1. Types of Number
Before solving any mathematical problem, you must understand the different types of numbers and how they relate to each other.
| Type | Definition | Examples |
|---|---|---|
| Natural Numbers | Positive counting numbers — used for counting objects. | 1, 2, 3, 4, 5, ... |
| Integers | All whole numbers — positive, negative, and zero. No fractions or decimals. | ..., −3, −2, −1, 0, 1, 2, 3, ... |
| Prime Numbers | A number greater than 1 with exactly two factors — 1 and itself. | 2, 3, 5, 7, 11, 13, 17, 19, 23, ... |
| Composite Numbers | A positive integer with more than two factors — not prime. | 4, 6, 8, 9, 10, 12, ... |
| Square Numbers | The result of multiplying an integer by itself. | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 |
| Cube Numbers | The result of multiplying an integer by itself three times. | 1, 8, 27, 64, 125, 216 |
| Rational Numbers | Any number that can be written as a fraction p/q where p and q are integers and q ≠ 0. | ½, 0.75, −3, 0, 2.333... |
| Irrational Numbers | Numbers that cannot be written as a fraction — their decimal expansion is non-terminating and non-repeating. | √2, √3, π, √5 |
• 1 is NOT a prime number — it has only one factor (itself).
• 2 is the ONLY even prime number.
• 0 is neither positive nor negative — it is an integer but not a natural number.
Factors and Multiples
Factors of 12: 1, 2, 3, 4, 6, 12
Multiples of 4: 4, 8, 12, 16, 20, ...
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
HCF — Highest Common Factor
The largest factor that divides exactly into two or more numbers.
Method — Prime Factorisation:
Write each number as a product of prime factors. The HCF is the product of the common prime factors using the lowest power of each.
LCM — Lowest Common Multiple
The smallest number that is a multiple of two or more numbers.
Method — Prime Factorisation:
Write each number as a product of prime factors. The LCM is the product of all prime factors using the highest power of each.
📐 Worked Example 1 — Find the HCF and LCM of 36 and 84
36 = 2² × 3²
84 = 2² × 3 × 7
Common factors: 2² and 3¹
HCF = 2² × 3 = 4 × 3 = 12
All factors: 2², 3², 7
LCM = 2² × 3² × 7 = 4 × 9 × 7 = 252
2. Fractions, Decimals and Percentages
Fractions
| Operation | Rule | Example |
|---|---|---|
| Add / Subtract | Find a common denominator first, then add/subtract numerators. | ⅓ + ¼ = 4/12 + 3/12 = 7/12 |
| Multiply | Multiply numerators together and denominators together. Simplify. | ⅔ × ¾ = 6/12 = ½ |
| Divide | Multiply by the reciprocal of the second fraction (flip and multiply). | ⅔ ÷ ¼ = ⅔ × 4/1 = 8/3 = 2⅔ |
| Mixed Numbers | Convert to improper fractions first, then operate. | 2½ + 1¾ = 5/2 + 7/4 = 10/4 + 7/4 = 17/4 = 4¼ |
Converting Between Fractions, Decimals and Percentages
| Convert | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide numerator by denominator. | 3/8 = 3 ÷ 8 = 0.375 |
| Decimal → Fraction | Write as fraction over appropriate power of 10. Simplify. | 0.45 = 45/100 = 9/20 |
| Fraction → Percentage | Multiply by 100. | 3/8 × 100 = 37.5% |
| Percentage → Decimal | Divide by 100. | 37.5% ÷ 100 = 0.375 |
| Percentage → Fraction | Write over 100. Simplify. | 45% = 45/100 = 9/20 |
Percentage Calculations
Finding a Percentage of an Amount
Multiply the amount by the percentage as a decimal.
Find 35% of $240:
0.35 × 240 = $84
Percentage Increase / Decrease
Percentage change = (Change ÷ Original) × 100
Price rises from $80 to $92:
Change = 12. Percentage = (12/80) × 100 = 15%
Reverse Percentage
Find the original value before a percentage change.
After 20% increase, price = $120. Find original:
Original = 120 ÷ 1.20 = $100
Compound Interest
Amount = P × (1 + r/100)ⁿ
where P = principal, r = rate %, n = years.
$500 at 8% for 3 years:
500 × (1.08)³ = 500 × 1.2597 = $629.86
📐 Worked Example 2 — Reverse Percentage
A jacket is sold for $68 after a 15% discount. Find the original price.
85% of Original = $68
Original = 68 ÷ 0.85 = $80
3. Standard Form (Scientific Notation)
Large Numbers (positive n)
Move the decimal point LEFT. Count the moves = n.
4,750,000 = 4.75 × 10⁶
302,000 = 3.02 × 10⁵
Small Numbers (negative n)
Move the decimal point RIGHT. Count the moves = n (negative).
0.000047 = 4.7 × 10⁻⁵
0.00302 = 3.02 × 10⁻³
Calculating with Standard Form
📐 Worked Example 3 — Multiplying in Standard Form
Calculate (3.2 × 10⁴) × (2.5 × 10³). Give your answer in standard form.
3.2 × 2.5 = 8.0
10⁴ × 10³ = 10⁷
Check: A = 8.0, which satisfies 1 ≤ A < 10 ✓
📐 Worked Example 4 — Dividing in Standard Form
Calculate (6.6 × 10⁵) ÷ (3 × 10⁻²). Give your answer in standard form.
6.6 ÷ 3 = 2.2
10⁵ ÷ 10⁻² = 10⁵⁻⁽⁻²⁾ = 10⁷
4. Ratio and Proportion
Dividing in a Given Ratio
📐 Worked Example 5 — Sharing in a Ratio
Divide $360 between Ali and Sara in the ratio 5 : 4.
5 + 4 = 9 parts
$360 ÷ 9 = $40 per part
Ali = 5 × $40 = $200
Sara = 4 × $40 = $160
Check: $200 + $160 = $360 ✓
Direct and Inverse Proportion
Direct Proportion
As one quantity increases, the other increases at the same rate.
y ∝ x → y = kx
Example: If 5 metres of fabric costs $15, find the cost of 8 metres.
k = 15/5 = 3. Cost = 3 × 8 = $24
Inverse Proportion
As one quantity increases, the other decreases at the same rate.
y ∝ 1/x → y = k/x → xy = k
Example: 4 workers take 15 days. How long for 6 workers?
k = 4 × 15 = 60. Time = 60/6 = 10 days
5. Rate — Speed, Density, and Pressure
Speed
Density
Pressure
Speed: km/h or m/s. If speed is in km/h and time in minutes → convert minutes to hours first.
Converting m/s to km/h: multiply by 3.6. Converting km/h to m/s: divide by 3.6.
📐 Worked Example 6 — Average Speed
A car travels 120 km in 1 hour 30 minutes. Find the average speed in km/h.
1 hour 30 minutes = 1.5 hours
Speed = 120 ÷ 1.5 = 80 km/h
6. Estimation, Rounding, and Approximation
Rounding Methods
| Method | Rule | Example: 0.036482 |
|---|---|---|
| Decimal Places (d.p.) | Count digits after the decimal point. Look at the next digit — if 5 or more, round up. | 2 d.p. → 0.04 | 3 d.p. → 0.036 | 4 d.p. → 0.0365 |
| Significant Figures (s.f.) | Start counting from the first non-zero digit. Zeros between non-zero digits count. Leading zeros do NOT count. | 1 s.f. → 0.04 | 2 s.f. → 0.036 | 3 s.f. → 0.0365 |
• Leading zeros (before first non-zero digit) are NOT significant: 0.0045 has 2 s.f.
• Zeros between non-zero digits ARE significant: 3.04 has 3 s.f.
• Trailing zeros after a decimal point ARE significant: 2.50 has 3 s.f.
• Trailing zeros in a whole number are ambiguous — use standard form to clarify: 3.00 × 10³ has 3 s.f.
Limits of Accuracy
If a value x is rounded to the nearest unit u:
Lower Bound = x − ½u Upper Bound = x + ½u
📐 Worked Example 7 — Limits of Accuracy
A length is measured as 8.4 cm to the nearest 0.1 cm. Write down the upper and lower bounds.
Upper Bound = 8.4 + 0.05 = 8.45 cm
Note: the upper bound uses strict inequality (<) not ≤.
Maximum result = Upper Bound of numerator ÷ Lower Bound of denominator
Minimum result = Lower Bound of numerator ÷ Upper Bound of denominator
7. Surds
Rules for Surds
√a × √b = √(ab) Example: √3 × √5 = √15
√a ÷ √b = √(a/b) Example: √12 ÷ √3 = √4 = 2
√(a²) = a Example: √25 = 5
(√a)² = a Example: (√7)² = 7
Simplifying Surds
📐 Worked Example 8 — Simplifying Surds
Simplify: (a) √48 (b) 3√2 × 4√6
48 = 16 × 3
√48 = √(16 × 3) = √16 × √3 = 4√3
3√2 × 4√6 = (3 × 4) × (√2 × √6) = 12 × √12
√12 = √(4 × 3) = 2√3
= 12 × 2√3 = 24√3
Rationalising the Denominator
📐 Worked Example 9 — Rationalising the Denominator
Rationalise: 6/√3
6/√3 × √3/√3 = 6√3/(√3 × √3) = 6√3/3
6√3/3 = 2√3
8. Number Sequences
A sequence is an ordered list of numbers following a pattern. The nth term formula allows you to find any term without listing all previous terms.
Types of Sequences
| Type | Pattern | Example | nth Term |
|---|---|---|---|
| Arithmetic | Add a fixed number (common difference d) each time. | 3, 7, 11, 15, ... (d = 4) | 4n − 1 |
| Geometric | Multiply by a fixed number (common ratio r) each time. | 2, 6, 18, 54, ... (r = 3) | 2 × 3ⁿ⁻¹ |
| Square Numbers | Sequence of n². | 1, 4, 9, 16, 25, ... | n² |
| Fibonacci | Each term is the sum of the two previous terms. | 1, 1, 2, 3, 5, 8, 13, ... | No simple formula |
Finding the nth Term of an Arithmetic Sequence
nth Term of Arithmetic Sequence
where a = first term, d = common difference
Simplified form: nth term = dn + (a − d)
📐 Worked Example 10 — nth Term
Find the nth term of the sequence: 5, 9, 13, 17, ...
d = 9 − 5 = 4
nth term = 4n + (5 − 4) = 4n + 1
📝 Exam Practice Questions
Q1 [2 marks] — Find the HCF and LCM of 60 and 90.
60 = 2² × 3 × 5
90 = 2 × 3² × 5
HCF = 2¹ × 3¹ × 5¹ = 30
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180
Q2 [2 marks] — Write 0.00000823 in standard form.
8.23 × 10⁻⁶
Q3 [3 marks] — A price increases from $240 to $276. Calculate the percentage increase.
Change = 276 − 240 = $36
Percentage increase = (36 ÷ 240) × 100 = 15%
Q4 [3 marks] — A length is measured as 12.6 cm correct to 1 decimal place. Write down the lower and upper bounds of this length.
Unit = 0.1 cm → half unit = 0.05 cm
Lower bound = 12.6 − 0.05 = 12.55 cm
Upper bound = 12.6 + 0.05 = 12.65 cm
So: 12.55 ≤ length < 12.65
Q5 [3 marks] — Simplify fully: √75 − √27
√75 = √(25 × 3) = 5√3
√27 = √(9 × 3) = 3√3
5√3 − 3√3 = 2√3
Q6 [2 marks] — Calculate (4.5 × 10³) + (6 × 10²). Give your answer in standard form.
Convert to ordinary numbers: 4500 + 600 = 5100
OR: 4.5 × 10³ = 45 × 10² → (45 + 6) × 10² = 51 × 10² = 5.1 × 10³
Q7 [3 marks] — The nth term of a sequence is 3n − 5. (a) Find the 8th term. (b) Find the value of n for which the nth term equals 40. (c) Is 50 a term in this sequence? Justify your answer.
(b) 3n − 5 = 40 → 3n = 45 → n = 15
(c) 3n − 5 = 50 → 3n = 55 → n = 55/3 = 18.33...
Since n is not a whole number (integer), 50 is NOT a term in this sequence.
Q8 [4 marks] — Ali invests $2,500 at a compound interest rate of 6% per year. Calculate the total amount after 4 years. Give your answer correct to the nearest dollar.
Amount = P × (1 + r/100)ⁿ
Amount = 2500 × (1.06)⁴
(1.06)⁴ = 1.26248...
Amount = 2500 × 1.26248 = 3156.19...
Amount = $3,156 (to nearest dollar)