Mathematics Intermediate

Integers are the set … −3, −2, −1, 0, 1, 2, 3 … They include positive numbers, zero, and negative numbers (numbers less than zero). On a number line, negative numbers sit to the left of zero.

• Adding a negative number is the same as subtracting: 5 + (−3) = 5 − 3 = 2.
• Subtracting a negative number is the same as adding: 5 − (−3) = 5 + 3 = 8.
• Two negatives multiplied give a positive: (−4) × (−2) = 8.
• A negative times a positive gives a negative: (−4) × 2 = −8.

Temperature example:
  Monday: −5 °C.  Rise of 9 °C.
  Tuesday: −5 + 9 = 4 °C.

When an expression contains several operations, we follow a strict order so everyone gets the same answer. The acronym PEMDAS (or BODMAS) helps us remember:

• Parentheses (brackets) first.
• Exponents (powers and roots) next.
• Multiplication and Division — left to right.
• Addition and Subtraction — left to right.

Evaluate: 3 + 4 × 2² − (6 ÷ 3)
  Step 1 Parentheses: 6 ÷ 3 = 2
  Step 2 Exponents:   2² = 4
  Step 3 Multiply:    4 × 4 = 16
  Step 4 Add/Sub:     3 + 16 − 2 = 17

A ratio compares two quantities: 3 : 5 means "3 for every 5." A rate compares two different units (e.g. 60 km per hour). A proportion states that two ratios are equal.

• To simplify a ratio, divide both parts by their GCF: 12 : 8 = 3 : 2.
• To solve a proportion, use cross-multiplication.

Proportion: solve for x
  3⁄4 = x⁄20
  Cross-multiply: 3 × 20 = 4 × x
  60 = 4x
  x = 15

Unit rates are useful for comparisons: a 500 mL bottle for $2.50 costs $0.005 per mL = $5.00 per litre.

A percentage is a fraction with denominator 100. The word "percent" means "per hundred" — the symbol is %.

• To convert a percentage to a decimal, divide by 100: 35% = 0.35.
• To find a percentage of an amount: multiply. 20% of 80 = 0.20 × 80 = 16.
• Percentage increase: (increase ÷ original) × 100.
• Percentage decrease: (decrease ÷ original) × 100.

A shirt costs $40 and is 25% off. What is the sale price?
  Discount = 25% of $40 = 0.25 × 40 = $10
  Sale price = $40 − $10 = $30

An exponent (or power) tells you how many times to multiply a base by itself. 5² = 5 × 5 = 25. The small raised number is the exponent; 5 is the base.

• Square: n² (e.g. 4² = 16).
• Cube: n³ (e.g. 2³ = 8).
• Square root √: the inverse of squaring. √49 = 7 because 7² = 49.
• Any non-zero number to the power 0 equals 1: 6⁰ = 1.

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
√36 = 6,  √100 = 10,  √144 = 12

A prime number has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11). Prime factorisation writes a number as a product of primes. Use a factor tree to find it.

• Greatest Common Factor (GCF): the largest factor shared by two numbers.
• Least Common Multiple (LCM): the smallest multiple shared by two numbers.

Prime factorise 36 and 48:
  36 = 2² × 3²
  48 = 2⁴ × 3

GCF = 2² × 3 = 12
LCM = 2⁴ × 3² = 144

GCF is used to simplify fractions; LCM is used to add fractions with different denominators.

In algebra, letters called variables stand for unknown numbers. An equation is a statement that two expressions are equal. We solve an equation by finding the value of the variable that makes it true.

• One-step equation: undo one operation. x + 7 = 15 → x = 15 − 7 = 8.
• Two-step equation: undo two operations in reverse PEMDAS order.

Solve: 2x + 5 = 17
  Subtract 5 from both sides: 2x = 12
  Divide both sides by 2:      x = 6
Check: 2(6) + 5 = 12 + 5 = 17  ✓

Golden rule: whatever you do to one side of an equation, do the same to the other side.

The coordinate plane is formed by two number lines that cross at right angles. The horizontal line is the x-axis; the vertical line is the y-axis. Their crossing point is the origin (0, 0).

• A point is given as an ordered pair (x, y): move x units along the x-axis, then y units up (or down if negative).
• The plane is divided into four quadrants (I through IV).

Quadrant I:   x > 0, y > 0   (top-right)
Quadrant II:  x < 0, y > 0   (top-left)
Quadrant III: x < 0, y < 0   (bottom-left)
Quadrant IV:  x > 0, y < 0   (bottom-right)

Area measures the space inside a 2-D shape (in square units). Volume measures the space inside a 3-D object (in cubic units).

• Rectangle: A = length × width
• Triangle: A = ½ × base × height
• Circle: A = π × r²  (π ≈ 3.14)
• Rectangular prism (box): V = length × width × height

Area of a triangle: base = 10 cm, height = 6 cm
  A = ½ × 10 × 6 = 30 cm²

Volume of a box: 5 cm × 4 cm × 3 cm
  V = 5 × 4 × 3 = 60 cm³

An angle is formed where two rays meet at a common point called the vertex. Angles are measured in degrees (°).

• Acute: less than 90°.
• Right: exactly 90°.
• Obtuse: between 90° and 180°.
• Straight: exactly 180°.
• Reflex: greater than 180°.
• Angles in a triangle always add up to 180°.
• Angles on a straight line add up to 180°; angles around a point add up to 360°.

Triangle angles: 50° + 70° + x = 180°
  x = 180° − 50° − 70° = 60°

Three measures describe the centre of a data set:

• Mean (average): sum of all values ÷ number of values.
• Median: the middle value when data is ordered. If two middle values exist, take their mean.
• Mode: the value that appears most often (a data set can have no mode, one mode, or more than one mode).

Data set: 4, 7, 7, 9, 3

Ordered: 3, 4, 7, 7, 9
Mean   = (3 + 4 + 7 + 7 + 9) ÷ 5 = 30 ÷ 5 = 6
Median = 7  (middle value)
Mode   = 7  (appears twice)

Probability measures how likely an event is to happen. It is always a number from 0 (impossible) to 1 (certain).

P(event) = number of favourable outcomes ÷ total equally-likely outcomes

• Probability can be written as a fraction, decimal, or percentage.
• P(event does NOT happen) = 1 − P(event happens).

Roll a fair six-sided die. What is P(rolling a 3)?
  Favourable outcomes: 1  (just the face showing 3)
  Total outcomes: 6
  P(3) = 1⁄6 ≈ 0.167 ≈ 16.7%

P(not rolling a 3) = 1 − 1⁄6 = 5⁄6