Fractions, Ratio, and Proportion

🍕 Imagine you and two friends share three pizzas equally. Or a map where every 1 cm stands for 50 km of real road. Or a sale where you save 30% on a pair of trainers. All of these everyday situations use the same powerful mathematical ideas: fractions, ratio, and proportion. In this lesson you will learn to multiply and divide fractions (even tricky mixed numbers!), write ratios using the a:b notation, and swap fluently between fractions, decimals, and percentages. Master these tools and you can solve problems that trip up many adults. Let us get started!

Multiplying fractions is the operation where you find a fraction OF another fraction — or of a whole number. The rule is beautifully simple: multiply the numerators (top numbers) together, then multiply the denominators (bottom numbers) together. So 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2. Always simplify your answer by dividing the numerator and denominator by their highest common factor (HCF) — the biggest number that divides exactly into both. Here HCF of 6 and 12 is 6, giving 1/2. You can also cross-cancel BEFORE multiplying: spot any numerator and denominator that share a common factor and divide them first to keep numbers smaller.

Mixed numbers (like 2 and 1/2) must be converted to improper fractions before you multiply or divide. An improper fraction has a numerator bigger than its denominator. To convert: multiply the whole number by the denominator, then add the numerator — keep the same denominator. So 2 and 1/2 becomes (2×2 + 1)/2 = 5/2. For 3 and 3/4: (3×4 + 3)/4 = 15/4. Once everything is an improper fraction, apply the normal multiplication rule, simplify, then convert back if needed: divide numerator by denominator to find the whole number, and the remainder becomes the new numerator.

Dividing fractions uses the keep-change-flip rule (also called KCF). Keep the first fraction exactly as it is. Change the division sign to multiplication. Flip the second fraction — this flipped version is called the reciprocal. Then multiply as normal. Example: 3/4 ÷ 1/2 → keep 3/4, change ÷ to ×, flip 1/2 to 2/1 → 3/4 × 2/1 = 6/4 = 3/2 = 1 and 1/2. Why does flipping work? Dividing by a number is the same as multiplying by its reciprocal — for instance, dividing by 2 is the same as multiplying by 1/2. KCF makes that swap automatic.

A ratio compares two or more quantities of the same kind. We write it using a colon: a:b. For example, if a smoothie uses 2 parts mango to 3 parts orange juice, the ratio is 2:3. Ratios do not have units — they just show the relationship. Like fractions, ratios can be simplified: 4:6 simplifies to 2:3 (divide both sides by 2). Scale on a map is a ratio: 1:50 000 means every 1 cm on the map equals 50 000 cm (500 m) in real life. To use a ratio to share an amount, first find the total number of parts, then divide the amount by that total to find one part, then multiply each share accordingly.

Fractions, decimals, and percentages are three different ways to write the same proportion. To convert a fraction to a decimal, divide the numerator by the denominator: 3/4 → 3 ÷ 4 = 0.75. To convert a decimal to a percentage, multiply by 100: 0.75 × 100 = 75%. To go from percentage back to a fraction, write it over 100 and simplify: 75/100 = 3/4. Being able to switch between all three forms lets you compare values quickly. Which is bigger, 3/8 or 35%? Convert 3/8 to a percentage: 3 ÷ 8 = 0.375 = 37.5%. So 3/8 is bigger than 35%. This kind of comparison comes up in data handling and financial problems constantly.

Proportion means that two ratios or rates are equal — it describes a multiplicative relationship between quantities. If 5 books cost £15, the proportion tells us every 1 book costs £3 (divide by 5). So 8 books cost £24 (multiply by 8). This unitary method — finding the value of one unit first — works for any proportion problem. Proportion problems also appear in recipes, speed-distance-time, and currency conversion. Always ask: what is the value of ONE? Then scale up or down. Check your answer makes sense: if you buy more, you should spend more; if you travel faster, you arrive sooner.