Maths 1A Ch1: Basic Mathematics

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Maths 1A Ch1: Basic Mathematics

Key points + Practice

The UT1 Maths paper covers Book 1A Chapters 1 to 3 (MCQ, short questions and conventional questions). Chapter 1, Basic Mathematics, revises whole-number operations and introduces factors, multiples, HCF and LCM.

➕ 1. Four Operations and Order of Operations

The four operations are addition, subtraction, multiplication and division.

Order of operations (very important):

1. Brackets ( )

2. Powers

3. Multiplication and division (from left to right)

4. Addition and subtraction (from left to right)

Examples:

3 + 4 × 2 = 3 + 8 = 11 (not 14)

(3 + 4) × 2 = 7 × 2 = 14

20 ÷ 4 × 2 = 5 × 2 = 10 (left to right; not 20 ÷ 8)

Distributive law: a × (b + c) = a × b + a × c

Example: 6 × 25 = 6 × (20 + 5) = 120 + 30 = 150

🔗 2. Factors and Multiples

Factors: whole numbers that divide a number exactly.

Example: the factors of 12 are 1, 2, 3, 4, 6 and 12.

Multiples: the results of multiplying a number by whole numbers.

Example: the multiples of 5 are 5, 10, 15, 20, ...

Prime numbers: whole numbers greater than 1 that have exactly two factors, 1 and the number itself.

Primes below 20: 2, 3, 5, 7, 11, 13, 17, 19.

Note: 1 is not a prime number; 2 is the smallest prime and the only even prime.

Composite numbers: numbers with more than two factors, e.g. 4, 6, 8, 9.

🧱 3. Prime Factorisation, HCF and LCM

Prime factorisation: writing a number as a product of prime factors.

Example: 36 = 2 × 2 × 3 × 3 = 2² × 3²

Short division:

36 ÷ 2 = 18 → 18 ÷ 2 = 9 → 9 ÷ 3 = 3 → 3 ÷ 3 = 1

So 36 = 2 × 2 × 3 × 3.

HCF (Highest Common Factor): take the common prime factors with the smaller index.

Example: 24 = 2³ × 3, 36 = 2² × 3²

HCF = 2² × 3 = 12

LCM (Least Common Multiple): take all prime factors with the larger index.

LCM = 2³ × 3² = 72

Relationship: HCF × LCM = product of the two numbers (for two numbers only)

Check: 12 × 72 = 864 = 24 × 36 ✓

🍕 4. Fractions

Reducing: divide the numerator and denominator by a common factor.

12/18 = 2/3 (divide both by 6)

Equivalent fractions and common denominators:

1/2 and 1/3 → 3/6 and 2/6 (common denominator 6)

Addition and subtraction: use a common denominator first.

1/2 + 1/3 = 3/6 + 2/6 = 5/6

3/4 − 1/6 = 9/12 − 2/12 = 7/12

Multiplication: multiply numerator by numerator, denominator by denominator.

2/3 × 3/5 = 6/15 = 2/5

Division: multiply by the reciprocal.

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

Fractions and decimals: 3/4 = 0.75; 0.6 = 6/10 = 3/5

本節生詞：simplest form

📝 例題

例題 1：Work out 12 + 3 × (8 − 5)²

例題 2：Find the HCF and LCM of 18 and 24.

例題 3：Work out 2/3 + 1/4 and give the answer in simplest form.

✏️ 練習 → 📕 錯題重練

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