Every whole number above 1 is built from primes. Prime factors let you find the highest common factor and lowest common multiple quickly.
🧱 Primes and prime factors
A prime has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, ...). Note that 1 is not prime and 2 is the only even prime. Every other whole number is a product of primes in exactly one way.
HCF = biggest shared factor = 6. LCM = smallest shared multiple = 36. Using primes: HCF takes the lowest powers, LCM the highest.
🤝 HCF and LCM
The HCF is the biggest number that divides both. Take the lowest power of each common prime. The LCM is the smallest number both divide into. Take the highest power of every prime that appears.
Making squares and cubes. A number is a perfect square when every power of a prime is even, and a perfect cube when every power is a multiple of 3.
72 = 2³ × 3²
the power of 2 is odd, so multiply by 2: 72 × 2 = 144 = 12²
the smallest multiplier that makes 72 a square is 2
72 = 2³ × 3² → to make a cube, multiply by 3: 72 × 3 = 216 = 6³
Divisibility rules. By 3 if the digit sum is divisible by 3; by 4 if the last two digits are; by 9 if the digit sum is divisible by 9; by 11 if the alternating digit sum is 0 or divisible by 11.
🎯 Exam focus: showing your method
When asked to “write as a product of prime factors”, give the answer in index form and show the factor tree or repeated division. Marks go to the method, even if the final number slips.
Question: Find the smallest number that both 12 and 18 divide into.
Write each as a product of primes, take the highest powers of each prime,
then multiply. LCM = 2² × 3² = 36.
Common mistakes. Calling 1 a prime number. Forgetting that 2 is prime (and the only even prime). Using the highest powers for the HCF, or the lowest for the LCM. Listing 1 and the number itself as “prime factors”. For “when will they next meet?” problems, using the HCF instead of the LCM.
Check. For two numbers, HCF × LCM = the product of the two numbers. If it does not, there is an error.
✏️ Quick check · 1 of 12
Prime or not
Which of these numbers is prime?
Try dividing by 3, 5, 7.
✏️ Quick check · 2 of 12
Prime factors
What is 60 as a product of prime factors?
Keep dividing by primes.
✏️ Quick check · 3 of 12
Which tool?
Two lighthouses flash every 12 s and 15 s. To find when they flash together again, find the:
Common multiple.
✍️ Type your answer · 4 of 12
Numbers 12 and 18.
What is the HCF of 12 and 18?
Enter an exact answer, without rounding.
Lowest power of each common prime.
✍️ Type your answer · 5 of 12
Numbers 4 and 6.
What is the LCM of 4 and 6?
Enter an exact answer, without rounding.
Highest power of every prime.
✍️ Type your answer · 6 of 12
Numbers 84 and 126.
What is the HCF of 84 and 126?
Enter an exact answer, without rounding.
Write both as prime factors.
✍️ Type your answer · 7 of 12
Two buses leave a station together at 9:00. One runs every 12 minutes and the other every 15 minutes.
After how many minutes do they next leave together?
Enter an exact answer, without rounding.
min
Find the LCM.
✍️ Type your answer · 8 of 12
The number 12.
How many factors does 12 have in total?
Enter an exact answer, without rounding.
List them in pairs.
✍️ Type your answer · 9 of 12
360 = 2³ × 3² × 5.
How many factors does 360 have? (Add 1 to each power and multiply.)
Never take 20% off the new price: it is 20% of the original. Divide by the multiplier (1.20 for +20%, 0.80 for −20%).
↩️ Reverse and compound
Reverse percentages: the final value is a percentage of the original, so divide by the multiplier. Compound change: multiply by the multiplier once for each year, which is the multiplier raised to the power of the number of years.
After a 20% rise the price is £60
original = 60 ÷ 1.2 = £50
£2000 at 5% compound for 2 years
2000 × 1.05² = £2205
Never take 20% off the new price to undo a 20% rise. The percentage is of the original amount.
🧪 Worked examples: successive changes and rates
Successive percentage changes. Multiply the multipliers. Percentages cannot be added or cancelled.
increase by 20% then decrease by 10%
× 1.20 × 0.90 = × 1.08
net change = +8% (not +10%)
Repeated change. Raise the multiplier to the power of the number of periods.
a car worth £10 000 loses 10% of its value each year
after 3 years: 10 000 × 0.9³ = 10 000 × 0.729 = £7290
Finding the rate. Divide the final amount by the original, then work out which multiplier raised to the power gives it.
£500 grows to £605 in 2 years at a compound rate
605 ÷ 500 = 1.21 = 1.1² → multiplier 1.1 → 10% per year
Simple against compound interest.
£2000 at 5% for 3 years
simple: 2000 × 0.05 × 3 = £300 interest
compound: 2000 × 1.05³ = £2315.25 → interest £315.25
compound earns £15.25 more because interest earns interest
Reverse percentages with VAT. A price of £240 including 20% VAT is 120% of the original: original = 240 ÷ 1.2 = £200. VAT = 240 − 200 = £40.
🎯 Exam focus: presentation and accuracy
Write the multiplier explicitly. It earns the method mark and makes the calculation transparent.
Increase £60 by 15% 60 × 1.15 = £69
Decrease £60 by 15% 60 × 0.85 = £51
Find the original (15% off gave £51) 51 ÷ 0.85 = £60
Common mistakes. Multiplying when you should divide in a reverse percentage question. Adding percentages for successive changes (+20% then −20% is not back where you started: 1.2 × 0.8 = 0.96, a 4% loss). Rounding the multiplier too early; keep full accuracy until the final answer. Giving money to the wrong number of decimal places.
Rounding. Money is given to the nearest penny (2 decimal places), unless the question says otherwise. State “to the nearest £” only if told to.
✏️ Quick check · 1 of 12
Multiplier
What multiplier represents an increase of 8%?
100% + 8%.
✏️ Quick check · 2 of 12
Multiplier
What multiplier represents a decrease of 15%?
100% − 15%.
✏️ Quick check · 3 of 12
Which operation?
To find the original price after a percentage increase, you:
Undo the multiplication.
✍️ Type your answer · 4 of 12
A price rises from £40 to £50.
What is the percentage increase?
Enter an exact answer, without rounding.
%
Change ÷ original.
✍️ Type your answer · 5 of 12
After a 20% increase, the price is £60.
What was the original price, in £?
Enter an exact answer, without rounding.
£
Divide by 1.2.
✍️ Type your answer · 6 of 12
A coat in a sale with 15% off costs £51.
What was the original price, in £?
Enter an exact answer, without rounding.
£
Divide by 0.85.
✍️ Type your answer · 7 of 12
£2000 is invested at 5% compound interest per year.
How much is there after 2 years, in £?
Enter an exact answer, without rounding.
£
Multiply by 1.05 twice.
✍️ Type your answer · 8 of 12
A car worth £10000 loses 10% of its value each year.
What is it worth after 2 years, in £?
Enter an exact answer, without rounding.
£
Multiply by 0.9 twice.
✍️ Type your answer · 9 of 12
A price is increased by 20% and then decreased by 10%.
What is the overall percentage change?
Enter an exact answer, without rounding.
%
Multiply the two multipliers.
✍️ Type your answer · 10 of 12
A car worth £10 000 loses 10% of its value each year.
What is it worth after 3 years, in £?
Enter an exact answer, without rounding.
£
Use the multiplier 0.9 three times.
✍️ Type your answer · 11 of 12
£500 invested at a compound rate becomes £605 after 2 years.
What is the rate of interest per year, as a percentage?
Enter an exact answer, without rounding.
%
Find the multiplier that squares to 1.21.
✏️ Quick check · 12 of 12
Reverse percentage
A television costs £240 including VAT at 20%. What was the price before VAT?
Divide by the multiplier.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
After a 50% increase a value is 90. The original value was:
Question 2 of 5
Decreasing £200 by 10% then by 10% again gives:
Question 3 of 5
Which formula gives the percentage change?
Question 4 of 5
Compound growth at 4% for 3 years uses the multiplier:
Question 5 of 5
A 20% rise followed by a 20% fall leaves the value:
Lesson 3 · Algebra
↔️ Linear Inequalities
Inequalities describe a range of answers rather than one. They are solved almost like equations, with one important exception.
⚖️ Solving inequalities
Solve like an equation, but reverse the inequality sign if you multiply or divide by a negative number. Solutions are often shown on a number line: an open circle means the end value is not included (> or <); a closed circle means it is included (≥ or ≤).
3x + 2 > 11
3x > 9
x > 3
−2x < 6
x > −3 (sign reversed)
Check one value from your answer in the original inequality.
🖼️ See it
Open circle for > and <, filled circle for ≥ and ≤. Multiplying or dividing both sides by a negative reverses the inequality sign.
📏 Double inequalities and regions
A double inequality such as −3 < 2x + 1 ≤ 9 is solved by doing the same operation to all three parts: −4 < 2x ≤ 8, then −2 < x ≤ 4. On a graph, y > 2x − 1 is the region above the line y = 2x − 1, drawn dashed because points on the line are not included.
Regions. Draw each boundary line, dashed for < or > and solid for ≤ or ≥. Test a point, such as (0, 0), to decide which side to shade.
y ≥ x solid line, shade above
y < 4 dashed line, shade below
x > 0 dashed line, shade to the right
The wanted region is a triangle bounded by x = 0, y = x and y = 4
🎯 Exam focus: conventions that lose marks
number line open circle for < or >, closed circle for ≤ or ≥
region “shade the UNWANTED region” unless told otherwise
answer give x < 4 (the inequality), not x = 4
integers list them, and check the end values against the signs
Common mistakes. Forgetting to reverse the sign when multiplying or dividing by a negative number: −2x < 6 gives x > −3. Writing the answer as −2 > x < 3. Including an end value that is not allowed by a strict inequality. Solving as an equation and stopping at x = 4.
Check. Pick one value inside your answer and one outside, and put both into the original inequality. The first should be true and the second false.
✏️ Quick check · 1 of 12
Solve
Solve 2x − 3 < 7.
Add 3, then divide by 2.
✏️ Quick check · 2 of 12
Reverse the sign
Solve −3x > 12.
Negative divisor.
✏️ Quick check · 3 of 12
Number line
An open circle at 3 with an arrow to the right shows:
Open means excluded.
✏️ Quick check · 4 of 12
Regions
The region y < 2x + 1 lies:
Think about the y-value.
✍️ Type your answer · 5 of 12
Solve 3x + 2 > 11 (so x > 3).
What is the smallest integer that satisfies it?
Enter an exact answer, without rounding.
x > 3 does not include 3.
✍️ Type your answer · 6 of 12
Solve −2x < 6 (so x > −3).
What is the smallest integer that satisfies it?
Enter an exact answer, without rounding.
Remember the reversed sign.
✍️ Type your answer · 7 of 12
Solve 5x − 4 ≤ 16.
What is the largest integer that satisfies it?
Enter an exact answer, without rounding.
Add 4, then divide by 5.
✍️ Type your answer · 8 of 12
−3 < 2x + 1 ≤ 9 gives −2 < x ≤ 4.
How many integers satisfy it?
Enter an exact answer, without rounding.
List the integers.
✍️ Type your answer · 9 of 12
Solve 5x − 3 < 2x + 9, which gives x < 4.
What is the largest integer value of x that satisfies it?
Enter an exact answer, without rounding.
x must be strictly less than 4.
✍️ Type your answer · 10 of 12
Consider −4 ≤ 3x − 1 < 8.
How many integer values of x satisfy it?
Enter an exact answer, without rounding.
Solve first, then list the integers.
✍️ Type your answer · 11 of 12
Solve (x + 3) ÷ 2 ≥ 4.
What is the smallest value of x?
Enter an exact answer, without rounding.
Clear the fraction first.
✏️ Quick check · 12 of 12
Reading a region
The region above the dashed line y = 2x − 1 is described by:
Dashed means strict.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
When must you reverse an inequality sign?
Question 2 of 5
x ≥ 2 is shown with:
Question 3 of 5
Solve x + 7 > 10.
Question 4 of 5
A dashed boundary line means:
Question 5 of 5
Solve 4x < −8.
Lesson 4 · Geometry
⭕ Circle Theorems
Angles in circles follow a small set of rules. Learn them as a toolkit: spot the pattern in the diagram, then apply the matching theorem.
📐 Angles at the centre and circumference
The angle at the centre is twice the angle at the circumference when both stand on the same arc. Angles in the same segment are equal. The angle in a semicircle (standing on a diameter) is 90°.
angle at circumference = 35°
angle at centre = 2 × 35 = 70°
triangle on a diameter
angle at the circumference = 90°
🖼️ See it
Name the theorem every time you use it in an answer.
🔶 Cyclic quadrilaterals and tangents
In a cyclic quadrilateral (all four corners on the circle) opposite angles add up to 180°. A tangent touches the circle once and is perpendicular to the radius at that point. Two tangents drawn from one outside point are equal in length.
opposite angle = 110°
the other = 180 − 110 = 70°
centre angle 100° in an isosceles triangle
base angles = (180 − 100) ÷ 2 = 40° each
Radii are equal, so any triangle with two radii as sides is isosceles. This unlocks many angle chases.
🧪 Worked examples: chords, tangents and the alternate segment
Chords. A line from the centre perpendicular to a chord bisects the chord. This makes a right-angled triangle with the radius as hypotenuse.
radius 13 cm, chord 24 cm
half the chord = 12 cm
distance from the centre² = 13² − 12² = 169 − 144 = 25
distance = 5 cm
Alternate segment theorem. The angle between a tangent and a chord equals the angle in the alternate segment (on the other side of the chord).
angle between the tangent and the chord AB = 50°
angle in the alternate segment (at any point P on the major arc) = 50°
Using algebra with cyclic quadrilaterals.
opposite angles (x + 20)° and (2x − 5)°
sum = 180: x + 20 + 2x − 5 = 180
3x + 15 = 180 → x = 55
angles are 75° and 105°
Angle chase. Mark every angle you can find on the diagram, and look for isosceles triangles (two radii), right angles (tangent and radius, or angle in a semicircle) and straight lines.
🎯 Exam focus: giving reasons
Each step in an angle chase needs its reason in words. Use these exact phrases:
• the angle at the centre is twice the angle at the circumference
• angles in the same segment are equal
• the angle in a semicircle is 90°
• opposite angles of a cyclic quadrilateral add up to 180°
• the tangent is perpendicular to the radius at the point of contact
• tangents from an external point are equal in length
• the perpendicular from the centre to a chord bisects the chord
• the angle between a tangent and a chord equals the angle in the alternate segment
Common mistakes. Writing “the angle is 90°” without saying why. Using a theorem when the angles are not on the same arc. Forgetting that the angle at the centre can be bigger than 180° (reflex). Confusing “same segment” with “opposite segments”.
Tangent tip. If a question gives a tangent, immediately draw the radius to the point of contact and mark the right angle.
✏️ Quick check · 1 of 12
Centre and circumference
The angle at the centre of a circle is how much bigger than the angle at the circumference on the same arc?
Doubling.
✏️ Quick check · 2 of 12
Semicircle
A triangle is drawn in a circle with one side a diameter. The angle opposite the diameter is:
Right angle.
✏️ Quick check · 3 of 12
Tangent
A tangent meets the radius at the point of contact at:
Perpendicular.
✍️ Type your answer · 4 of 12
An angle at the circumference is 35°.
What is the angle at the centre on the same arc, in degrees?
Enter an exact answer, without rounding.
°
Double it.
✍️ Type your answer · 5 of 12
Opposite angles of a cyclic quadrilateral. One angle is 110°.
What is the opposite angle, in degrees?
Enter an exact answer, without rounding.
°
They sum to 180°.
✍️ Type your answer · 6 of 12
A triangle has two radii as sides and a 100° angle between them at the centre.
What is each base angle, in degrees?
Enter an exact answer, without rounding.
°
It is isosceles.
✍️ Type your answer · 7 of 12
Two tangents are drawn from point P. One is 8 cm long.
How long is the other, in cm?
Enter an exact answer, without rounding.
cm
They are equal.
✍️ Type your answer · 8 of 12
The angle at the centre standing on an arc is 124°.
What is the angle at the circumference on that arc, in degrees?
Enter an exact answer, without rounding.
°
Halve it.
✍️ Type your answer · 9 of 12
A chord of length 24 cm is in a circle of radius 13 cm.
How far is the chord from the centre of the circle, in cm?
Enter an exact answer, without rounding.
cm
Use a right-angled triangle with the radius as hypotenuse.
✍️ Type your answer · 10 of 12
The angle between a tangent and a chord is 50°.
What is the angle in the alternate segment, in degrees?
Enter an exact answer, without rounding.
°
Name the theorem for tangent and chord.
✍️ Type your answer · 11 of 12
A cyclic quadrilateral has opposite angles (x + 20)° and (2x − 5)°.
What is x?
Enter an exact answer, without rounding.
Opposite angles add to 180°.
✏️ Quick check · 12 of 12
Chords
A line is drawn from the centre of a circle, perpendicular to a chord. What does it do to the chord?
Think of the isosceles triangle formed by two radii.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Angles in the same segment are:
Question 2 of 5
In a cyclic quadrilateral, opposite angles add to:
Question 3 of 5
The angle in a semicircle is:
Question 4 of 5
A triangle with two radii as sides is:
Question 5 of 5
Tangents from the same external point are:
Lesson 5 · Geometry
➡️ Vectors & Translations
A vector describes a movement: how far across and how far up. Column vectors make translations and journeys easy to add and compare.
📦 Column vectors
A column vector is written with two numbers: the top is the horizontal move (right is +) and the bottom is the vertical move (up is +). We write it here as (x, y) meaning x across and y up. Add and subtract component by component. A scalar multiple such as 3a multiplies both parts.
a = (2, 3) b = (4, −1)
a + b = (6, 2)
a − b = (−2, 4)
3a = (6, 9)
🖼️ See it
Join the vectors head to tail: the resultant goes from the start to the end. a + b = (3 + 1, 2 + 3) = (4, 5).
📏 Size, translation and journeys
The magnitude (length) of (x, y) is √(x² + y²). A translation by (x, y) moves every point x across and y up. The vector from A to B is AB = B − A, and going the other way, BA = −AB.
magnitude of (3, 4) = √(9 + 16) = 5
translate (1, 2) by (3, −4) → (4, −2)
A(1, 2), B(5, 5): AB = (4, 3), |AB| = 5
🧪 Worked examples: midpoints, parallel vectors and proofs
Position vectors and midpoints. The position vector of a point is its vector from the origin. The midpoint of AB is the average of the two position vectors.
Parallel vectors. Two vectors are parallel when one is a scalar multiple of the other.
(6, 9) = 3 × (2, 3) so they are parallel
(2, k) is parallel to (6, 9) when k = 3
for a vector (5, 12): |v| = √(25 + 144) = 13
Vector geometry. Write each journey as a sum of known vectors and simplify.
OA = a OB = b
AB = −a + b = b − a
M is the midpoint of AB:
OM = a + ½(b − a) = ½(a + b)
Collinear points. A, B and C lie on a straight line when AB and BC are parallel and share the point B. Show AB = k BC.
🎯 Exam focus: notation and proof
In print, vectors are bold (a); in handwriting they are underlined or have an arrow. Write AB with the arrow notation, never as a plain letter pair.
“Show that A, B and C are collinear”
1 find AB and BC (or AB and AC)
2 show one is a multiple of the other: AB = 2 BC
3 state that they share the point B, so A, B and C lie on a line
Common mistakes. Getting the direction wrong: AB = OB − OA (end minus start), and BA = −AB. Adding vectors that should be subtracted. Writing the magnitude as the sum of the components: |(3, 4)| = 5, not 7. Forgetting that translations are described by a column vector, not by a reflection or rotation.
Ratio tip. A point P that divides AB in the ratio 1 : 2 is one third of the way from A: OP = a + ⅓ AB.
✏️ Quick check · 1 of 12
Reading a vector
In a column vector, a negative top number means the move is:
The top is the horizontal part.
✏️ Quick check · 2 of 12
Opposite vector
If AB = (3, 2) then BA is:
Opposite direction.
✏️ Quick check · 3 of 12
Translation
A translation changes the:
It slides the shape.
✍️ Type your answer · 4 of 12
a = (2, 3) and b = (4, −1).
What is the y-component of a + b?
Enter an exact answer, without rounding.
Add the bottom numbers.
✍️ Type your answer · 5 of 12
a = (2, 3).
What is the y-component of 3a?
Enter an exact answer, without rounding.
Multiply both components by 3.
✍️ Type your answer · 6 of 12
The vector (3, 4).
What is its magnitude?
Enter an exact answer, without rounding.
Use Pythagoras.
✍️ Type your answer · 7 of 12
The point (1, 2) is translated by (3, −4).
What is the new x-coordinate?
Enter an exact answer, without rounding.
Add the top numbers.
✍️ Type your answer · 8 of 12
A is (1, 2) and B is (5, 5).
What is the length of AB?
Enter an exact answer, without rounding.
Subtract coordinates then use Pythagoras.
✍️ Type your answer · 9 of 12
A is the point (1, 2) and B is the point (5, 8).
What is the y-coordinate of the midpoint of AB?
Enter an exact answer, without rounding.
Average the y-coordinates.
✍️ Type your answer · 10 of 12
The vector (2, k) is parallel to the vector (6, 9).
What is k?
Enter an exact answer, without rounding.
Find the scalar multiple first.
✍️ Type your answer · 11 of 12
A vector is (5, 12).
What is its magnitude?
Enter an exact answer, without rounding.
Use Pythagoras.
✏️ Quick check · 12 of 12
Vector AB
OA = a and OB = b. Which expression gives AB?
End minus start.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The vector (4, −3) moves a point:
Question 2 of 5
(2, 5) + (1, −2) =
Question 3 of 5
2 × (3, −1) =
Question 4 of 5
If A = (2, 1) and B = (6, 4), then AB =
Question 5 of 5
Two vectors are equal if they have the same:
Lesson 6 · Geometry
🔍 Similar Shapes & Enlargement
Maps, models and photos are all enlargements. Similar shapes have the same angles, and their lengths, areas and volumes scale in a predictable way.
🔍 Enlargement and scale factor
An enlargement multiplies every length by the same scale factor k, measured from a centre of enlargement. Angles do not change. A scale factor between 0 and 1 makes the shape smaller. From the origin, (x, y) → (kx, ky).
scale factor 3 from (0, 0): (2, 3) → (6, 9)
sides 4, 6, 8 → similar shape with shortest side 10
k = 10 ÷ 4 = 2.5 → longest side 8 × 2.5 = 20
🖼️ See it
Every coordinate is multiplied by 3. The sides are 3 times as long, but the angles do not change.
📦 Area and volume factors
Shapes are similar when their angles are equal and their sides are in the same ratio. If the length scale factor is k, the area factor is k² and the volume factor is k³.
k = 4: area 5 cm² → 5 × 16 = 80 cm²
jugs 10 cm and 20 cm tall: k = 2, volume × 8
300 ml → 2400 ml
Do not multiply an area by k. Square k for area and cube it for volume.
✏️ Quick check · 1 of 7
Angles
When a shape is enlarged, its angles:
Similar shapes keep their angles.
✏️ Quick check · 2 of 7
Area
Two similar shapes have a length scale factor of 3. The area scale factor is:
Square it.
✏️ Quick check · 3 of 7
Similar
Which pair of rectangles is similar?
Check the same factor works for both sides.
✏️ Quick check · 4 of 7
Fraction
A scale factor of ½ makes the image:
Multiply each length by ½.
✍️ Type your answer · 5 of 7
Triangle A has sides 4, 6 and 8 cm. Triangle B is similar, and its shortest side is 10 cm.
How long is B’s longest side, in cm?
Enter an exact answer, without rounding.
cm
Find the scale factor first.
✍️ Type your answer · 6 of 7
A shape with area 5 cm² is enlarged by scale factor 4.
What is the area of the image, in cm²?
Enter an exact answer, without rounding.
cm²
Square the scale factor.
✍️ Type your answer · 7 of 7
Two similar jugs are 10 cm and 20 cm tall. The small jug holds 300 ml.
How many millilitres does the large jug hold?
Enter an exact answer, without rounding.
ml
Cube the scale factor.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
(2, 3) is enlarged by scale factor 3 from the origin. The image is:
Question 2 of 5
Similar shapes have:
Question 3 of 5
All lengths are doubled. Volumes are multiplied by:
Question 4 of 5
A model car is 1/20 of real size. A real door is 1.2 m wide. The model door is:
Question 5 of 5
The area scale factor is 25. The length scale factor is:
Lesson 7 · Statistics
📊 Grouped Data & the Estimated Mean
When there is a lot of data, we put it into classes. We lose the exact values, but we can still estimate the mean and find the modal class.
📊 Grouped frequency tables
Data is sorted into classes such as 10 ≤ t < 20, which includes 10 but not 20. The modal class is the class with the highest frequency. The median class is the class that holds the middle value, found by adding up the frequencies as you go.
class frequency
0 ≤ x < 10 3
10 ≤ x < 20 5 ← modal class
20 ≤ x < 30 2
total 10 → the median is between the 5th and 6th values → 10 ≤ x < 20
🖼️ See it
The tallest bar shows the modal class: 10 ≤ x < 20.Estimated mean = 140 ÷ 10 = 14.
🧮 Estimating the mean
We do not know the exact values, so we use each class midpoint as a typical value. Estimated mean = Σ(midpoint × frequency) ÷ total frequency.
Divide by the total frequency, not by the number of classes.
✏️ Quick check · 1 of 7
Midpoint
What is the midpoint of the class 20 ≤ x < 30?
Add the ends and halve.
✏️ Quick check · 2 of 7
Modal
The modal class is the class with:
Most often.
✏️ Quick check · 3 of 7
Estimate
Why is a mean from grouped data only an estimate?
What information is lost?
✏️ Quick check · 4 of 7
Boundary
A time of exactly 20 s goes in which class?
Read the ≤ and < signs.
✍️ Type your answer · 5 of 7
Classes: 0 ≤ x < 10 (frequency 3), 10 ≤ x < 20 (frequency 5), 20 ≤ x < 30 (frequency 2).
What is the estimated mean?
Enter an exact answer, without rounding.
Multiply each midpoint by its frequency.
✍️ Type your answer · 6 of 7
Four classes in order have frequencies 4, 9, 6 and 1. The median is about the 10th value.
Which class number (1, 2, 3 or 4) contains it?
Enter an exact answer, without rounding.
Add up the frequencies as you go.
✍️ Type your answer · 7 of 7
The class 40 ≤ m < 60 has frequency 6.
What is midpoint × frequency for this class?
Enter an exact answer, without rounding.
Find the midpoint first.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The class 150 ≤ h < 160 has midpoint:
Question 2 of 5
Frequencies: 0–10: 2, 10–20: 7, 20–30: 4. The modal class is:
Question 3 of 5
Estimated mean = Σ(midpoint × frequency) divided by:
Question 4 of 5
Which class contains 9.99?
Question 5 of 5
Grouping data makes patterns easier to see but loses:
Lesson 8 · Probability
🎯 Independent Events & Relative Frequency
Some events do not affect each other at all. Others we can only estimate by doing an experiment many times.
🎯 Independent events
Two events are independent if one happening does not change the probability of the other, like tossing a coin and rolling a dice. For independent events, P(A and B) = P(A) × P(B).
P(heads and an even number) = ½ × ½ = ¼ = 0.25
P(late on Monday and on Tuesday) = 0.1 × 0.1 = 0.01
P(two tails) = ½ × ½ = ¼
🖼️ See it
With few throws the estimate jumps around. With many throws it settles down, so more trials give a more reliable estimate.
🔁 Relative frequency
When outcomes are not equally likely, we estimate a probability by experiment: relative frequency = number of times it happened ÷ number of trials. More trials give a more reliable estimate. If the results are far from what a fair object would give, it may be biased.
pin lands point up 42 times in 120 throws → 42 ÷ 120 = 0.35
expected 6s in 300 fair rolls = 300 ÷ 6 = 50
90 sixes in 300 rolls → the dice is probably biased
Multiply for “and” only when the events are independent.
✏️ Quick check · 1 of 7
Independent
Two events are independent if:
Does one affect the other?
✏️ Quick check · 2 of 7
Multiply
For independent events, P(A and B) equals:
And means multiply here.
✏️ Quick check · 3 of 7
Reliable
Which number of trials gives the most reliable estimate of a probability?
More is better.
✏️ Quick check · 4 of 7
Bias
A dice lands on 6 in 90 of 300 rolls. A fair dice would give about 50. The dice is:
Compare with the expected number.
✍️ Type your answer · 5 of 7
A fair coin is tossed and a fair dice is rolled.
What is P(heads and an even number), as a decimal?
Enter an exact answer, without rounding.
Multiply the probabilities.
✍️ Type your answer · 6 of 7
A drawing pin lands point up 42 times in 120 throws.
What is the relative frequency of point up, as a decimal?
Enter an exact answer, without rounding.
Successes ÷ trials.
✍️ Type your answer · 7 of 7
Each day, independently, P(the train is late) = 0.1.
What is P(late on two days in a row), as a decimal?
Enter an exact answer, without rounding.
Multiply.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
A and B are independent with P(A) = 0.5 and P(B) = 0.3. P(A and B) is:
Question 2 of 5
A spinner landed on blue 18 times in 60 spins. The estimated P(blue) is:
Question 3 of 5
P(red) = 0.2. In 250 spins you expect about this many reds:
Question 4 of 5
Two fair coins are tossed. P(two tails) is:
Question 5 of 5
Doing more trials makes the relative frequency:
Final Challenge
🏆 16-Question Challenge Quiz
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
0 / 16
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · Number Theory: Primes, HCF & LCM
The prime factorisation of 72 is:
Q2 · Number Theory: Primes, HCF & LCM
Which statement about 1 is correct?
Q3 · Reverse & Compound Percentages
A population of 5000 grows by 10% per year. After 2 years it is:
Q4 · Reverse & Compound Percentages
A 20% rise followed by a 20% fall leaves the value:
Q5 · Linear Inequalities
Which integers satisfy −1 ≤ x < 3?
Q6 · Linear Inequalities
Solve 4x < −8.
Q7 · Circle Theorems
A cyclic quadrilateral has angles 80°, 95° and x°. If x° is opposite the 95° angle, then x is:
Q8 · Circle Theorems
Tangents from the same external point are:
Q9 · Vectors & Translations
The magnitude of (5, 12) is:
Q10 · Vectors & Translations
Two vectors are equal if they have the same:
Q11 · Similar Shapes & Enlargement
An enlargement with scale factor 2 makes every length:
Q12 · Similar Shapes & Enlargement
The area scale factor is 25. The length scale factor is:
Q13 · Grouped Data & the Estimated Mean
The midpoint of a class is used because it:
Q14 · Grouped Data & the Estimated Mean
Grouping data makes patterns easier to see but loses:
Q15 · Independent Events & Relative Frequency
The expected number of times an outcome happens equals: