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Chapter 2 of 10

🔢 Numbers, Ratio & Percentage

6 min+15 XP

The Number strand covers prime factorisation, HCF/LCM, indices, standard form, ratio, proportion, percentage, rate and speed. These topics appear in both Paper 1 and Paper 2 and are fundamental to all other strands. Weakness in numbers will cripple your ability to handle algebra, geometry, and statistics. This chapter provides comprehensive coverage with detailed worked examples for every subtopic.

Prime Factorisation, HCF & LCM

Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers. This fundamental result is called the Fundamental Theorem of Arithmetic. We express the factorisation using index (power) notation for clarity and efficiency. For example, instead of writing 72 = 2 x 2 x 2 x 3 x 3, we write 72 = 2^3 \times 3^2.

The primary reason we learn prime factorisation is to find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two or more numbers. The HCF is the largest number that divides evenly into all given numbers, while the LCM is the smallest number that all given numbers divide into evenly. These concepts have practical applications in problems involving repeated events, sharing equally, and finding common denominators.

To find HCF: Take the product of all common prime factors, each raised to the lowest power that appears in any factorisation.
To find LCM: Take the product of all prime factors, each raised to the highest power that appears in any factorisation.

🔢Prime Factorisation, HCF & LCM

Express numbers as products of prime factors using index notation. Find HCF by taking lowest powers of common primes; find LCM by taking highest powers of all primes.

Prime factorisation (index notation)
n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}
HCF (product of common primes, lowest powers)
\text{HCF}(a,b) = \prod (\text{common primes})^{\min(\text{powers})}
LCM (product of all primes, highest powers)
\text{LCM}(a,b) = \prod (\text{all primes})^{\max(\text{powers})}
Relationship between HCF and LCM
\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

📝Worked Example 1: Finding HCF and LCM

Q
Question

Find the HCF and LCM of 72 and 120.

📝Worked Example 2: HCF/LCM Word Problem

Q
Question

Two buses leave the same station at 8:00 am. Bus A returns every 18 minutes and Bus B returns every 24 minutes. When will they next be at the station together?

🎯Key Takeaway
Master the 7 laws of indices and standard form — they appear in almost every Paper 1. For HCF/LCM, always use prime factorisation in index notation. For percentages, remember: reverse percentage DIVIDES, not multiplies. Average speed = total distance / total time (not the average of speeds). For proportion, identify the model first (direct, inverse, or squared) then find k. For number patterns, always calculate first and second differences to determine if the sequence is linear or quadratic. For fractions without a calculator, convert mixed numbers to improper fractions first and always find the LCD.

What the full chapter covers

  • Prime Factorisation, HCF & LCM

  • Laws of Indices (Exponents)

  • Standard Form (Scientific Notation)

  • Ratio & Proportion

  • Percentage, Simple & Compound Interest

  • Rate, Speed, Distance & Time

  • Number Properties and Surds

  • More on Proportion

  • Number Patterns & Sequences

  • Estimation & Significant Figures

  • Fractions Without a Calculator

  • Paper 1 Number Skills Checklist

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