area of 7.2𝜋 𝑐𝑚2 , calculate the angle of the sector.
𝜃
6cm
The sector is formed into a cone. Gold is poured into the cone. Calculate the
volume of gold in the cone.
The gold cone is sold. The person who buys the cone doesn’t like pointy things and so melts it into a spherical shape.
Calculate the radius of the sphere.
After sometime the gold sphere is sold to an Egyptian. Being an Egyptian and a lover of pyramids, the sphere is melted and
formed into a square based pyramid. The pyramid is the same height as the cone. Calculate the dimensions of the base of
the pyramid.
The gold square based pyramid is finally sold to the bank who melt it into a standard gold bar shape – a trapezium prism
(well actually gold bars are not true trapezium prisms, but for this exercise we will say they are).
Calculate the value of 𝑎
a 𝑐𝑚
1 𝑐𝑚
4 𝑐𝑚
2.5 𝑐𝑚
CHAPTER 8 IMPORTANT POINTS – Introduction to Probability
Make sure you understand the key words in your Vocabulary
Booklet.
Theoretical Probability =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑖𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
e.g. A coin has two possible outcomes (Head or Tail). So
1
probability of a Head = P(H)= 2 one represents the 1 Head & 2
represents the two possible outcomes.
The probability means in theory we expect one Head for every
2 trials (trials in this case is the number of times the coin is
flipped).
Possibility Diagrams or Sample Spaces help you to determine all
the possible outcomes of an investigation. (Exs 8.2)
Independent Events – are events where one outcome does not
influence another outcome. E.g. if you flip a coin twice, the
outcome from the first throw does not affect the outcome on the
second throw.
This is called a combination of events –
𝑃(event 𝐴 occurs and then event 𝐵 occurs = 𝑃(𝐴&𝐵)
= 𝑃 𝐴 ×𝑃 𝐵
Mutually Exclusive Events – if two events cannot occur at the
same time then they are mutually exclusive. E.g. there are red,
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠
Experimental Probability =
This is also
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
blue and green coloured disks in a bag. If you draw a disk out of
referred to as Relative Frequency – the frequency of something the bag you can only draw out one colour. You can’t draw 2 or 3
occurring with respect to the number of attempts.
colours out if you are only drawing one colour at a time.
We do experiments if we are unable to show the probability
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
theoretically.
Note if the question asks for event A & B (combination of events)
then multiply the probabilities. If the question asks for one event
All probabilities of an investigation add to 1. If 𝐴 is the event of
or another event, then add the probabilities.
a possible occurrence (e.g. a 5 on the roll of a dice, raining
today), then 𝐴’ is the complement of event 𝐴 i.e. not 𝐴 (e.g. not (Exs 8.3)
a 5 on the roll of a dice, not raining today). 𝑃 𝐴′ = 1 − 𝑃 𝐴 .
(Exs 8.1)
Here are 36 numbers. Your task is to use the information about numbers from Chapter 1 (especially the table you
completed for these numbers) and answer the following probability questions.
3
3
-6, -5, -4, -3, -2 -1, 0, 0. 6, 1, 2, 3, 𝜋, 4, 5, 192, 6, 7, 8, 9, 111, 11, 12, 13, 14, 15.098, 17, 19, 23, 552, 25, 17576, 27,
36, 49, 64, 125
If a number is chosen at random calculate the following probabilities –
1.
2.
3.
4.
5.
6.
7.
8.
9.
P(it is a rational number)
P(it is a square number)
P(it is not prime)
P( it is prime and also a square number)
P(it is a cube number and also a square number)
P(it is 15)
P(its even and is not in any other category)
P(it is a part of the Fibonacci Sequence)
P(it is a triangle number)
If two numbers are chosen at random calculate the following probabilities –
1.
2.
3.
4.
5.
6.
P(odd, even)
P(irrational, irrational)
P(recurring, negative)
P(not prime, prime)
P(square, cube)
P(at least one number is irrational)
Here are 36 numbers. Your task is to use the information about numbers from Chapter 1 (especially the table you
completed for these numbers) and answer the following probability questions.
3
3
-6, -5, -4, -3, -2 -1, 0, 0. 6, 1, 2, 3, 𝜋, 4, 5, 192, 6, 7, 8, 9, 111, 11, 12, 13, 14, 15.098, 17, 19, 23, 552, 25, 17576, 27,
36, 49, 64, 125
If a number is chosen at random calculate the following probabilities –
1.
2.
3.
4.
5.
6.
7.
8.
9.
P(it is a rational number)
P(it is a square number)
P(it is not prime)
P( it is prime and also a square number)
P(it is a cube number and also a square number)
P(it is 15)
P(its even and is not in any other category)
P(it is a part of the Fibonacci Sequence)
P(it is a triangle number)
If two numbers are chosen at random calculate the following probabilities –
1.
2.
3.
4.
5.
6.
P(odd, even)
P(irrational, irrational)
P(recurring, negative)
P(not prime, prime)
P(square, cube)
P(at least one number is irrational)
CHAPTER 9 IMPORTANT POINTS – Sequences & Sets
Know and understand all the terms in you Vocabulary Booklet.
In a linear sequence you can find the rule or term-to-term rule
(whatever you are adding or subtracting to each term to
continue the sequence). The rule is the difference between
each term and in a linear sequence it is constant. (Exs 9.1)
The position-to-term rule finds the value of the term at any
position. It is also known as the 𝑛𝑡ℎ term (𝑡𝑛 ), formula or
equation of the sequence. (Exs 9.2)
Sets are a list or collection of objects which share the same
characteristics. A set is represented by {} and its members or elements
are inside these curly brackets.
Sets are usually given letters e.g. Set 𝐴 is the set of even numbers
between 1 & 10 𝐴 = 2, 4, 6, 8 or it can be written in set builder
notation
𝐴 = {𝑥: 𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 1 & 10}
Set 𝐴 is all values of 𝑥 such that 𝑥 is an even number between 1 & 10.
(Exs 9.10)
You must understand and be able to use these symbols and notation –
∅
If you are given a pattern and you have to find the rule or 𝑛𝑡ℎ
∈
term, always put the information to help you. (Exs 9.3)
∉
𝑛(𝐴) (Exs 9.7)
The 𝑛𝑡ℎ term can also be written like this 𝑢𝑛 (Exs 9.4)
𝑈 or ξ
3
Rational Numbers can be written as a fraction e.g. 3 =
0.5 = 𝐴′
1
∩
1
1
0. 3 = 3. 0. 3 is a recurring decimal. You must be able to show ∪
2
algebraically a recurring decimal equals a fraction.
⊂
⊄
Irrational Numbers are numbers where the decimal part never
⊆
⊈ (Exs 9.8)
repeats e.g. 𝜋, 3 (Exs 9.5)
CHAPTER 9 IMPORTANT POINTS – Sequences & Sets continued
Venn Diagrams – When drawing a Venn Diagram, always,
always draw a rectangle and put this symbol 𝑈 or this
symbol ξ on the outside of it to represent the universal set.
You must be able to shade and/or recognise disjoint sets
unions, intersections, complements and combinations of
these. Recall the little trick I showed you it identify the area
you have to shade.
(Exs 9.9)
Here are 36 numbers. Your task is to use the information above to represent these numbers in a Venn diagram.
3
3
-6, -5, -4, -3, -2 -1, 0, 0. 6, 1, 2, 3, 𝜋, 4, 5, 192, 6, 7, 8, 9, 111, 11, 12, 13, 14, 15.098, 17, 19, 23, 552, 25, 17576, 27, 36,
49, 64, 125
Answers
Answers
U - Real Numbers
Natural Numbers
Primes
Odd
Square
Even
Irrational Numbers
Answers
U - Real Numbers
Natural Numbers
Primes
Odd
Square
Even
Irrational Numbers
CHAPTER 10 IMPORTANT POINTS – Straight Lines & Quadratics continued
Straight Lines
𝑦 = 𝑚𝑥 + 𝑐
𝑚 is
𝑐 is
Know how to determine 𝑚 & 𝑐 from an equation and a graph.
(Exs 10.4)
Coordinates of the origin is
What is another name of a straight line?
Parallel lines have the same ___________
𝑥 = 4 is parallel to what axes? (Exs 10.2)
𝑦 = −4 is parallel to what access? (Exs 10.2)
Positive gradients slope up to the ______
Negative gradients slope down to the _______
Two gradients are perpendicular if 𝑚1 × 𝑚2 = −1 (Exs 10.5)
Given one gradient, know how to calculate the gradient of a line
perpendicular to it. (Exs 10.5)
Know how to calculate the gradient from a straight line on a
graph (Exs 10.4)
Know how to calculate the gradient of the line from two points
(Exs 10.4)
Know how to calculate the midpoint of a straight line segment
(Exs 10.7)
Know how to calculate the length of a line segment (Exs 10.7)
Know how to determine the equation of a line from a straight
line on the graph (Exs 10.3)
Know how to determine the equation of a line from two points (Exs
10.4)
To sketch linear graphs find the 𝑥-intercepts & 𝑦-intercepts. To find
𝑥-intercept ___= 0.
To calculate 𝑦-intercept 𝑥 =_____.
(Exs 10.6)
CHAPTER 10 IMPORTANT POINTS – Straight Lines & Quadratics continued
Quadratics
What is a quadratic expression
Know how to expand two or more sets of brackets
Factorise these expressions
2𝑥 2 + 21𝑥 + 27
(Exs 10.8)
Remember (𝑥 − 𝑎) 2 represents two brackets are multiplied
together 𝑥 − 𝑎 𝑥 − 𝑎 (Exs10.9)
7𝑥 2 − 21𝑥 − 6
Know how to factorise quadratics (when the squared term has a
coefficient of one and greater than one) and how to check your 4𝑥 2 + 8𝑥 + 3
answer using the calculator. If they don’t match find out where
you went wrong. (Exs 10.10 only has coefficient of one on the
squared term. On the right are questions where the coefficient
is not one)
6𝑥 2 + 13𝑥 + 6
Know how to factorise quadratics which are the difference of
two squares 𝑎2 − 𝑏 2 = 𝑎 − 𝑏 𝑎 + 𝑏 (Exs 10.11)
9𝑥 2 + 6𝑥 − 8
Solving Quadratics means you are finding solutions, 𝑥intercepts, the zeros, roots, solving for the variable. The
equation always equals zero i.e 𝑦 = 0 (Exs 10.12)
CHAPTER 11 IMPORTANT POINTS – Pythagoras’ Theorem & Similar Shapes
Again ensure you understand all the terms in your vocabulary
booklet.
Pythagoras’ Theorem is a formula which shows the relationship
between three sides of a right angled triangle.
What is this relationship?
If you need to calculate the long side (hypotenuse) of a right
angled triangle you add the squares of the other two sides.
To calculate a short side of a right angled triangle subtract the
square of the other two sides.
You can use Pythagoras’ Theorem to check if a triangle is a tight
angled triangle. Check to see if a triangle with side lengths 11,
12, 13 is a right angled triangle.
(Exs 11.1 & 11.2)
Similar Shapes – Two objects are mathematically similar if the
have the same shape (angles) and proportions (lengths are
increased or decreased by the same multiple/scale factor).
In mathematically similar shapes the ratio of all corresponding
sides are the same. In triangles, corresponding sides have the
same angle on each end.
These ratios can be used to find missing side lengths.
(Exs 11.3 & 11.4)
If the scale factor between mathematically similar shapes is 𝑝 and the
area of one shape is 𝐴1 , then the scale factor for area is 𝑝2 and the
area of the other shape (𝐴2 ) is 𝐴2 = 𝑝2 × 𝐴1
𝐴𝑟𝑒𝑎 1
Note - 𝐴𝑟𝑒𝑎 2 gives the area scale factor. The length scale factor is
𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 (Exs 11.5)
If the scale factor between mathematically similar shapes is 𝑝 and the
volume of one shape is 𝑉1 , then the scale factor for volume is 𝑝3 and
the volume of the other shape (𝑉2 ) is 𝑉2 = 𝑝3 × 𝑉1
𝑉𝑜𝑙𝑢𝑚𝑒 1
Note - 𝑉𝑜𝑙𝑢𝑚𝑒 2 gives the volume scale factor. The length scale factor is
3
𝑡ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 (Exs 11.6)
Congruency – Objects, shapes, figures are congruent if all side lengths
and angles are equal. If objects are congruent we can find missing
angles or side lengths. In triangles the tests for congruency are SSS,
SAS, ASA, RHS. (Exs 11.7 & 11.8)
CHAPTER 12 IMPORTANT POINTS – Averages & Measures of Spread
Understand all the vocabulary in your vocabulary booklet.
There are 3 averages (measures of central tendency) –
mean, median, mode. Know how to find these from lists
(Exs 12.1), frequency tables (Exs 12.3) and grouped
frequency tables (Exs 12.4 remember to find the midpoint
of the class interval). Understand the influence extreme
values have on the mean. (Exs 12.1)
There are several measures of spread (range & InterQuarile
Range IQR). It is a measure of how consistent the data is.
The smaller the range or IQR, the consistent the data is.
Using the averages and range is useful for making
comparisons between sets of data. (Exs 12.2)
Percentiles refer to what data is above or below a certain
percentage. There are 3 common ones we use – Lower
Quartile (𝑄1 , 25%), Median (𝑄2 50%) and Upper Quartile
(𝑄3 75%). Know how to find quartiles from lists, and Stem &
Leaf plots. (Exs 12.5)
Box & Whisker Plots – Know how to draw and interpret box and
whisker plots.
What is the 5 figure summary?
CHAPTER 13 IMPORTANT POINTS – Understanding Measurement
Know how to convert between metric measurements for
length, mass and capacity.
Know how to convert between different units for area (e.g. 𝑐𝑚2
to 𝑚𝑚2 and between different units for volume (𝑐𝑚3 to 𝑚𝑚3 .
(Exs 13.1)
Lets say we know a quantity has been rounded to the nearest
hundredth and it is 0.32. Take half of what the quantity has been
rounded to. Half of 0.01 is 0.005. So the –
Lower Bound is 0.32 − 0.005 = 0.315
Upper Bound is 0.32 + 0.005 < 0.325
Be able to work with differences in time, speed, distance, time,
and read timetables. (Exs 23.2 & 13.3)
Lets say we know a quantity has been rounded to the nearest
thousandth and it is 0.960. Take half of what the quantity has
been rounded to. Half of 0.001 is 0.0005. So the –
Lower Bound is 0.960 − 0.0005 = 0.9595
Upper Bound is 0.960 + 0.0005 < 0.9605
(Exs 13.4 & 13.5)
Lower Bounds and Upper Bounds – If we know a number has
been rounded to a given place e.g. 1dp, nearest 100, 3sf, then
there are a range of values that could round up to the number
(of which there is a Lower Bound) or round down to that
number (of which there is an Upper Bound).
Lets say we know a quantity (𝑞) has been rounded to the
nearest hundred and it is 300. Take half of what the quantity
has been rounded to. Half of 100 is 50. So the –
Lower Bound is 300 − 50 = 250
Upper Bound is 300 + 50 = 350 If you are doing a calculation
this is the upper bound. If you are writing an inequality use <
e.g. 250 ≤ 𝑞 < 350
Conversion Graphs – Know how to read conversion graphs.
Always, Always understand what each graduation represents
before you read off the values. (Exs 13.6)
Money Exchange – Know how to convert from one currency to
another. (Exs 13.7)
CHAPTER 14 IMPORTANT POINTS – Further Solving of Equations & Inequalities
Simultaneous Equations - Two equations can share a point
or points simultaneously. If the equations were drawing on
the graph the intersection of the two graphs represents the
point they simultaneously share. So solving simultaneous
equations means you are finding the value of these points.
It is finding the value of the variables.
You solve simultaneous equations graphically (Exs 14.1), or
algebraically by substitution or elimination (Exs 14.2).
Linear Inequalities – Inequality means not equal, so these
inequality symbols <, >, ≤, ≥ lets us know there is a range
of values a variable can take. These can be shown on a
number line (Exs 14.3). Remember to shade in the circle
when the variable can equal the quantity and leave the
circle open when the variable does not equal the quantity.
Solving Linear Equations – Solving linear inequalities is like
solving normal linear equations except you are using an
inequality symbol instead of an equals sign.
Remember when rearranging to isolate the variable and you
have to multiply or divide by a negative the reverse the
inequality symbol. (Exs 14.4)
Linear Programming – Start by knowing how to shade an area of
graph that is above or below a linear graph. For instance if you are
given 2𝑥 − 3𝑦 ≥ 12
1) Draw the line on the graph (in this case a solid line since the
inequality is equal to. If the inequality was not equal to it
would be a dotted line.
2) Pick a point above or below the line and substitute into the
inequality. If the answer is ≥ 12 this is the required region, do
not shade. Shade the other region. If the answer is ≤ 12 this is
the unwanted region, so shade.
(Exs 14.5)
The above involves shading for just one linear graph. Next is to
shade a system of graphs. To do this plot one graph at a time and
shade the area which is not required. After doing this for each
linear graph in the system if equations you will have an unshaded
area. This is the required area. (Exs 14.6)
You maybe asked to state what the maximum and minimum
possible values are. (Exs 14.7)
CHAPTER 14 IMPORTANT POINTS – Further Solving of Equations & Inequalities cont
Completing the Square – A quadratic can be written in 3
different ways –
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Standard form
𝑦 = 𝑎(𝑥 + 𝑝)(𝑥 + 𝑞) and 𝑦 = 𝑎(𝑥 + 𝑏)2 factored form.
Why is this factor form? Because each expression is a factor
of 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 = 𝑎(𝑥 + ℎ)2 +𝑘 Vertex Form or Completed Square Form.
Why is it called vertex form? Because you can find the
vertex from this form. 𝑥 ordinate of the vertex is the
negative of whatever you add or subtract in the brackets,
and the 𝑦 ordinate is whatever you add or subtract after the
brackets.
Why is it called completed square form, because you have
to complete the square of 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Ensure you
know how to do this. (Exs 14.8)
Fining 𝑥-intercepts
From the factor form you can solve for 𝑥, by equating the
equation to 0.
From the vertex form you can solve for 𝑥, by equating the
equation to 0.
You can also use the quadratic formula
−𝑏± 𝑏2 −4𝑎𝑐
Quadratic Formula – 𝑥 =
where does , 𝑏 & 𝑐come from?
2𝑎
This is another way to solve for 𝑥. Other ways of saying solve for 𝑥 is –
calculate the solutions, calculate the roots, calculate the zeros,
calculate the 𝑥-intercepts, for what values of 𝑥 does the equation
equal zero.
If a quadratic equals a constant, linear function or a quadratic
function, form a new quadratic equal to zero and solve for 𝑥. You may
have to found the 𝑦 values if you are required to calculate the
intersection of the two graphs.
(Exs 14.9)
Factorising a quadratic when the coefficient on the squared term is
not 1 – know how to do this. (Exs 14.10 & 14.11)
Algebraic Fractions – have variables either in the numerator,
denominator or both.
Simplify the fractions as you would any other fraction by cancelling
own between numerator and denominator. (Exs 14.12)
Multiply and Divide algebraic fractions as you would any other
multiplication or division of fractions. (Exs 14.13)
Add and subtract algebraic fractions as you would any other
multiplication or division of fractions. (Exs 14.14)
CHAPTER 15 IMPORTANT POINTS – Scale Drawings, Bearings & Trigonometry
Scale Drawings – know how to read a map scale or a
drawing scale and find actual lengths from scaled lengths
and scaled lengths from actual lengths. (Exs 15.1)
Drawing a Diagram to Scale – Understand what angle of
elevation and angle of depression are. Given a scale and
measurements know how to draw a diagram to scale.
(Exs 15.2)
Bearings – A bearing is a measurement of an amount
turning from a North Point. They are measured in a
clockwise direction and written as 3 figures. E.g. 038°
means when facing the North Point turn through 38° in a
clockwise direction.
If you are told a person/object travels on a bearing of 238°
from A to B, then the person is standing at A and turns
toward B.
Remember 2 or more North Points from different places are
parallel. So remember the rules for angles in parallel lines –
especially co-interior angles (co-interior angles add to
180°).
(Exs 15.3)
Trigonometric Ratios – sine, cosine & tangent ratios which are
used in right angle triangles.
Remember these formula and know how to rearrange them –
𝑂
𝐴
𝑂
sin 𝜃 =
cos 𝜃 =
tan 𝜃 =
𝐻
𝐻
𝐴
𝑆𝑂𝐻 𝐶𝐴𝐻 𝑇𝑂𝐴
When labelling the sides of a right angled triangle remember
𝑂, 𝐴 & 𝐻 are placed with respect to the angle you are going to
use/find. (Exs 15.4 & 15.5)
To find the angle given two sides take the inverse of the function
𝑂
𝐴
𝑂
−1
−1
−1
𝜃 = sin
𝜃 = cos
𝜃 = tan
𝐻
𝐻
𝐴
(Exs 15.6)
Ensure your calculator is in degrees (D)
Finding missing sides – rearrange the formula to find the missing
sides (Exs 15.7 & 15.8). Remember if you draw a perpendicular
from a point on an isosceles or equilateral triangle to the side
opposite the point the perpendicular bisects the line. Also two
radii and a chord in a circle create an isosceles triangle.
CHAPTER 15 IMPORTANT POINTS – Scale Drawings, Bearings & Trig continued
Sine, Cosines & Tangents of angles greater than 90°
Know how to read angles and sine, cos & tan of angles from their
respective graphs. (Exs 15.9)
Remember sine graph start at (0,0), cosine graphs start at (0,1) and
tangent graphs have asymptotes. Identify the graphs below –
Sine Rule – Know how to apply the sine rule to find missing angles or
sides when you have a non right-angled triangle.
sin 𝐴
sin 𝐵
sin 𝐶
=
=
Use the formula this way up to find a missing
𝑎
𝑏
𝑐
angle. You need to use two at onetime. Substitute in the information
you know.
𝒂
𝒃
𝒄
=
=
Use the formula this way up to find a missing
sin 𝐴
sin 𝐵
sin 𝐶
length. You need to use two at onetime. Substitute in the information
you know. (Exs 15.10)
Remember side a is opposite angle A, b is
opposite angle B and c is opposite angle
C.
Cosine Rule – Use this rule if two sides are given and you have to
find the third side. Or three sides are given and you have to find the
missing angle. The given angle or the required angle must be
between two given sides. E.g. if sides a & b are given then the
required angle or given angle must be at C.
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
(Exs 15.11)
𝑎2 − 𝑏 2 − 𝑐 2
cos 𝐴 =
−2𝑏𝑐
CHAPTER 15 IMPORTANT POINTS – Scale Drawings, Bearings & Trig continued
Area of non right-angled triangles – Apply this formula:
1
𝐴 = 𝑎𝑏 sin 𝐶
2
(Exs 15.12)
Trigonometry and 3D shapes – To solve these problems
identify the triangles you need to find the required angle.
See the example on the right.
First you need triangle ABD to find length BD, then you need
triangle SDB to find <DBS.
(Exs 15.13)
CHAPTER 16 IMPORTANT POINTS – Scatter Diagrams & Correlation
Ensure you understand the terms in your vocabulary booklet.
Scatter Diagrams – These are a comparison of two quantities
(or bivariate data, which means two sets of data vary). E.g.
comparing a person’s height to their hand width. We would
expect as the person grew taller (independent variable) their
hands would be bigger (dependent variable). In other words the
size of someone’s hands depends on how tall they are –this is
the relationship between the two variables.
You maybe asked to write down the relationship between the
two variables.
Another example – if you track a car’s worth over time
(independent variable) you may find it looses value (dependent
variable). In other words the value of the car depends on how
old it is – this is the relationship between the two variables.
Know how to draw and interpret scatter diagrams.
Know the different types of correlation in a scatter diagram
The correlation can be weak or strong. If it weak then say any
predictions maybe unreliable.
Also give answers in context with the data. E.g. There is a strong
negative correlation therefore it is possible to make good
predictions between hot pie sales and temperature.
Know how to draw a line of best fit and interpret the line of best
fit – you can use the line of best fit to predict values. This is
interpolation – using the values between the end points of the
scatter diagram.
Extrapolation, making predictions outside the given data by
extending the line of best fit is risky because we don’t know what
the data is going to do. It may not follow the trend we see on the
scatter diagram. (Exs 16.1)
CHAPTER 17 IMPORTANT POINTS – Managing Money
Ensure you understand the keywords ion your vocabulary
booklet.
When earning money you don’t necessarily get 𝑥 amount of
dollars for the time you worked. There are often deductions
(money taken from your earnings) such as: tax; meals if you
eat the food provided); insurance, superannuation
payments, money saved for your retirement; money owed
from a salary advance. Sometimes along with your salary
you may have other income such as: housing allowance; car
allowance; phone allowance; overtime at time and a half or
double time. (Exs 17.1, 17.2, 17.3)
Borrowing & Investing – When you borrow money you have
to pay interest. When you invest money interest is paid to
you. There are two types of interest – simple & compound.
Simple Interest – This involves adding the interest amount
(𝐼) to the original amount (Principal, 𝑃) at a set interest rate
(R), over a set time period (T). Simple interest is calculated
using the formula below. Know how to rearrange this
formula to find either 𝑃, 𝑅 𝑜𝑟 𝑇.
𝑃𝑅𝑇
𝐼=
100
Add the interest to the principal to calculate your return
(money you get back). (Exs 17.4)
Hire Purchase (HP) – If you can’t afford to pay cash for something
then you maybe able to pay a deposit, take the product and pay
the ret of the money in weekly or monthly instalments. These
instalments include interest because you are borrowing money to
pay for the product. (Exs 17.5)
Compound Interest – earns you more money if you are investing.
If you are borrowing you have to pay more back. The formula
below calculates the amount of return. If you require the interest
received or paid then subtract the principal (𝑃) from the value (𝑉)
of the amount returned for an investment or value (𝑉) paid for
borrowings.
𝑅 𝑛
1 + 100 OR 𝑉 = 𝑃 1 + 𝑟 𝑛
𝑉=𝑃
where 𝑃 is the principal (initial amount), 𝑅 is the interest rate and
𝑛 is the number of time interest is calculated. In the second
formula 𝑟 is the percentage rate expressed as a decimal.
Note the compound interest formula is an exponential formula
because the variable is an exponent (indices). (Exs 17.6)
CHAPTER 17 IMPORTANT POINTS – managing Money continued
Exponential Growth & Decay –When the exponent in a
formula is a variable we have exponential growth or decay
Graphically they look like this -
Algebraically they are expressed as a formula –
For growth: 𝑦 = 𝑎(1 + 𝑟) 𝑛
For decay: 𝑦 = 𝑎(1 − 𝑟)𝑛
where 𝑎 is the original value or principal, 𝑟 is the rate of
change expressed as a decimal and 𝑛 is the number of time
period. (Exs 17.7)
𝑚𝑜𝑛𝑒𝑡𝑟𝑦 𝑝𝑟𝑜𝑓𝑖𝑡
× 100%
𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒
𝑚𝑜𝑛𝑒𝑡𝑎𝑟𝑦 𝑙𝑜𝑠𝑠
Percentage Loss - % 𝑙𝑜𝑠𝑠 = 𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒 × 100%
Percentage Profit - % 𝑝𝑟𝑜𝑓𝑖𝑡 =
(Exs 17.8)
Calculating Selling Price, Cost Price & Markup – We have talked
about multipliers. The multiplier is 100%+% increase expressed as
a decimal. E.g. The % markup on an item is 8%. The multiplier is 100% + 8% = 108% = 1.08 so multiply the cost price by this
multiplier to calculate the selling price.
The % reduction on an item is 12.5%. The multiplier is –
100% − 12.5% = 87.5% = 0.875 so multiply the cost price by
this multiplier to calculate the selling price.
To find the original amount or cost price divide the selling price by
the multiplier. (Exs 17.9)
Discount – Same method as reductions – see above example. Exs
17.10.
CHAPTER 18 IMPORTANT POINTS – Curved Graphs (non-Linear Graphs)
Know the meaning of all the words in your vocabulary
booklet.
Know how fill in a table of values and plot quadratic graphs
(parabolas). (Exs 18.1 & 18.2)
Sketching parabolas – When sketching a parabola there are
some important features you may have to calculate –
Identify the shape of the graph. If the squared term is
positive then it is concave up. If the squared term is
negative then it is concave down.
Axis of symmetry (dotted line on the graph) and its equation
𝑏
𝑥 = − 2𝑎 .
Turning Point (vertex) – since the axis of symmetry passes
through the turning point, then use the 𝑥 value as the 𝑥
ordinate and then find the 𝑦 ordinate.
The vertex can also be found by completing the square (see
the second tile for Chp 14)
𝑥-intercepts – Are found when 𝑦 = 0, in other words the
quadratic equation equal zero. There are 3 ways to do this –
(see the second tile of Chp 14)
𝑦-intercepts – are found when 𝑥 = 0, in other words
substitute zero for 𝑥 and find 𝑦.
Once you have required features sketch a smooth curve and
label these features. (Exs 18.3)
Reciprocal Graphs (hyperbolas) – This is where an equation has
the variable in the denominator. It can either be 𝑥 or an
expression involving 𝑥. When you see an expression involving 𝑥 or
just 𝑥 in the denominator state these cannot equal zero and solve
for what 𝑥 cannot equal.
5
E.g. 𝑦 = 𝑥+2 so write 𝑥 + 2 ≠ 0 ∴ 𝑥 ≠ −2.
Reciprocal graphs have two parts to them, they are symmetrical
and in opposite quadrants, and they approach an asymptote.
What is that again?!
Vertical asymptote is the value 𝑥 cannot equal – in the above
equation it is 𝑥 = −2 and this is the equation of the vertical
asymptote (vertical dotted line).
Vertical asymptote is found by understanding what the graph is
approaching as 𝑥 approaches infinity - in the above equation as 𝑥
gets larger in value then the value of 𝑦 gets closer and closer to
zero and the equation of the asymptote is therefore 𝑦 = 0 (𝑥axis).
Exs 18.4 More notes on
the next tile
CHAPTER 18 IMPORTANT POINTS – Curved Graphs cont.
Using Graphs to Solve Quadratic Equations Exs 18.5
It is like solving simultaneous equations – you are finding out what
𝑥-ordinates two graphs share.
Solving Equations Graphically – generally you are given an
equation e.g. 𝑦 = 𝑥 2 − 2𝑥 − 7 or 𝑥 2 − 2𝑥 − 7 = 0 or
𝑥 2 − 2𝑥 − 7 = 3 or 𝑥 2 − 2𝑥 = 1
For 𝑦 = 𝑥 2 − 2𝑥 − 7 you maybe asked to solve the equation
graphically for 𝑦 = some number. So plot the graph and see
where the line 𝑦 = some number crosses the graph. Read off the
𝑥-intercepts. What 𝑦-ordinates do they share?
For 𝑥 2 − 2𝑥 − 7 = 0 the equation tells you what 𝑦 equals, so do
the same as above.
For 𝑥 2 − 2𝑥 − 7 = 3 the equation tells you what 𝑦 equals so
repeat above. What 𝑦-ordinates do they share?
For 𝑥 2 − 2𝑥 = 1 these types of equations occur when you are
given an equation such as 𝑦 = 𝑥 2 − 2𝑥 − 7 and perhaps the
graph and you have to use the graph to solve 𝑥 2 − 2𝑥 = 1.
Therefore you need to rearrange 𝑥 2 − 2𝑥 = 1 into
𝑥 2 − 2𝑥 − 7 = something. To obtain −7 on the left hand sode
you need to subtract 7 fom both sides –
𝑥 2 − 2𝑥 − 7 = 1 − 7
𝑥 2 − 2𝑥 − 7 = −6.
Now use your graph to solve the
equation. What 𝑦-ordinates do they share?
CHAPTER 18 IMPORTANT POINTS – Curved Graphs cont.
Using Graphs to Solve Linear and Non-linear Simultaneous
Equations, and Two Non-linear Equations. Exs 18.6
It is like solving simultaneous equations – you are finding
out what 𝑥-ordinates and 𝑦-ordinates two graphs share.
Pretty simple really – plot the two graphs and write down
the co-ordinates of their intersection.
Understand what this means - a ≤ 𝑥 ≤ 𝑏
Plotting Cubic Graphs Exs 18.7
Draw up a table of values and plot the pairs of co-ordinates.
Ensure you draw a smooth curve through these points.
Understand the two different shapes of a cubic.
Sketching Cubic Graphs Exs 18.7
These graphs give you
a general idea of the
orientation and shape
of the cubic you are to
sketch. Also find the 𝑦intercept (𝑥 = 0) and
the 𝑥-intercept. Easy if
the cubic is factorised
otherwise use
differentiation. You can
also solve cubic
equations graphically
Combination of Curves Exs 18.8
Use a table of values to plot these complex curves, such as,
3
𝑦 = 𝑥 3 − 2𝑥 2 − 𝑥
You mat be asked to solve these types of equations.
Exponential Graphs Exs 18.9
These graphs have the variable as a power of some number 𝑦 = 𝑎 𝑥 . A positive power represents exponential growth and
a negative power represents exponential decay. Exponential
equations are used to model many real life situations. E.g.
population changes, compound interest.
Know how to find the 𝑦-intercepts and the 𝑥-intercepts.
Gradient of a Curve Exs 18.10
The gradient on a curve is always changing. A tangent to a
point on the curve is used to find the gradient at that point.
Now chose two
easy points as
close to point a
as possible and
calculate the
gradient
CHAPTER 18 IMPORTANT POINTS – Curved Graphs cont.
Differentiation Exs 18.11, 18.12, 18.13, 18.14
Taking the derivative of a function gives you the gradient
function of that function. In other words using the gradient
function you can calculate the gradient at any point on a
curve, or given a gradient calculate the 𝑥-ordinate of the
point where the gradient occurs and hence the tangent to
the curve at that point. Know how to do this.
Remember how to find the derivative of a function.
𝑑𝑦
For 𝑦 = 𝑎𝑥 𝑛 then = 𝑛𝑎𝑥 𝑛−1 which is the gradient
𝑑𝑥
function of 𝑦.
e.g. 𝑦 = 2𝑥 3 + 5.5𝑥 2 − 10𝑥 + 6 Derive the gradient
function and
1) Calculate the gradient of the curve at 𝑥 = 3
𝑑𝑦
2) The gradient of the curve is 𝑑𝑥 = 𝑚 = 7 Derive the
equation of the tangent at this point.
Turning Points Exs 18.15
Turning points occur when the gradient changes direction,
e.g. from negative to positive (this gives a minimum point)
or positive to negative (this gives a maximum point).
At these points the gradient is always zero.
𝑑𝑦
Knowing this then equate 𝑑𝑥 to zero.
For 𝑦 = 2𝑥 3 + 5.5𝑥 2 − 10𝑥 + 6 calculate the coordinates of all
turning points. State whether they are a maximum or a minimum.
We can use he turning points to maximise a situation or minimise
a situation e.g.
CHAPTER 19 IMPORTANT POINTS – Symmetry & Circle Geometry
A chord splits a circle into two
segments – major and minor
segments.
And into two arcs – major and
minor arcs.
The largest chord is the
diameter.
Two radii and a chord
form an isosceles triangle.
If two perpendicular bisectors from the
centre are of equal length, then the
chords are equal.
If two chords are equal then the
perpendicular bisectors form the centre
are equal.
A perpendicular bisector
from the centre of the
circle to the chord bisects
the chord. It is also the
shortest point between
the centre and the chord.
A
A chord and a tangent
meet at right angles
O
Two tangents come from
a point P. The lengths PA
& PB are equal.
P
<APB + <BOA = 1800
Why?
B
A
B
𝑥
O
𝑥
O
𝑥
P
O
2𝑥
𝑥
A
A cyclic quadrilateral has all
its vertices on the
circumference of the circle.
Opposite angles in a cyclic
quadrilateral add to 1800.
O
𝑦
The angles on the
circumference formed by
the red lines and the
green lines, which come
from the same points, A
& B (or arc or chord), are
equal.
In the cyclic quadrilateral one side
is extended to form an external
angle.
Angle 𝑥 is formed by the two red
lines. Angle 𝑦 is formed by the two
green lines. Angle 𝑥 = angle 𝑦.
The exterior angle in a cyclic
quadrilateral is equal to the interior
angle opposite it.
B
The angle on the centre formed by
the green lines and the angle at the
circumference formed by the red lines
come from the same points A & B (or
arc or chord). The angle at the centre
is twice the angle at the
circumference. <AOB = 2(<APB)
The diameter is the largest chord.
An angle formed by drawing lines
from the end points of the diameter
to a point P on the circumference is a
right angle (900).
O
𝑦
O
𝑥
Angle 𝑥 is formed by the two red
lines (a tangent and a chord). Angle
𝑦 is formed by the two green
chords. Angle 𝑥 =angle 𝑦.
Alternate segment theorem –
Angles between a tangent and a
chord equal the angle in the
alternate segment.
CHAPTER 20 IMPORTANT POINTS – Histograms & Cumulative Frequency Graphs
Histograms represent continuous data. What you collected
your data on is always on the horizontal axis. There are two
types of Histograms –
Histograms with equal intervals (vertical axis is frequency) &
Histograms with unequal intervals (vertical axis is frequency
density).
From a table of values you have to find frequency Density, the
formula found by rearranging f = cw × 𝑓𝑑 therefore 𝑓
𝑓𝑑 =
𝑐𝑤
Equal interval (class width) histograms Exs 20.1
Unequal interval (class width) histograms Exs 20.2
Cumulative Frequency Tables & Cumulative Frequency Curves are
used to represent data up to a particular class boundary.
Cumulative means a quantity increases as more is added.
From the cumulative frequency graph you can find the median,
IQR, and percentiles.
You are given the first 2
Amount spent
No.of
Upper
Cumulative
on books
students – boundary
Frequency
columns and you need
frequency
to show the data as a
(𝒇)
Note bar charts are for discrete data and have gaps
cumulative Frequency
0 ≤ 𝑥 ≤ 10
between the bars.
0
graph.
10 ≤ 𝑥 ≤ 20
4
1. write down the
The area of the bar in the unequal width histogram give the
20 ≤ 𝑥 ≤ 30
upper boundary for
8
frequency for the interval. E.g. the 45 to 70 age group
each interval.
interval has a class width (𝑐𝑤) of 25 and a frequency density 30 ≤ 𝑥 ≤ 40
12
2. Fill in the cumulative
(𝑓𝑑) of 2, therefore frequency (𝑓) for that class interval is
40 ≤ 𝑥 ≤ 50
11
frequency. (See next
f = cw × 𝑓𝑑 = 25 × 2 = 50 people.
50 ≤ 𝑥 ≤ 60
slide)
5
CHAPTER 20 IMPORTANT POINTS – Histograms & Cumulative Frequency Graphs
continued.
Amount spent
on books ($)
No.of
students –
frequency
(𝒇)
Upper
boundary
0 ≤ 𝑥 ≤ 10
0
≤10
0
10 ≤ 𝑥 ≤ 20
4
≤20
0+4=4
8
≤30
4+8=12
30 ≤ 𝑥 ≤ 40
12
≤40
12+12=24
40 ≤ 𝑥 ≤ 50
11
≤50
35
50 ≤ 𝑥 ≤ 60
5
≤60
40
Total
40
20 ≤ 𝑥 ≤ 30
Now to graph this
information. What you
are collecting data on
(surveying) in this case
the cost of books, goes
on the horizontal axis.
Anything to do with
frequency goes on the
vertical axis.
Cumulative
Frequency
Note the final
cumulative
frequency value is
always the same as
the total frequency.
So what does the
table mean? E.g. the
3rd row means - 12
people spent up to
$30 dollars on
books.
How many people
spent up to $50?
How many people
spent $40 or less on
books?
Students Spending on books
45
Cumulative
Frequency
40
35
30
25
20
15
10
5
45
40
Cumulative
Frequency
𝑄1 (lower quartile) is
found from 25% of the
total frequency, so 𝑄1 =
$28
𝑄3 (upper quartile) is
found from 75% of the
total frequency, so 𝑄3 =
$45.
𝐼𝑄𝑅 (Inter Quartile range)
is the middle 50% of the
data. 𝐼𝑄𝑅 = 𝑄3 = 𝑄1 =
45 − 28 = $17
Students Spending on books
0
35
0
30
25
20
15
10
5
0
0
10
20
30
40
50
60
70
Amount
spent ($)
10
20
30
40
50
60
You may be asked about
percentiles –
1. 87.5% of students paid
up to what amount. 87.5
% of 40 is 35. Use the
70
graph to read off $50.
From the graph we can find the median Amount
spent ($)
(𝐶). The median is in the middle 50% of
2. How many people
the total frequency (50% of 40 is 20).
spent less than $22. Use
Draw a line to the graph from 20 then
the graph to read off 5
read 𝑄2 = $38.
people.
Never give the cumulative frequency
value as a Quartile value.
CHAPTER 21 IMPORTANT POINTS – Ratio, Rate & Proportion
Ratio is the comparison of quantities in a given order. E.g.
1. the ratio of boys to girls in the school 𝑏: 𝑔 = 5: 3.
2. To make lemon squash you need 80 𝑚𝑙 of lemon juice and 1.2 𝑙
of water. To form a ratio of juice to water the units must be the
same 1.2 𝑙 = 1200 𝑚𝑙 ∴ 𝐽: 𝑊 = 80: 1200 = 1: 15. Always simplify
ratios.
Know how to write ratios in their simplest form. Remember the
units should be the same. Exs 21.1
Know how to find a missing quantity in a given ratio e.g.
An alloy is made of 3 parts copper and 7 parts silver. If I have 15𝑔 of
silver, how much copper do I need to make the alloy? Set up the
ratios underneath each other –
C : S
3 : 7
𝑥 : 15
Cross multiply - 𝑥 × 7 = 3 × 15
3×15
𝑥=
7
3
∴ 𝑥 = 6 𝑔 of silver
7
Exs 21.2
Know how to divide a quantity into a given ratio.
Nancy, David & Amanda share some money in the ratio 14:7:3.
Nancy receives £84 more than David. How much do they each
receive? Set up the ratios underneath each other –
N
: D : A
Let D receive 𝑥 amount, ∴ N receives 𝑥 + 84
14
: 7 : 3
𝑥 + 84 : 𝑥 : 𝑦
Since everything changes in the same proportion then
𝑥+84
𝑥
=
Rearrange and simplify
14
7
𝑥 + 84 = 2𝑥
⇒ 𝑥 = 84
∴ Nancy receives 84+84= £168; David £84;
84
Amanda has 3 parts. One part = = £12
7
∴ Amanda receives 3 × 12 = £36
Exs 21.3
Scales: A scale shows by what proportion a model is larger or smaller than the
real life object. Scales are usually written as 1: 𝑛 e.g. 1:25,000. There are no
units, so if you measure 1cm on the model then this represents 25,000 𝑐𝑚 in
real life. Know how to convert from 𝑚𝑚 & 𝑐𝑚 to 𝑚 & 𝑘𝑚.
Exs 21.4
Rates: are one quantity per another quantity e.g. km/h; people per 𝑘𝑚2 ; heart
rate beats per minute. Know how to rearrange the formula for speed
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑠𝑝𝑒𝑒𝑑 =
to make distance or time the subject.
Exs 21.5
𝑡𝑖𝑚𝑒
Know how to –
construct and interpret Distance-Time Graphs.
Exs 21.6
calculate speed from distance-time graphs
Exs 21.7
construct and interpret Speed-Time Graphs.
Exs 21.8
Proportion also Exs 21.12
Direct Know how to find the constant of proportionality and find missing
quantities.
Exs 21.9
In Direct or Inverse proportion Know how to find the constant of
proportionality and find missing quantities.
Exs 21.11
More ratios using the unitary method
Exs 21.10
Know how to increase & decrease a quantity in a given ratio.
Exs 21.13
CHAPTER 22 IMPORTANT POINTS – More Equations Formulae & Functions
Form linear equations from given information and solve for the
unknown quantity.
From worded Qs you need to be able to form an equation and
solve. E.g. Bill is 12 years old. Two years ago I was 5.9 time his
age. How old am I now?
Exs 22.1 & Exs 22.2
Form quadratic equations from given information and solve for
the unknown quantity.
e.g. The length of a rectangle is 9cm longer than its width. The
area is 36𝑐𝑚2 , calculate the length of each side. Exs 22.3
Know how to rearrange formulae to make a different letter the
subject.
e.g. rearrange A = 𝜋𝑟 2 to make 𝑟 the subject of the formula
Exs 22.4 & Exs 22.5
Functions & Function Notation A function is another name for
the equations 𝑦 equals. Understand what
𝑓 𝑥 , 𝑓 5 , 𝑓 𝑥 = 7, 𝑦 = 𝑓(𝑥) represent and how to
substitute into a function and solve functions.
Also understand this type of notation e.g. 𝑓: 𝑥 → 2𝑥 Exs 22.6
Composite functions - A composite function is new function
made up of two or more functions. Understand what these
represent - 𝑓 ∘ 𝑔 = 𝑓 𝑔 𝑥 = 𝑓𝑔 𝑥 ; 𝑔𝑔(𝑥) and 𝑓𝑔(4) for
example.
This is basically substituting 𝑥 in a function with a number or
another function.
Exs 22.7
Finding the inverse of a function
This is very easy. Swap the 𝑥 for 𝑦 and 𝑦 for 𝑥, then solve for
𝑦. Your answer should be of the form
𝑓 −1 𝑥 = (the answer to 𝑦 =)
𝑓 −1 (x) means the inverse function of 𝑓.
Exs 22.8
Try these questions –
1
𝑓 𝑥 = 3𝑥 − 7
Calculate
i) 𝑓(−6)
i)
𝑔(8)
ii) 𝑓 𝑥 = 9
iii) 𝑔 𝑥 = 51
𝑔: 𝑥 → 𝑥 2 − 49
v) 𝑓 −1 𝑥
vi) 𝑓𝑓 −1 (𝑥)
CHAPTER 23 IMPORTANT POINTS – Transformations & Vectors
Transformations The 4 transformations are –
If you do this question then you should be able to transform and
object. You are transforming the original shape each time.
Vectors A vector has …………………….. and ………………………………….
A vector is described by a column vector e.g. 𝑦𝑥
A line with an arrow is used to represents a vector. The vector can be labelled
be labelled with a (a bold lower case letter), (a lower case letter with a
squiggle underneath it), or 𝐴𝐵 (two capital letters with an arrow above them to
show which direction to go).
All the above can be used to write a vector expression or equation e.g. –
6 37 − −17
=
12
𝒂 + 𝟐𝒃=c
𝐴𝐵 = 𝐴𝑀 + 𝑀𝐵
If you can do this question then you should be able to describe
transformations. Describe these transformations.
Know how to - write a vector on a grid as a column vector; Exs 23.5 Q1
Draw a vector from a column vector; Exs 23.5 Q2
Multiply vectors by a scalar; Exs 23.6
Add & subtract Vectors from diagrams, using column vectors and/or correct
notation; Exs 23.7
Calculate the magnitude of a vector. 𝐴𝐵 means calculate the magnitude of
vector 𝐴𝐵 Exs 23.8
Position vectors Exs 23.8
Follow vector paths to calculate new vectors between two points. Always follow
known vector paths and write all new paths on the diagram with arrows Exs
23.8
Combining transformations Know how to transform an object using a
combination of transformations. Exs 23.9
Using the grid and transformation, top left, transform the object with the first
transformation and then transform that image with he next transformation and
CHAPTER 24 IMPORTANT POINTS – Probability Tree Diagrams & Venn Diagrams
Probabilities can be represented by a tree diagram or a
Venn Diagram. Remember in Venn Diagrams the regions
either have elements, number of elements or probabilities.
You know how to draw these – the hard part is to decide
which to use.
As a rule of thumb, if in the information for the question
they use 𝐴 𝐵 𝐴 given 𝐵 then use a Venn Diagram.
If only probabilities given and an experiment is repeated,
then use a tree diagram.
Examples – some of these a very tricky! Answers slide 66+.
1. In Grade 2 the probability the boys are sitting still, no
matter what the girls are doing is 0.4. The probability the
girls are sitting still is 0.7. The probability the girls are sitting
still given the boys are sitting still is 0.95. Determine the
probability:
a) both boys and girls are sitting still;
b) Just the boys are sitting still;
c) Just the girls are sitting still;
d) None of the boys or girls are sitting still
e) The boys are sitting still given the girls are sitting still;
f) At least one of the sexes are sitting still;
g) Both are sitting still or none of them are sitting still.
2. 400 families are surveyed which showed 90% had a TV and 60%
a computer. Every family had at least one of these. One of the
families is selected at random and it is found they have a
computer. Calculate the probability they also have a TV.
3. 50 students went bushwalking. 23 were sunburnt, 22 were
bitten by ants, and 5 were bitten and sunburnt. Determine the
probability a randomly selected student is:
a) Escaped being bitten;
b) Was either bitten or sunburnt;
c) Was neither bitten or sunburnt;
d) Was bitten given he or she was sunburnt;
e) Was sunburnt, given he or she was not bitten.
4. In a certain town three newspapers are published. 20% of the
population read paper 𝐴. 16% read 𝐵. 14% read 𝐶. 8% read
𝐴 & 𝐵. 5% read 𝐴 & 𝐶. 4% read 𝐵 & 𝐶. 2% read all three
newspapers. A person is selected at random. Determine the
probability the person reads –
a) none of the papers;
b) at least one of the papers;
c) Exactly one of the papers;
d) Either 𝐴 or 𝐵;
e) A, given the person reads at least one paper;
f) 𝐶, given that the person either read 𝐴 or 𝐵 or both.
CHAPTER 24 IMPORTANT POINTS – Probability Tree Diagrams & Venn Diagrams
5. Urn 𝐴 contains 2 red and 3 blue balls, and urn 𝐵 contains
4 red and 1 blue ball. Peter selects an urn by flipping a coin,
and takes a ball from that urn.
a) Determine the probability it is red.
b) Given the probability it is red, what is the probability it
came from urn 𝐵?
6. The probability Greta’s mother takes her shopping is
0.4. When Greta goes shopping with her mother she gets an
ice-cream 70% of the time. When Greta does not go
3
shopping with her mother she gets an ice-cream 10 of the
time.
Determine the probability of:
a) Greta’s mother buying her an ice-cream when shopping.
b) Greta went shopping with her mother, given her mother
buys her an ice-cream.
7. A bag contains 6 green discs, 4 yellow discs, and 2 pink discs.
A disc is randomly selected without replacement, and a second is
chosen. Determine the probability green is selected first, given
yellow is randomly selected second.
1. In Grade 2 the probability the boys are sitting still, no matter what the
girls are doing is 0.4. The probability the girls are sitting still is 0.7. The U
probability the girls are sitting still given the boys are sitting still is 0.95.
Determine the probability:
a) both boys and girls are sitting still; P 𝐺 𝐵 =
G
𝑃(𝐺∩𝐵)
= 0.95,
𝑃(𝐵)
∴ 𝑃 𝐺 ∩ 𝐵 = 0.95 × 0.4 = 0.38
b) Just the boys are sitting still; 0.02
c) Just the girls are sitting still; 0.32
d) None of the boys or girls are sitting still 0.28
e) The boys are sitting still given the girls are sitting still;
𝑃(𝐵 ∩ 𝐺)
0.38
P 𝐵𝐺 =
=
= 0.543
𝑃(𝐺)
0.38 + 0.32
f) At least one of the sexes are sitting still; 0.28 is the probability of no
boy or girl is sitting still, so the P(at least one of the sexes is sitting still is
1 − 0.28 = 0.72 or 0.02+0.38+0.32=0.72
g) Both are sitting still or none of them are sitting still.
P(both sitting still or none of them sitting still)= 0.38 + 0.28 = 0.66
0.4 − 0.38
= 0.02
0.7 − 0.38
= 0.32
0.38
1 − 0.32 − 0.38 − 0.02 = 0.28
T
C
160
200
40
0
2. 400 families are surveyed which showed 90% had a TV and 60% a
computer. Every family had at least one of these. One of the families is
selected at random and it is found they have a computer. Calculate the
probability they also have a TV.
𝑃(𝑇 ∩ 𝐶) 200
𝑃 𝑇𝐶 =
=
𝑃(𝐶)
240
𝑇 = 0.9 × 400 = 360
𝐶 = 0.6 × 400 = 240
OR
𝑃 𝑇𝐶 =
360+240−400
200
5
=
=
240
240
6
3. 50 students went bushwalking. 23 were sunburnt, 22 were
bitten by ants, and 5 were bitten and sunburnt. Determine the
probability a randomly selected student is:
10+18
28
a) Escaped being bitten; 50 = 50
U
S
B
40
b) Was either bitten or sunburnt; 50
c)
10
Was neither bitten or sunburnt; 50
d) Was bitten given he or she was sunburnt;
23 − 5
= 18
5
23
18
5
22 − 5
= 17
18
e) Was sunburnt, given he or she was not bitten. 10+18 = 28
Answer
𝑛 𝑈 = 50 𝑛 𝑆 = 23 𝑛 𝐵 = 22 𝑛 𝑆 & 𝐵 = 5
4. In a certain town three newspapers are published. 20% of
the population read paper 𝐴. 16% read 𝐵. 14% read 𝐶. 8% read
𝐴 & 𝐵. 5% read 𝐴 & 𝐶. 4% read 𝐵 & 𝐶. 2% read all three
newspapers. A person is selected at random. Determine the
probability the person reads –
a) none of the papers;
b) at least one of the papers;
c) Exactly one of the papers;
d) Either 𝐴 or 𝐵;
e) A, given the person reads at least one paper;
f) 𝐶, given that the person either read 𝐴 or 𝐵 or both.
50 − 18 − 5 − 17 = 10
U
A
B
C
CHAPTER 24 IMPORTANT POINTS – Probability Tree Diagrams & Venn Diagrams
5. Urn 𝐴 contains 2 red and 3 blue balls, and urn 𝐵 contains
4 red and 1 blue ball. Peter selects an urn by flipping a coin,
and takes a ball from that urn.
a) Determine the probability it is red.
b) Given the probability it is red, what is the probability it
came from urn 𝐵?
6. The probability Greta’s mother takes her shopping is
0.4. When Greta goes shopping with her mother she gets an
ice-cream 70% of the time. When Greta does not go
3
shopping with her mother she gets an ice-cream 10 of the
time.
Determine the probability of:
a) Greta’s mother buying her an ice-cream when shopping.
b) Greta went shopping with her mother, given her mother
buys her an ice-cream.
7. A bag contains 6 green discs, 4 yellow discs, and 2 pink discs.
A disc is randomly selected without replacement, and a second is
chosen. Determine the probability green is selected first, given
yellow is randomly selected second.