Ratio
Ratios, those deceptively simple beasts that say “A : B” but can hide in shadows until they pounce on your exam paper. You might think that is just sharing things, but ratios show up in disguise inside algebra, geometry, speed problems, and more. Master it, and you’ll spot relationships everywhere, even in your snack distribution!
By the time you’re 16, ratios aren’t just “share sweets in the ratio 2:3” You may be expected to:
- Simplify ratios, including with units.
- Split totals with ratio statements, sometimes algebraic.
- Switch between ratios and fractions gracefully.
- Tangle with compound ratios, linking multiple ratios.
- Translate ratio into algebraic form (2x : 3x or x : (x + 4)).
- Embed ratio reasoning in contextual problems – mixing, geometry, scaling, speed, etc.
In short, you’ll need ratio agility, not one, repetitive, cut and paste method.
‘Ratio’ means reasoning, from the Latin ‘ratiō’ meaning ‘reckoning’ or ‘reason’.
Simplifying (and unit wrangling)
Simplifying is your warm-up. But more advanced papers can throw in a simple but frustrating twist: mismatched units. Suppose you see:
“The ratio of walking time to resting time is 1 hour : 30 minutes.”
If you naively say 1:30, you’ll be wrong. First convert to a common unit: 60 : 30 = 2 : 1. Tick! That’s the “simple form.” Always check units before simplifying: it’s a tiny detail that can cost a valuable mark.
Exam tip: Always check units before simplifying – it’s an easy trap to dodge.
Sharing a quantity in a ratio
Classic territory: “Split £360 into the ratio 3:5.” That gives parts of £135 and £225. Fine.
But if instead the question says:
“Two friends share money in the ratio (x + 2) : (2x – 1). If the total is £90, find x.”
Now you’re in algebraic ratio land. You convert those ratio parts into expressions, set up (x+2) + (2x-1) = 90 scaled by the “unit part,” and off you go. That’s exactly the kind of question examiners enjoy.
Ratio to fraction
Ratios and fractions are two faces of the same coin. If the ratio is 2:3, the first quantity is:
2 / (2 + 3) = 2 / 5 of the whole.
Examiners also love to reverse it. You might be told, “A is 3 sevenths of the total. What’s the ratio of A : B?” That immediately gives 3 : 4. No arithmetic needed, just a clear concept and you’ve bagged some nice and easy marks!
Here’s where ratios start gossiping behind your back. Suppose:
Where does the idea of a ratio come from? The Greeks first formalised ratio in geometry – Euclid’s Elements treats ratios as fundamental comparisons between magnitudes. Centuries later, mathematicians extended ratio into algebra, rate, and proportion – laying foundations for modern physics, economics, and beyond. When you solve a ratio problem on your maths paper, you’re walking in the footsteps of Euclid, hopefully with fewer confusing parchment diagrams for your sanity.
Compound ratios
- A : B = 3 : 4
- B : C = 2 : 5
To link A and C, you make the B parts match:
- Scale the first ratio by 2 → 6 : 8
- Scale the second by 4 → 8 : 20
Then A : C = 6 : 20 = 3 : 10. That kind of composite thinking shows up in areas like scaling, map questions, and similar shapes.
Algebraic (symbolic) ratios
When you see ratios involving x, you’re doing mini algebra. Examples:
“The ratio of two numbers is 4 : 5. Their difference is 9. What are the numbers?”:
Set them as 4x and 5x. Then 5x – 4x = 9, so x = 9. The numbers are 36 and 45.
Also in geometry:
“A triangle’s sides are in the ratio x : (x + 3) : (2x – 1), and its perimeter is 72 cm. Find its side lengths.”
You write x + (x + 3) + (2x – 1) = 72, solve for x, then scale. That kind of question blends ratio, algebra, and geometry.
Famously the golden ratio φ (1.618…) appears in sunflowers, spirals, art, and galaxies.
Ratio, proportion and variation
Ratio is a close relative of proportion. In direct proportion, you might see y ∝ x, so y = kx. That’s saying the ratio y : x is constant (equal to k:1).
In inverse proportion, as one quantity increases, the other shrinks so that their product is fixed — the ratio flips in behaviour.
Even when a question doesn’t shout “ratio,” you’ll often recognise it hiding behind y = kx or y = k over x
Ratios in context
This is where examiners challenge you. Ratio will come wrapped in geometry, similarity, mixtures, rates, and more. Here are some forms:
- Similar shapes: If the area ratio is 9 : 25, then the side ratio is √9 : √25 = 3 : 5.
- Mixture problems: “Mix sugar and water in ratio 2:5. To make 1.4 litres, how much sugar is needed?” You’d compute (2/7) x 1.4.
- Map or scale: “On a map, 1 cm represents 50 km. Two towns are 6 cm apart. What’s their real separation?” You apply the scale ratio.
- Speed, work, rates: Ratio reasoning can underlie more advanced rate problems – distance : time, work rates, etc.
The trick is: convert to the same units, express unknowns in ‘parts’, then translate back. Clear structure will get you marks!
Relevance and applications
So what’s the point of learning about ratios? Well, ratios are everywhere – from recipes to DNA sequencing, from architectural models to financial portfolios. Engineers use them to size gears and pulleys; artists to maintain aspect ratios; chemists to mix reactants. Even sports statistics use ratios to compare players’ performance. Ratio is the language of comparison, and once you understand it well, you’ll see it everywhere – and certainly use it outside of exams.
Musical harmony comes from octaves being in the frequency ratio 2:1, and perfect fifths 3:2.
At first glance, ratio seems comfortable — just share, simplify, convert. But many of the most competent maths students find it to be a bit of a shape-shifter: algebra disguised as arithmetic, geometry hiding in scale, mixtures in everyday life. If you can see through the disguises, then you’re mastering one of the most flexible, useful ideas in GCSE Maths. And that sets you up beautifully for scope, proportion, calculus, premier league football – whatever comes next.