Fractions, Factors, and Number Theory Basics

Why can't you split a pizza evenly among 3 people using decimal slices? Because 1/3 is 0.333... forever — decimals cannot represent it exactly. Fractions can. They are not an outdated notation from grade school; they are the only exact way to express many rational numbers.

What fractions represent

A fraction a/b means “a parts out of b equal pieces.” The top number (numerator) is how many pieces you have; the bottom (denominator) is how many pieces make a whole. The fraction 3/4 means three out of four equal parts — 75% of the whole.

Adding fractions: the common denominator

You can only add fractions when the denominators match. To add 1/3 + 1/4, find a common denominator (12), convert both fractions, and add:

1/3 + 1/4
= 4/12 + 3/12
= 7/12

Rule: a/b + c/d = (ad + bc) / bd

Multiplying and dividing

Multiplication is straightforward: multiply across. 2/3 × 4/5 = 8/15. Division flips the second fraction and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

Simplification and the GCF

A fraction is in simplest form when the numerator and denominator share no common factor other than 1. To simplify, divide both by their Greatest Common Factor (GCF). For 18/24: GCF(18, 24) = 6, so 18/24 = 3/4.

The GCF is found efficiently using the Euclidean algorithm, which repeatedly divides and takes remainders:

GCF(48, 18):
48 = 2 × 18 + 12
18 = 1 × 12 + 6
12 = 2 × 6  + 0  ← remainder is 0, so GCF = 6

Prime factorisation

Every integer greater than 1 can be written as a product of prime numbers in exactly one way (ignoring order). This is the Fundamental Theorem of Arithmetic and it is why primes are called the “atoms” of number theory.

Prime factorisation of 360:
360 = 2 × 180
    = 2 × 2 × 90
    = 2 × 2 × 2 × 45
    = 2 × 2 × 2 × 3 × 15
    = 2 × 2 × 2 × 3 × 3 × 5
    = 2³ × 3² × 5

Prime factorisation is the backbone of the GCF and LCM. To find the GCF, take the lowest power of each shared prime. For the LCM, take the highest power of every prime that appears.

LCM: the least common multiple

The LCM of two numbers is the smallest number that both divide into evenly. It answers scheduling questions: if Bus A comes every 12 minutes and Bus B every 8 minutes, when do they arrive together? LCM(12, 8) = 24 — every 24 minutes.

The LCM and GCF are connected: LCM(a, b) = (a × b) / GCF(a, b). This means you never need to compute both from scratch.

Ratios and proportions

A ratio compares two quantities: 3:2 means “for every 3 of this, there are 2 of that.” A proportion says two ratios are equal: 3/4 = 6/8. Cross-multiplying is the standard test: 3 × 8 = 4 × 6 = 24, so they are proportional.

Percentages: fractions of 100

A percentage is simply a fraction with denominator 100. 45% = 45/100 = 9/20. The word “percent” literally means “per hundred” (Latin per centum). Three essential percentage operations:

1. What is X% of Y? — Multiply: 0.X × Y
2. X is what percent of Y? — Divide: (X / Y) × 100
3. Percent change — ((new − old) / old) × 100

Why this matters beyond school

• Cooking — Scaling a recipe from 4 servings to 6 is a ratio problem: multiply every ingredient by 6/4 = 3/2.
• Finance — Interest rates, tax brackets, and discounts are all percentage calculations.
• Cryptography — RSA encryption relies on the difficulty of factoring very large numbers into primes.

Fractions are not a limitation of mathematics — they are a feature. They give you exact answers where decimals can only approximate.