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Mathematics - Coggle Diagram
Mathematics
Algebra
Chapter 2
2.1
Letters for unkown values
Variable: A letter used to represent an unknown number (like x or y).
Constant: A fixed number on its own (like 5 or -3).
Coefficient: The number multiplied by a variable (in 4x, the coefficient is 4).
Term: A single number, variable, or numbers/variables multiplied together (like 3x, -5y², or 7).
Expression: A group of terms without an equals sign (like 3x + 2y - 5).
Equation: A math statement with an equals sign (like 3x + 2 = 11).
Formula: A rule written using math symbols, usually with an equals sign (like Area = length × width).
2.2
Substitution
Rule: Replace the letters with the given numbers and calculate the value.
CRITICAL IGCSE TIP: Always put negative numbers in brackets when substituting to avoid sign errors!
Example: If x = -3 and y = 2, find the value of 2x² - y.
Step 1: 2(-3)² - (2) (Notice the brackets!)
Step 2: 2(9) - 2
Step 3: 18 - 2 = 16
2.3
Simplifying expressions
Collecting Like Terms: You can only add or subtract terms that have the exact same variables raised to the same powers.
Example: Simplify 3a + 2b - a + 5b - 4
Group the a's: 3a - a = 2a
Group the b's: 2b + 5b = 7b
Constants stay alone: -4
Answer: 2a + 7b - 4
Multiplying/Dividing Terms: Multiply the numbers and combine the variables.
Example: 3x × 4y = 12xy
Example: 10x²y ÷ 2x = 5xy
2.4
Brackets
Expanding Brackets (Multiplying out): Multiply the term outside the bracket by every term inside.
Rule: a(b + c) = ab + ac
Example: 3(x + 4) = 3x + 12
Example with negatives: -2(x - 5) = -2x + 10 (Watch those signs, baby!)
Factorizing (Single Bracket): The reverse of expanding. Find the Highest Common Factor (HCF) of the terms and put it outside.
Example: Factorize 6x + 9
HCF of 6 and 9 is 3.
Answer: 3(2x + 3)
Example: Factorize 4x² - 8x
HCF of 4x² and 8x is 4x.
Answer: 4x(x - 2)
2.5
Indices
These use the exact same laws we did in Chapter 1, but now with letters!
Multiplication: x^a × x^b = x^(a+b)
Example: x³ × x⁴ = x⁷
Division: x^a ÷ x^b = x^(a-b)
Example: y⁵ ÷ y² = y³
Power of a Power: (x^a)^b = x^(a×b)
Example: (m²)³ = m⁶
Combining with Coefficients: Handle the numbers and the letters separately.
Example: 2x³ × 3x⁴ = (2 × 3)(x^(3+4)) = 6x⁷
Chapter 6
6.1
Solving equation
The Golden Rule: Whatever you do to one side of the equals sign, you MUST do the exact same thing to the other side to keep it balanced.
Basic Equations: Use inverse operations to isolate the letter (x).
If it's added, subtract it. If it's multiplied, divide it.
Example: 3x + 5 = 20 -> 3x = 15 -> x = 5.
Equations with Brackets: Expand the brackets first, then solve.
Example: 2(x + 3) = 14 -> 2x + 6 = 14 -> 2x = 8 -> x = 4.
Equations with Fractions: Multiply the whole equation by the denominator to get rid of the fraction.
Example: x/4 = 3 -> multiply both sides by 4 -> x = 12.
Unknowns on Both Sides: Move all the letter terms to one side, and all the number terms to the other side.
Example: 5x - 2 = 3x + 8 -> 5x - 3x = 8 + 2 -> 2x = 10 -> x = 5.
6.2
Factorising algebraic expression
Single Brackets (Review): Always look for the Highest Common Factor (HCF) first.
Example: 10x + 15 -> 5(2x + 3).
Difference of Two Squares (Important for IGCSE!): If you see two squared terms being subtracted, it factorises into two brackets.
Formula: a^2 - b^2 = (a + b)(a - b)
Example: x^2 - 9 -> (x + 3)(x - 3).
Quadratic Expressions (x^2 + bx + c): You need to find two numbers that multiply to give the last number (c) and add to give the middle number (b).
Example: x^2 + 5x + 6. (What multiplies to 6 and adds to 5? 2 and 3).
Answer: (x + 2)(x + 3).
Harder Quadratics (ax^2 + bx + c): These are trickier. You can use the "split the middle term" method or trial and error with the factors of 'a' and 'c'.
Example: 2x^2 + 7x + 3 -> (2x + 1)(x + 3).
6.3
Rearranging formula
The Goal: Change the "subject" of the formula. This means making a different letter the boss (getting it completely alone on one side of the equals sign).
The Method: It is exactly the same as solving equations! Use inverse operations and keep both sides balanced.
Basic Rearranging:
Example: Make x the subject of y = mx + c.
Subtract c: y - c = mx
Divide by m: x = (y - c) / m.
Tricky Rearranging (Exam Traps!):
When the subject is squared: If x^2 = y, then x = square root of y.
When the subject is under a square root: If y = square root of x, square both sides to get y^2 = x.
When the subject is in the denominator (bottom of a fraction): If y = 5/x, multiply both sides by x first to get xy = 5. Then divide by y to get x = 5/y.
Chapter 14
14.1
Simultaneous linear equation
Goal: Find the values of two unknowns (usually x and y) that work for both equations at the same time.
Elimination Method:
If the numbers in front of a letter are the same, subtract the equations.
If one is positive and one is negative (like 3x and -3x), add the equations.
This cancels out one letter, letting you solve for the other.
Substitution Method:
Rearrange one equation to make a letter the subject (like y = 2x + 1).
Plug that whole expression into the other equation wherever you see that letter.
14.2
Linear equation
The Golden Rule: You solve these exactly like normal equations, EXCEPT if you multiply or divide by a negative number, you MUST flip the inequality sign! (e.g., < becomes >).
Number Lines:
Use an open circle (o) for < or > (the number is not included).
Use a closed/filled circle (•) for <= or >= (the number is included).
14.3
Regions in a plane
This is about shading areas on a graph based on inequalities.
The Lines:
Dashed/dotted line means < or > (strict inequality).
Solid line means <= or >= (includes the line).
Shading: It is usually easier to shade the region you DO NOT want, and leave the correct region blank. To check which side is correct, pick a test point (like 0,0) and see if it makes the inequality true.
14.4
Completing the square
Goal: Turn a quadratic like x^2 + bx + c into the form (x + p)^2 + q.
Why do it? It makes it super easy to find the turning point (vertex) of the graph, which is at (-p, q).
The Method:
Take half of the 'b' number. That goes in the bracket: (x + b/2)^2.
Subtract that number squared, then add the original 'c'.
Example: x^2 + 6x + 5 -> (x + 3)^2 - 9 + 5 -> (x + 3)^2 - 4. Turning point is (-3, -4).
14.5
Quadratic formula
The Formula: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
The Discriminant (b^2 - 4ac): This little part under the square root tells you how many solutions the equation has without even solving it!
If it is positive (> 0): Two real roots (crosses x-axis twice).
If it is zero (= 0): One repeated root (touches x-axis once).
If it is negative (< 0): No real roots (graph floats above or below the x-axis).
14.6
Factorising quadratic when coefficient is not 1
These look like 2x^2 + 7x + 3.
The Method (Splitting the middle term):
Multiply 'a' and 'c' together (2 x 3 = 6).
Find two numbers that multiply to give 6, but add to give 'b' (which is 7). Those numbers are 6 and 1.
Split the middle term: 2x^2 + 6x + 1x + 3.
Factorise in pairs: 2x(x + 3) + 1(x + 3).
Final answer: (2x + 1)(x + 3).
14.7
Algebraic fractions
Simplifying: ALWAYS factorise the top and bottom first, then cancel out the matching brackets. Do not just cancel individual letters!
Adding/Subtracting: You MUST find a common denominator first, just like with normal numbers. Cross-multiply the numerators to combine them.
Multiplying: Multiply tops together, multiply bottoms together.
Dividing: Use the "Keep, Change, Flip" rule. Keep the first fraction, change divide to multiply, flip the second fraction upside down.
Chapter 22
22.1
Make equations
The Golden Rule: Read the problem carefully and decide what your unknown is. Always start by writing "Let x = ..." (for example, "Let x be the number of apples").
Translating Words to Math:
"Sum" or "more than" means add (+).
"Difference" or "less than" means subtract (-).
"Product" or "times" means multiply (x).
"Is" or "equals" means the equals sign (=).
The Method:
Step 1: Define your unknown (x).
Step 2: Write expressions for the other parts of the problem using x.
Step 3: Put it all together into an equation and solve for x.
Step 4: Always check your answer by plugging it back into the original word problem to see if it makes sense!
22.2
Use & Rearrange Formulae
Substitution: When plugging numbers into a formula, ALWAYS put negative numbers in brackets.
Example: If v = u + at, and u = -5, a = 2, t = 3. Then v = (-5) + (2)(3) = -5 + 6 = 1.
Rearranging (Changing the Subject): The goal is to get one letter completely alone on one side of the equals sign. Use inverse operations and keep both sides balanced.
Tricky Exam Traps:
If the subject is squared: (like x^2 = y). You must square root both sides. So x = square root of y.
If the subject is under a square root: (like y = square root of x). You must square both sides. So y^2 = x.
If the subject is on the bottom of a fraction: (like y = 5/x). Multiply both sides by x first to get it off the bottom (xy = 5). Then divide by y to get it alone (x = 5/y).
22.3
Functions & Function Notation
What is a function? Think of it like a machine. You put a number in (the input, x), the machine does a specific rule to it, and exactly one number comes out (the output).
Notation: f(x) just means "the function f applied to x". It is exactly the same as y.
Example: If f(x) = 2x + 3, then f(4) means plug 4 in for x. f(4) = 2(4) + 3 = 11.
Composite Functions: This is when you put one function inside another, like fg(x) or f(g(x)).
The Rule: Always do the inside function first!
Example: If f(x) = 2x and g(x) = x + 1. Find fg(3).
Step 1: Do g(3) first. 3 + 1 = 4.
Step 2: Plug that answer into f. f(4) = 2(4) = 8.
Inverse Functions: Written as f^-1(x). This is the function that "undoes" the original function.
How to find it:
Step 1: Replace f(x) with y. (y = 2x + 3)
Step 2: Swap x and y. (x = 2y + 3)
Step 3: Rearrange to make y the subject. (x - 3 = 2y, so y = (x - 3) / 2)
Step 4: Replace y with f^-1(x). So f^-1(x) = (x - 3) / 2.
Chapter 11
11.1
Pythagoras theorem
The Golden Rule: This ONLY works for right-angled triangles.
The Formula: a^2 + b^2 = c^2
'c' is always the hypotenuse (the longest side, opposite the right angle).
'a' and 'b' are the two shorter sides.
Finding the Hypotenuse (longest side): Square the two shorter sides, add them together, and then square root your answer.
Finding a Shorter Side: Square the hypotenuse, subtract the square of the other shorter side, and then square root your answer.
3D Pythagoras (Bonus for IGCSE!): To find the longest diagonal inside a box (cuboid), the formula is: d^2 = x^2 + y^2 + z^2 (where x, y, and z are the length, width, and height).
11.2
Similar triangle
Similar means the same shape, but a different size.
All matching angles are exactly the same.
All matching sides are in the exact same ratio (proportional).
How to find a missing side:
Find the Scale Factor by dividing a known matching side by its partner (Big side / Small side).
Multiply or divide the other side by that scale factor to find the missing length.
11.3
Similar shapes
The rules are the same as triangles, but for any shape.
THE BIG IGCSE TRAP (Memorize this, baby!):
If the Length Scale Factor is k...
The Area Scale Factor is k^2 (k squared).
The Volume Scale Factor is k^3 (k cubed).
Example: If you double the lengths of a shape (scale factor 2), the area doesn't double—it quadruples (2^2 = 4). The volume octuples (2^3 = 8). Examiners test this all the time!
11.4
Congruence
Congruent means exactly the same shape AND exactly the same size. They are identical twins, even if one is flipped or rotated.
The 4 Tests for Congruent Triangles (You must know these acronyms!):
SSS (Side-Side-Side): All three sides are the same length.
SAS (Side-Angle-Side): Two sides and the angle between them are the same.
ASA (Angle-Side-Angle): Two angles and the side between them are the same.
RHS (Right angle-Hypotenuse-Side): Only for right-angled triangles. The hypotenuse and one other matching side are the same.
Shape
Chapter 3
3.1
Lines and angles
Types of angles:
Acute: less than 90 degrees
Right angle: exactly 90 degrees
Obtuse: between 90 and 180 degrees
Reflex: greater than 180 degrees
Basic Angle Facts:
Angles on a straight line add up to 180 degrees.
Angles around a point add up to 360 degrees.
Vertically opposite angles (formed when two lines cross) are equal.
Parallel Lines (The F, Z, and C rules):
Corresponding angles (F-shape) are equal.
Alternate angles (Z-shape) are equal.
Co-interior or Allied angles (C-shape or U-shape) add up to 180 degrees.
3.2
Triangles
The three interior angles of any triangle always add up to 180 degrees.
Exterior angle rule: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Types of triangles:
Equilateral: 3 equal sides, all angles are 60 degrees.
Isosceles: 2 equal sides, 2 equal base angles.
Right-angled: has one 90-degree angle.
Scalene: no equal sides or angles.
Area formula: 1/2 x base x height
3.3
Quadrilaterals (4-sided)
The interior angles of any quadrilateral always add up to 360 degrees.
Key Area Formulas:
Square: side x side (or side^2)
Rectangle: length x width
Parallelogram: base x vertical height
Trapezium: 1/2 x (a + b) x h (where a and b are the two parallel sides)
Rhombus and Kite: 1/2 x d1 x d2 (where d1 and d2 are the diagonals)
3.4
Polygons
Regular polygon: All sides are equal, and all angles are equal.
Exterior angle sum: The exterior angles of ANY polygon always add up to 360 degrees.
Interior angle sum formula: (n - 2) x 180 (where n is the number of sides)
For Regular Polygons:
Each exterior angle = 360 / n
Each interior angle = 180 - exterior angle
3.5
Circles
Key parts: Radius (r), Diameter (d = 2r), Circumference (the edge), Chord (a line connecting two points on the edge), Arc (part of the circumference), Sector (like a pizza slice), Tangent (a line touching the circle at exactly one point).
Key Formulas:
Circumference = 2 x pi x r (or pi x d)
Area of a full circle = pi x r^2
Sectors and Arcs (The "Pizza Slice" rules):
Arc length = (angle / 360) x 2 x pi x r
Area of a sector = (angle / 360) x pi x r^2
Important Circle Theorems for IGCSE:
The angle at the center is double the angle at the circumference (from the same arc).
The angle in a semicircle is always 90 degrees.
Angles in the same segment are equal.
Opposite angles in a cyclic quadrilateral (a 4-sided shape inside a circle) add up to 180 degrees.
A tangent is always perpendicular (90 degrees) to the radius at the point of contact.
3.6
Construction
You will need to know how to use a compass and ruler for these.
Perpendicular bisector: A line that cuts another line exactly in half at a 90-degree angle.
Angle bisector: A line that cuts an angle exactly in half.
Locus: The path or rule that a point follows (e.g., "the locus of points exactly 5cm from point A" is a circle with a radius of 5cm).
Chapter 7
7.1
Perimeter & area
Perimeter: The total distance around the outside of a shape. Just add up all the side lengths!
Circumference: The perimeter of a circle = 2 x pi x r (or pi x d)
Area Formulas (MUST KNOW THESE!):
Rectangle: length x width
Square: side x side (or side^2)
Triangle: 1/2 x base x vertical height
Parallelogram: base x vertical height (NOT the slanted side!)
Trapezium: 1/2 x (a + b) x h (where a and b are the parallel sides)
Circle: pi x r^2
Compound Shapes: Break the shape into simpler pieces (like rectangles and triangles), find the area of each piece, then add or subtract them as needed.
7.2
3D
3D Shapes you need to know:
Cube: 6 square faces
Cuboid (rectangular prism): 6 rectangular faces
Prism: A 3D shape with the same cross-section all the way through (like a triangular prism)
Cylinder: Circular cross-section
Pyramid: Triangular faces meeting at a point
Cone: Circular base, tapering to a point
Sphere: Perfectly round 3D shape
Nets: A 2D pattern that can be folded to make a 3D shape. You might be asked to draw these or identify them.
Plans and Elevations:
Plan view: What you see from directly above (bird's eye view)
Front elevation: What you see from the front
Side elevation: What you see from the side
7.3
Surface area & volume solids
Volume Formulas (MUST KNOW THESE!):
Cube: side^3
Cuboid: length x width x height
Prism: area of cross-section x length
Cylinder: pi x r^2 x h
Pyramid: 1/3 x area of base x vertical height
Cone: 1/3 x pi x r^2 x h
Sphere: 4/3 x pi x r^3
Surface Area Formulas:
Cube: 6 x side^2
Cuboid: 2(lw + lh + wh)
Cylinder: 2 x pi x r^2 + 2 x pi x r x h (two circles + the curved rectangle)
Cone: pi x r^2 + pi x r x l (base circle + curved surface, where l is slant height)
Sphere: 4 x pi x r^2
Important Tips:
Volume is always measured in cubic units (cm^3, m^3, etc.)
Surface area is always measured in square units (cm^2, m^2, etc.)
For compound solids, break them into simpler shapes and add or subtract volumes
Watch out for units! Make sure everything is in the same units before calculating
Chapter 10
10.1
Straight line graph
The Equation of a Straight Line: y = mx + c
m = the gradient (slope) of the line.
c = the y-intercept (where the line crosses the vertical y-axis).
Finding the Gradient (m) from two points:
Formula: m = (y2 - y1) / (x2 - x1).
Basically: change in y divided by change in x (rise over run).
Parallel Lines: Lines that never cross. They always have the exact same gradient (m).
Perpendicular Lines: Lines that cross at a perfect 90-degree angle. Their gradients multiply together to make -1.
Example: If one line has a gradient of 2, the perpendicular line has a gradient of -1/2.
Drawing a Graph: You can either make a table of values (plug in x numbers to find y numbers) or start at the y-intercept (c) and use the gradient (m) to draw the rest of the line.
10.2
Quadratic expression & equation
Standard Form: ax^2 + bx + c = 0 (where a is not zero).
The Graph Shape: It is called a parabola.
If 'a' is positive, it is a U-shape (smiley face).
If 'a' is negative, it is an n-shape (sad face).
Solving Quadratics (Finding x): This means finding the values of x that make the equation equal zero.
Method 1: Factorising. Put it into double brackets, like (x + 2)(x - 3) = 0. Then x = -2 or x = 3.
Method 2: The Quadratic Formula (MUST MEMORIZE FOR EXAM!). If you can't factorise, use this formula:
x = (-b ± √(b² - 4ac)) / 2a
Tip: The ± means you do the calculation twice: once with a plus, and once with a minus, to get your two answers.
Key Features of a Quadratic Graph:
Roots (x-intercepts): Where the graph crosses the x-axis. These are the answers you get when you solve the equation.
y-intercept: Where the graph crosses the y-axis. This is always the 'c' value from your equation.
Turning Point (Vertex): The very bottom of the U-shape or the very top of the n-shape. You can find the x-coordinate of the turning point using the formula: x = -b / 2a.
Chapter 15
15.1
A scale tells you the ratio between the drawing and real life (e.g., 1:50,000 means 1 cm on the map = 50,000 cm in real life).
Exam Trap: Watch your units! You usually have to convert centimeters to meters or kilometers at the end. (100 cm = 1 m, 100,000 cm = 1 km).
15.2
How to solve: Always draw a little North arrow at the point you are measuring from. Use parallel line rules (co-interior angles add to 180) to find missing angles.
Back Bearings: To find the bearing of A from B when you know the bearing of B from A, just add or subtract 180 degrees.
The 3 Golden Rules of Bearings:
They are always measured from North.
They are always measured clockwise.
They are always written as three digits (e.g., 045 degrees, not 45 degrees).
15.3
SOH CAH TOA
ONLY for right-angled triangle
First, label your sides: Hypotenuse (longest side, opposite the right angle), Opposite (across from the angle you are looking at), Adjacent (next to the angle).
SOH: Sin(angle) = Opposite / Hypotenuse
CAH: Cos(angle) = Adjacent / Hypotenuse
TOA: Tan(angle) = Opposite / Adjacent
Tip: Use the formula triangle. If you want to find the angle, use the inverse buttons on your calculator (sin^-1, cos^-1, tan^-1).
15.4
Exact ratio
Sin(30) = 1/2
Cos(60) = 1/2
Tan(45) = 1
Sin(60) = sqrt(3) / 2
Cos(30) = sqrt(3) / 2
15.6
Sin Cos Tan angles >90 degree
To solve these, use the CAST diagram (or "All Students Take Calculus"). It tells you which ratios are positive in each quadrant.
0 to 90 (Quadrant 1): All are positive.
90 to 180 (Quadrant 2): Only Sin is positive.
180 to 270 (Quadrant 3): Only Tan is positive.
270 to 360 (Quadrant 4): Only Cos is positive.
15.7
Sine and Cosine rule
Use these for NON-right-angled triangles!
Sine Rule: Use when you have a matching "pair" of an angle and its opposite side.
Formula for sides: a / sinA = b / sinB = c / sinC
Formula for angles: sinA / a = sinB / b = sinC / c (flip it upside down to find an angle).
Cosine Rule: Use when you have two sides and the included angle (to find a side), or three sides (to find an angle).
Formula for a side: a^2 = b^2 + c^2 - 2bc cosA
Formula for an angle: cosA = (b^2 + c^2 - a^2) / 2bc
15.8
Formula: Area = 1/2 ab sinC
For non-right-angled triangles, if you know two sides and the angle between them:
15.9
Trigo in 3D shape
Step 1: Flatten the Base (Find the floor diagonal)
Ignore the height of the 3D shape for a second and just look at the flat floor.
Draw the diagonal line across the floor to create a flat, right-angled triangle.
Use Pythagoras (a^2 + b^2 = c^2) to find the length of this floor diagonal.
In our example: 4^2 + 3^2 = 25. The square root of 25 is 5 cm.
Step 2: Build the Vertical Triangle (Find the 3D diagonal)
Now, stand that floor diagonal up to make a new, vertical right-angled triangle inside the shape.
The base of this new triangle is the floor diagonal you just found (5 cm).
The vertical side is the height of the shape (12 cm).
Use Pythagoras again to find the longest diagonal (the hypotenuse).
In our example: 5^2 + 12^2 = 169. The square root of 169 is 13 cm.
Step 3: Find the Angle (Use SOH CAH TOA)
Look at that exact same vertical triangle from Step 2.
Identify which sides you have in relation to the angle you want to find (Opposite, Adjacent, or Hypotenuse).
Choose your ratio (SOH, CAH, or TOA) and use the inverse button on your calculator (sin^-1, cos^-1, or tan^-1) to find the angle.
In our example: We had the Opposite (12) and Adjacent (5). We used TOA (12 / 5 = 2.4). The inverse tan of 2.4 is 67.4 degrees.
15.5
Solving problems
Angle of Elevation: The angle looking UP from the horizontal (like looking up at a tree).
Angle of Depression: The angle looking DOWN from the horizontal (like looking down from a cliff).
Exam Trap: The angle of depression from the top is equal to the angle of elevation from the bottom (alternate angles).
Chapter 19
19.1
Symmetry in 2D
Line Symmetry (Reflection Symmetry): A shape has line symmetry if you can fold it along a line and both halves match perfectly.
The fold line is called the "line of symmetry" or "mirror line."
Example: A square has 4 lines of symmetry. An equilateral triangle has 3 lines of symmetry.
Rotational Symmetry: A shape has rotational symmetry if it looks the same after being rotated (turned) less than 360 degrees.
Order of Rotational Symmetry: The number of times the shape fits onto itself during a full 360-degree turn.
Example: A square has rotational symmetry of order 4 (it looks the same at 90, 180, 270, and 360 degrees).
Exam Tip: If a shape only looks the same after a full 360-degree turn, it has rotational symmetry of order 1 (which means it has NO rotational symmetry).
19.2
Symmetry in 3D
Plane Symmetry: A 3D shape has plane symmetry if you can slice it with a flat plane and both halves are mirror images.
Example: A cube has 9 planes of symmetry.
Rotational Symmetry in 3D: A 3D shape has rotational symmetry if it looks the same after being rotated around an axis (a line through the shape).
Order of Rotational Symmetry: The number of times the shape fits onto itself during a full 360-degree turn around the axis.
Example: A cylinder has infinite rotational symmetry around its central axis (it looks the same at any angle).
19.3
Symmetry properties in circle
Circles have infinite lines of symmetry (any line through the center).
Circles have infinite rotational symmetry (they look the same at any angle).
Key Symmetry Facts:
Equal chords are equidistant from the center.
The perpendicular bisector of a chord passes through the center of the circle.
Tangents from the same external point to a circle are equal in length.
19.4
Circle theorem
Theorem 1: Angle at the Center
The angle at the center is double the angle at the circumference (from the same arc).
Formula: Angle at center = 2 x Angle at circumference
Theorem 2: Angle in a Semicircle
The angle in a semicircle is always 90 degrees (a right angle).
Theorem 3: Angles in the Same Segment
Angles in the same segment are equal (if they are subtended by the same arc).
Theorem 4: Cyclic Quadrilateral
Opposite angles in a cyclic quadrilateral (a 4-sided shape inside a circle) add up to 180 degrees.
Theorem 5: Tangent and Radius
A tangent is always perpendicular (90 degrees) to the radius at the point of contact.
Theorem 6: Two Tangents
Tangents from the same external point to a circle are equal in length.
Theorem 7: Alternate Segment Theorem
The angle between a tangent and a chord is equal to the angle in the alternate segment.
Data
Chapter 4
4.1
Collecting and classifying data
Primary data: Data you collect yourself (e.g., doing a survey).
Secondary data: Data collected by someone else (e.g., from a website or book).
Qualitative data: Descriptive data, not numbers (e.g., eye color, favorite food).
Quantitative data: Numerical data (e.g., height, number of siblings).
Discrete data: Data that can only take specific, exact values. You count it. (e.g., number of students in a class, shoe size).
Continuous data: Data that can take any value within a range. You measure it. (e.g., height, weight, time).
Population: The entire group you are studying.
Sample: A smaller, representative part of the population used to gather data.
4.2
Organising data
Tally charts: Using tally marks (groups of 5, with the 5th mark crossing the previous 4) to count data quickly.
Frequency tables: A table showing how often each value or group of values occurs.
Grouped data: When you have a lot of continuous data, you group it into "class intervals" (e.g., 0-10, 10-20, 20-30).
Two-way tables: Used to show the relationship between two different categories of data (like a grid). You can find totals by adding across rows and down columns.
4.3
Charts to display data
Bar charts: Used for categorical or discrete data. The bars must have gaps between them.
Pie charts: Used to show parts of a whole.
Crucial Formula: Angle for a slice = (Frequency of that item / Total frequency) x 360.
Tip: Always check that your angles add up to 360 degrees!
Line graphs: Best for showing continuous data changing over time (e.g., temperature over a week).
Scatter graphs: Used to show the relationship between two sets of data.
Positive correlation: As one variable goes up, the other goes up.
Negative correlation: As one variable goes up, the other goes down.
No correlation: No relationship between the variables.
Line of best fit: A straight line drawn through the data points to show the general trend. It should have roughly the same number of points above and below it.
Stem-and-leaf diagrams: A way to organize numbers where the "stem" is the first digit(s) and the "leaf" is the last digit.
Exam Trap: You MUST include a key! (e.g., Key: 2 | 5 means 25).
The leaves must be ordered from smallest to largest.
Chapter 16
Intoduction
Bivariate data: Data that involves two variables (that's what "bi" means - two). For example, height and weight, or study time and test scores.
Scatter Diagram (Scatter Plot): A graph where you plot two sets of data against each other to see if there is a relationship between them.
One variable goes on the x-axis (usually the independent variable - the one you control).
One variable goes on the y-axis (usually the dependent variable - the one that changes as a result).
Correlation
Positive Correlation: As one variable increases, the other variable also increases. The points go UP from left to right.
Example: Height and shoe size (taller people tend to have bigger feet).
Negative Correlation: As one variable increases, the other variable decreases. The points go DOWN from left to right.
Example: Time spent playing video games and test scores (more gaming might mean lower scores).
No Correlation: There is no pattern or relationship between the variables. The points are scattered randomly.
Example: Shoe size and IQ (they have nothing to do with each other).
Strength of Correlation
Strong Correlation: The points are tightly clustered around an imaginary line.
Weak Correlation: The points are more spread out, but you can still see a general trend.
Perfect Correlation: All points lie exactly on a straight line (this is super rare in real life!).
Line of best fit
A straight line drawn through the scatter plot to show the general trend of the data.
How to draw it:
It should follow the general direction of the points.
There should be roughly the same number of points above the line as below the line.
Ignore any outliers (weird points that are far away from the rest).
The line does NOT have to go through the origin (0,0).
Using the Line of Best Fit:
You can use it to estimate values that you didn't measure.
Interpolation: Estimating a value WITHIN the range of your data (this is reliable).
Extrapolation: Estimating a value OUTSIDE the range of your data (this is risky and can be unreliable!).
Chapter 21
21.1
Ratio
What is a ratio? A way of comparing two or more quantities. Written as a:b or "a to b".
Simplifying Ratios: Just like fractions! Divide all parts by the same number until you can't divide anymore.
Example: 12:8 simplifies to 3:2 (divide both by 4).
Sharing in a Ratio:
Step 1: Add the parts of the ratio together to get the total number of parts.
Step 2: Divide the total amount by the total number of parts to find the value of ONE part.
Step 3: Multiply each part of the ratio by that value.
Example: Share $100 in the ratio 3:2. Total parts = 5. One part = $100/5 = $20. Answer: $60 and $40.
Dividing a quantity into a given ratio: If you know one part, work backwards to find the whole amount.
21.2
Ratio & Scale
Scale drawings and maps use ratios to represent real distances.
Scale notation: 1:50,000 means 1 cm on the map = 50,000 cm in real life.
How to calculate:
Real distance = Map distance x Scale factor
Map distance = Real distance / Scale factor
Exam Trap: Watch your units! You usually need to convert centimeters to meters or kilometers at the end.
100 cm = 1 m
100,000 cm = 1 km
21.3
Rates
What is a rate? A ratio that compares two quantities with different units (like speed = distance/time).
Common rates:
Speed: km/h or m/s
Flow rate: liters/minute
Price: dollars/kg
Unitary method: Find the value of ONE unit first, then multiply to find what you need.
Example: If 5 kg of apples cost $12, then 1 kg costs $12/5 = $2.40, so 8 kg costs $2.40 x 8 = $19.20.
21.4
Kinematic Graphs (Distance-Time and Speed-Time Graphs)
Distance-Time Graphs:
Gradient (slope) = Speed
Horizontal line = Stationary (not moving)
Steeper line = Faster speed
Curved line = Changing speed (acceleration or deceleration)
Speed-Time Graphs:
Gradient (slope) = Acceleration
Horizontal line = Constant speed
Area under the graph = Distance traveled
Exam Tip: To find the distance, calculate the area under the speed-time graph (use triangles, rectangles, and trapeziums).
21.5
Proportion
Direct Proportion: When one quantity increases, the other increases at the same rate.
Example: The more hours you work, the more money you earn.
If you double one, you double the other.
Inverse (Indirect) Proportion: When one quantity increases, the other decreases.
Example: The more workers you have, the less time it takes to complete a job.
If you double one, the other is halved.
Finding the constant of proportionality (k):
For direct proportion: y = kx, so k = y/x
For inverse proportion: y = k/x, so k = xy
21.6
Direct & Inverse Proportion
Direct Proportion:
y is directly proportional to x: y x
Formula: y = kx (where k is the constant of proportionality)
Graph: A straight line through the origin (0,0)
Direct Proportion to Powers:
y x² means y = kx²
y ∝ x³ means y = kx³
y ∝ √x means y = k√x
Inverse Proportion:
y is inversely proportional to x: y ∝ 1/x
Formula: y = k/x (where k is the constant of proportionality)
Graph: A curved line that gets closer to the axes but never touches them
Inverse Proportion to Powers:
y ∝ 1/x² means y = k/x²
Numbers
Chapter 1
1.1
Numbers
Natural numbers (ℕ): 1, 2, 3, 4, ... (counting numbers, no zero)
Whole numbers: 0, 1, 2, 3, ... (natural numbers + zero)
Integers (ℤ): ..., -3, -2, -1, 0, 1, 2, 3, ... (positive and negative whole numbers)
Rational numbers (ℚ): Numbers that can be written as a fraction p/q where q ≠ 0 (e.g., ½, 0.75, -3)
Irrational numbers: Cannot be written as a simple fraction (e.g., √2, π, 3)
Real numbers (ℝ): All rational and irrational numbers combined
Prime numbers: Only divisible by 1 and itself (2, 3, 5, 7, 11...)
Square numbers: 1, 4, 9, 16, 25... (n²)
Cube numbers: 1, 8, 27, 64... (n³)
1.2
Multiples & Factors
Multiple: A number multiplied by an integer (e.g., multiples of 3: 3, 6, 9, 12, 15...)
Factor: A number that divides exactly into another number (e.g., factors of 12: 1, 2, 3, 4, 6, 12)
HCF (Highest Common Factor): The largest factor shared by two or more numbers
Method: List factors of each number → find the largest common one
Example: HCF of 12 and 18 → factors of 12: {1,2,3,4,6,12}, factors of 18: {1,2,3,6,9,18} → HCF = 6
LCM (Lowest Common Multiple): The smallest multiple shared by two or more numbers
Method: List multiples or use prime factorization
Example: LCM of 4 and 6 → multiples of 4: {4,8,12,16...}, multiples of 6: {6,12,18...} → LCM = 12
1.3
Prime numbers
Prime number: A number greater than 1 with exactly two factors (1 and itself)
First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
️ 2 is the only even prime number!
1 is NOT a prime number (only has one factor)
Prime Factorization: Writing a number as a product of prime numbers
Use a factor tree or repeated division
Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Finding HCF using prime factors: Multiply the lowest power of common prime factors
60 = 2² × 3 × 5, 72 = 2³ × 3² → HCF = 2² × 3 = 12
Finding LCM using prime factors: Multiply the highest power of all prime factors
60 = 2² × 3 × 5, 72 = 2³ × 3² → LCM = 2³ × 3² × 5 = 360
1.4
+ve & -ve
Adding:
Same signs → add and keep the sign: (-3) + (-5) = -8
Different signs → subtract and keep sign of larger number: (-7) + 4 = -3
Subtracting: Change the sign of the second number, then add
5 - (-3) = 5 + 3 = 8
(-4) - 6 = (-4) + (-6) = -10
Multiplying & Dividing:
(+) × (+) = + | (-) × (-) = + | (+) × (-) = - | (-) × (+) = -
Same signs → positive | Different signs → negative
(-3) × (-4) = 12 | (-15) ÷ 3 = -5
1.5
Indices
Multiplication Law: When multiplying terms with the same base, add the powers.
Formula: am×an\=am+nam×an\=am+n
Example: 23×24\=23+4\=2723×24\=23+4\=27
Division Law: When dividing terms with the same base, subtract the powers.
Formula: am÷an\=am−nam÷an\=am−n
Example: 56÷52\=56−2\=5456÷52\=56−2\=54
Power of a Power Law: When raising a power to another power, multiply the powers.
Formula: (am)n\=am×n(am)n\=am×n
Example: (32)3\=32×3\=36(32)3\=32×3\=36
Zero Index Law: Any non-zero number raised to the power of 0 is always 1.
Formula: a0\=1a0\=1
Example: 70\=170\=1 (Note: 0000 is undefined!)
Negative Index Law: A negative power means you flip the base to the denominator to make it positive.
Formula: a−n\=1ana−n\=an1
Example: 2−3\=123\=182−3\=231\=81
Fractional Index Law (1): A power of 1nn1 is the same as taking the nn-th root.
Formula: a1/n\=ana1/n\=na
Example: 81/3\=83\=281/3\=38\=2
Fractional Index Law (2): A power of mnnm means you take the nn-th root first, then raise it to the power of mm.
Formula: am/n\=(an)mam/n\=(na)m
Example: 163/4\=(164)3\=23\=8163/4\=(416)3\=23\=8
1.6
BODMAS
B — Brackets ( )
I/O — Indices/Orders (powers and roots)
D — Division ÷
M — Multiplication ×
A — Addition +
S — Subtraction −
⚠️ D and M are equal priority → work left to right ⚠️ A and S are equal priority → work left to right
Example: 3 + 4 × 2² = 3 + 4 × 4 = 3 + 16 = 19 (NOT 28!)
Example: (5 + 3) × 2 - 4² = 8 × 2 - 16 = 16 - 16 = 0
1.7
Rounding & Estimating
Rounding Rules:
Look at the digit after the place you're rounding to
If it's 5 or more → round up
If it's 4 or less → round down
Types of Rounding:
Decimal places (d.p.): Count digits after the decimal point
3.14159 to 2 d.p. = 3.14
Significant figures (s.f.): Count from the first non-zero digit
0.004567 to 2 s.f. = 0.0046
12345 to 3 s.f. = 12300
Estimating:
Round all numbers to 1 significant figure, then calculate
Example: Estimate 48.7 × 2.1 ≈ 50 × 2 = 100
Used to check if your answer is reasonable ✅
Chapter 5
5.1
Fractions
Numerator: The top number (how many parts you have).
Denominator: The bottom number (how many parts make a whole).
Equivalent fractions: You can make equivalent fractions by multiplying or dividing the top and bottom by the exact same number.
Simplifying (Canceling): Divide the top and bottom by their Highest Common Factor (HCF) until you cannot divide anymore. This is your simplest form.
Mixed numbers to improper fractions: Multiply the whole number by the denominator, add the numerator, and put that over the original denominator.
Example: 2 and 1/3 -> (2 x 3) + 1 = 7. So, 7/3.
Improper fractions to mixed numbers: Divide the top by the bottom. The answer is your whole number, and the remainder is your new top number.
Example: 7/3 -> 7 divided by 3 is 2 with a remainder of 1. So, 2 and 1/3.
5.2
Operations with fractions
Adding and Subtracting: You MUST have a common denominator. Find the Lowest Common Multiple (LCM) of the bottom numbers, adjust the top numbers to match, and then add or subtract the tops.
Example: 1/2 + 1/3 -> Change to 3/6 + 2/6 = 5/6.
Multiplying: Multiply the top by the top, and the bottom by the bottom. You can simplify at the end, or "cross-cancel" before you multiply to make the numbers smaller.
Example: 2/3 x 3/4 = 6/12, which simplifies to 1/2.
Dividing: Use the "Keep, Change, Flip" (KCF) rule. Keep the first fraction exactly the same. Change the division sign to multiplication. Flip the second fraction upside down (this is called the reciprocal). Then multiply.
Example: 1/2 divided by 1/4 -> Keep 1/2, change to multiply, flip 1/4 to 4/1. -> 1/2 x 4/1 = 4/2 = 2.
5.3
Percentages
"Percent" literally means "out of 100".
Converting:
Percentage to Decimal: Divide by 100 (move the decimal point two places to the left). 45% = 0.45.
Decimal to Percentage: Multiply by 100. 0.45 = 45%.
Fraction to Percentage: Divide the top by the bottom, then multiply by 100.
Finding a percentage of an amount: Turn the percentage into a decimal and multiply it by the amount.
Example: 20% of 50 -> 0.20 x 50 = 10.
Percentage Increase and Decrease (The Multiplier Method):
To increase by 15%, your multiplier is 1.15 (100% + 15%).
To decrease by 15%, your multiplier is 0.85 (100% - 15%).
Example: Increase 200 by 15% -> 200 x 1.15 = 230.
Percentage Change Formula: (Change in value / Original value) x 100.
Reverse Percentages (Finding the original amount): If a shirt costs $120 after a 20% increase, that $120 represents 120% of the original price (multiplier 1.2). To find the original, divide: 120 / 1.2 = $100.
5.4
Standard form
Also called Scientific Notation. It is used for writing very large or very small numbers neatly.
The Format: A x 10^n
Rule 1: 'A' must be a number that is 1 or greater, but less than 10. (1 <= A < 10).
Rule 2: 'n' must be a whole number (positive or negative).
Converting to Standard Form: Move the decimal point until you have a number between 1 and 10. Count how many places you moved.
If you moved left (for big numbers), the power is positive.
If you moved right (for tiny decimals), the power is negative.
Example: 4500 -> move decimal 3 places left -> 4.5 x 10^3.
Example: 0.0045 -> move decimal 3 places right -> 4.5 x 10^-3.
Calculations in Standard Form:
Multiplying: Multiply the 'A' numbers together, and ADD the powers of 10.
Dividing: Divide the 'A' numbers, and SUBTRACT the powers of 10.
Adding/Subtracting: You MUST make the powers of 10 the same first, then you can add or subtract the 'A' numbers.
Chapter 8
8.1
Basic probability
The Probability Scale: Probability is always a number between 0 and 1.
0 means impossible (it will never happen).
1 means certain (it will definitely happen).
0.5 (or 1/2) means an even chance.
The Basic Formula: P(event) = Number of favorable outcomes / Total number of possible outcomes.
The Complement Rule (The "NOT" rule): The probability of an event NOT happening is 1 minus the probability of it happening.
Formula: P(not A) = 1 - P(A).
Example: If the chance of rain is 0.3, the chance of NO rain is 1 - 0.3 = 0.7.
8.2
Space diagram
What it is: A grid or table that lists all the possible outcomes when you do two things at once (like rolling two dice, or flipping two coins).
How to use it:
The total number of boxes in the grid is your denominator (bottom number).
Count the boxes that match what the question is asking for to get your numerator (top number).
Exam Tip: Always draw the grid neatly and fill it in carefully so you don't miss any outcomes!
8.3
Independent & mutually exclusive events
This is the most important part of the chapter, baby! Examiners love testing this.
Mutually Exclusive Events: Events that CANNOT happen at the same time.
Example: Rolling a 3 and rolling a 5 on a single dice roll. You can't get both.
The OR Rule (Addition): If two events are mutually exclusive, you ADD their probabilities to find the chance of one OR the other happening.
Formula: P(A or B) = P(A) + P(B).
Independent Events: Events where the outcome of the first one does NOT affect the outcome of the second one.
Example: Flipping a coin twice. Getting heads on the first flip doesn't change the 50/50 chance of the second flip.
The AND Rule (Multiplication): If two events are independent, you MULTIPLY their probabilities to find the chance of one AND the other happening.
Formula: P(A and B) = P(A) x P(B).
Tree Diagrams (Your best friend for this section!):
Used for two or more events in a row.
Golden Rule for Trees: Multiply along the branches (for AND). Add the final probabilities together at the ends (for OR).
Check: The probabilities on the branches from any single point must always add up to 1!
Chapter 9
9.1
Rational
Term-to-term rule: How to get from one number to the next (e.g., "add 3" or "multiply by 2").
Position-to-term rule (nth term): A formula to find any number in the sequence based on its position (n).
Arithmetic (Linear) Sequences: The numbers go up or down by the same amount each time.
Formula: nth term = dn + c
How to find it: 'd' is the common difference (the amount it goes up by). 'c' is the adjustment (what you need to add or subtract to the first term to make the formula work).
Example: 5, 8, 11, 14... (goes up by 3, so 3n. To get 5, we add 2). nth term = 3n + 2.
Quadratic Sequences: The difference between the numbers isn't constant, but the second difference is. The nth term will have an n^2 in it.
9.2
Rational & irrational
Rational numbers: Any number that can be written as a simple fraction (p/q, where q is not zero). This includes whole numbers, terminating decimals (like 0.5), and recurring decimals (like 0.333...).
Irrational numbers: Numbers that CANNOT be written as a simple fraction. Their decimals go on forever without repeating.
Examples: Pi (3.14159...), e, and most square roots (like sqrt(2) or sqrt(3)).
9.3
Surds
What is a surd? A square root that cannot be simplified to a whole number. It is an irrational number left in root form to be exact.
Multiplying Surds: sqrt(a) x sqrt(b) = sqrt(a x b)
Example: sqrt(2) x sqrt(3) = sqrt(6)
Dividing Surds: sqrt(a) / sqrt(b) = sqrt(a / b)
Example: sqrt(10) / sqrt(2) = sqrt(5)
Simplifying Surds: Look for the largest square number that divides into the number under the root.
Example: Simplify sqrt(12). 12 = 4 x 3. sqrt(4) is 2. So, sqrt(12) = 2 x sqrt(3).
Rationalizing the Denominator: Getting rid of the surd on the bottom of a fraction (examiners love this!).
Simple: 1 / sqrt(a) -> multiply top and bottom by sqrt(a). Result: sqrt(a) / a.
Harder (Conjugate): If the bottom is (a + sqrt(b)), multiply top and bottom by (a - sqrt(b)). This uses the difference of two squares to remove the root from the bottom.
9.4
Sets
Notation:
{ } means a set.
∈ means "is an element of" (is inside the set).
∉ means "is not an element of".
Venn Diagrams: Circles inside a box used to show relationships between groups.
The Box (Universal Set, ξ or U): Everything being considered.
Key Set Operations:
Union (A ∪ B): Everything in A OR B (or both). Think "Add them together".
Intersection (A ∩ B): Only the things that are in BOTH A AND B. Think "The overlap in the middle".
Complement (A' or A with a line over it): Everything NOT in A.
Subsets (⊂): When all elements of one set are inside another set.
Chapter 12
12.1
Types of averages
Mean: The "average" everyone usually thinks of. Add up all the numbers and divide by how many numbers there are.
Median: The exact middle number when all the numbers are listed in order from smallest to largest.
Mode: The number that appears the most often (the most popular one).
12.2
Comparisons using range and average
Range: This is NOT an average. It measures the "spread" of the data.
Formula: Highest value minus Lowest value.
How to compare two sets of data in an exam:
Compare the averages (usually the mean) to see which group is generally higher or lower.
Compare the ranges to see which group is more consistent. A smaller range means the data is more consistent/reliable. A larger range means the data is more spread out.
12.3
Calculation
When data is in a frequency table, you have to multiply the value by its frequency.
Mean Formula: (Sum of all (Value x Frequency)) divided by (Total Frequency).
Simple way to write it: Total of (fx) / Total of (f).
Median Position: To find where the median is, use the formula: (n + 1) / 2. (Where 'n' is the total frequency). Count through the frequencies to find that position.
Mode: The value with the highest frequency.
Range: Highest value minus Lowest value.
12.4
Estimating mean and findng modal
Exam Trap! When data is in groups (like 10-20, 20-30), we don't know the exact numbers. Therefore, any mean we calculate is an ESTIMATE. Examiners love asking "Why is this only an estimate?" Your answer: "Because we used the midpoints, not the exact data values."
Midpoint Formula: (Lower bound + Upper bound) / 2.
Estimated Mean Formula: (Sum of all (Midpoint x Frequency)) divided by (Total Frequency).
Modal Class: The group (class interval) that has the highest frequency.
Median Class: The group that contains the (n + 1) / 2 th value.
12.5
Quartiles
Quartiles split your ordered data into four equal quarters.
Lower Quartile (Q1): The middle of the bottom half of the data.
Position formula: (n + 1) / 4.
Upper Quartile (Q3): The middle of the top half of the data.
Position formula: 3(n + 1) / 4.
Interquartile Range (IQR): This measures the spread of the middle 50% of your data. It is great because it ignores extreme outliers (weirdly high or low numbers).
Formula: Q3 - Q1.
Chapter 13
13.1
Understanding units
Metric Length:
1 km = 1000 m
1 m = 100 cm
1 cm = 10 mm
Metric Mass: 1 kg = 1000 g
Metric Capacity: 1 litre = 1000 ml
THE BIG IGCSE TRAPS (Memorize these!):
Area is squared, so you multiply the conversion factor twice.
1 m = 100 cm, so 1 m^2 = 100 x 100 = 10,000 cm^2.
Volume is cubed, so you multiply the conversion factor three times.
1 m = 100 cm, so 1 m^3 = 100 x 100 x 100 = 1,000,000 cm^3.
13.2
Time speed density
Speed, Distance, Time Triangle:
Speed = Distance / Time
Distance = Speed x Time
Time = Distance / Speed
Tip: Watch your units! If speed is in km/h and time is in minutes, you MUST convert the time to hours first.
Density, Mass, Volume Triangle:
Density = Mass / Volume
Mass = Density x Volume
Volume = Mass / Density
13.3
Upper and lower bound
When a number is rounded or measured, it has a range of possible actual values.
The Golden Rule: The bound is always half of the unit it was rounded to.
Example: A length is measured as 10 cm to the nearest cm.
Lower bound (smallest it could actually be) = 10 - 0.5 = 9.5 cm.
Upper bound (largest it could actually be) = 10 + 0.5 = 10.5 cm.
Note: The actual value is always strictly less than the upper bound (9.5 <= x < 10.5).
Calculating with Bounds (Exam Trap!):
To find the MAXIMUM possible answer for a division (A / B), use the biggest top and smallest bottom: Max A / Min B.
To find the MINIMUM possible answer for a division (A / B), use the smallest top and biggest bottom: Min A / Max B.
13.4
Conversion graphs
These are straight line graphs used to convert between two different units (like miles to kilometers, or Celsius to Fahrenheit).
How to use them: Draw a straight line from the value on one axis until you hit the diagonal line, then read across to the other axis.
The Gradient: The steepness (gradient) of the line tells you the conversion rate.
Example: If the line goes up 2 units for every 1 unit across, the conversion rate is 2.
13.5
Exchanging currencies
Exchange rates tell you how much one currency is worth in another.
The Rule:
To convert FROM the base currency (the "1" in the rate) TO the other currency, you MULTIPLY.
To convert FROM the other currency BACK TO the base currency, you DIVIDE.
Example: Rate is 1 USD = 4 MYR.
To change 50 USD to MYR: 50 x 4 = 200 MYR.
To change 200 MYR to USD: 200 / 4 = 50 USD.
Chapter 17
17.1
Earning money
Wages: Usually paid by the hour.
Formula: Hourly rate x Number of hours worked.
Overtime: Often paid at a higher rate, like "time and a half" (1.5 times the normal hourly rate) or "double time" (2 times the normal rate).
Salary: A fixed amount paid for a year, usually paid monthly.
Formula: Annual salary / 12 = Monthly pay.
Commission: Paid a base salary plus a percentage of the sales they make.
Formula: Base salary + (Percentage x Total sales).
Piecework: Paid a fixed amount for every item they make or task they complete.
Formula: Rate per item x Number of items produced.
17.2
Borrow & Invest money
Simple Interest: Interest is only calculated on the original amount of money (the principal). It stays the same every year.
Formula: Interest = Principal x Rate x Time (I = P x r x t). Remember to write the rate as a decimal or divide by 100!
Total amount at the end = Principal + Total Interest.
Compound Interest: Interest is calculated on the original amount PLUS any interest already earned. The interest grows every year.
Formula: Total Amount = P(1 + r)^n
(P = Principal, r = interest rate as a decimal, n = number of years/time periods).
Exam Trap: Watch your time units! If the interest rate is "per year" but the time is given in months, you MUST convert the months into years first.
17.3
Buying & Selling
Profit and Loss:
Profit = Selling Price - Cost Price.
Loss = Cost Price - Selling Price.
Percentage Profit or Loss Formula: (Profit or Loss / Cost Price) x 100. (Always divide by the original Cost Price, never the selling price!)
Discounts:
Discount amount = Percentage x Original Price.
Sale Price = Original Price - Discount amount.
Multiplier Method (Faster!): If an item is 20% off, you are paying 80% of the price. Just multiply the original price by 0.80.
Tax (like VAT or GST):
Adding tax: Multiply the price by (1 + tax rate). For example, 10% tax means multiplying by 1.10.
Reverse tax: If a price already includes tax, divide by the multiplier to find the original price before tax.
Hire Purchase (Buying on credit/installments):
Total Hire Purchase Cost = Deposit + (Monthly payment x Number of months).
Extra Cost (Interest paid) = Total Hire Purchase Cost - Original Cash Price.