Carry the idea of a ratio from numbers to the lengths of segments — the alphabet of similarity.
Which cup of sugar-water tastes sweeter — the one with more sugar, or the one with the bigger ratio of sugar to water? It is always the ratio that decides. A ratio compares two quantities by division, and the moment we let those quantities be the lengths of segments, the whole language of shape swings open. Four segments whose lengths pair off perfectly form a proportion — a balanced equation you can cross-multiply — and that little piece of algebra is exactly what, three lessons from now, will let us say two triangles are the same shape. Along the way we even meet one magical cut, the golden ratio, that has fascinated artists for two thousand years.
A ratio a : b compares two quantities of the same kind by division — it is just the quotient ab (with b ≠ 0). When the two quantities share a unit, the units cancel and the ratio is a pure number with no unit at all. We first met ratios back in Stage 4 (Ratio & Proportion); here we put them to geometric work.
When two ratios are equal we write a proportion: a : b = c : d. Think of a drink mixed 2 parts syrup to 5 parts water — ratio 2 : 5. Pour a second glass twice the size, 4 : 10, and it tastes identical, because 2 : 5 = 4 : 10. Doubling both parts never changes the flavour; it never changes the ratio.
A ratio compares by division: a : b means a ÷ b. Equal ratios form a proportion, a : b = c : d. Multiplying both parts of a ratio by the same number leaves it unchanged — that is why 2 : 5 = 4 : 10 = 6 : 15.
Now let the two quantities be lengths. Lay one segment against another like a little ruler and ask: how many times does CD fit into AB? That number is the ratio of the two segments, AB : CD = ABCD. Because both are measured in the same unit, the unit cancels and the answer is a plain number.
Suppose AB = 6 cm and CD = 4 cm. Then AB : CD = 6 : 4. Reduce it just as you would a fraction — divide both by their greatest common divisor, 2 — to get the tidy form 3 : 2. As a decimal that is 1.5: AB is one-and-a-half times as long as CD.
The ratio of two segments is unit-free, so as long as you measure both the same way the unit makes no difference: 6 cm : 4 cm = 60 mm : 40 mm = 3 : 2. Always reduce to lowest terms.
Four lengths a, b, c, d are called proportional when the first pair has the same ratio as the second pair: a : b = c : d. The outer two, a and d, are the extremes; the inner two, b and c, are the means (read the proportion a : b = c : d left to right — the means sit in the middle).
Here is the single piece of algebra that powers the rest of Stage 17. Start from ab = cd and multiply both sides by b·d. The fractions clear, and you are left with
a : b = c : d ⟺ a·d = b·c.
In words: the product of the extremes equals the product of the means. This is the famous cross-multiplication, and it turns a question about ratios into a question about ordinary multiplication.
Are 2, 3, 4, 6 proportional? Check the ratios: 2 : 3 and 4 : 6 = 2 : 3 — equal. Or cross-multiply: extremes 2 · 6 = 12, means 3 · 4 = 12. The two products match, so 2 : 3 = 4 : 6 ✓. They are proportional.
A proportion is like a seesaw in perfect balance: there are several honest moves you can make that keep it level. Every one of them is really the same fact — ad = bc — written a new way. Starting from ab = cd (all four nonzero):
| Move | You may rewrite it as | Why |
|---|---|---|
| cross-multiply | a·d = b·c | clear the fractions |
| invert | b : a = d : c | flip both sides |
| alternate | a : c = b : d | swap the means |
| add (componendo) | (a + b) : b = (c + d) : d | add 1 to each side |
| subtract | (a − b) : b = (c − d) : d | subtract 1 from each side |
Let us prove alternation in full, so you see it is nothing but algebra. We are given ab = cd. Multiply both sides by b: that gives a = bcd. Now divide both sides by c: ac = bd. That is exactly a : c = b : d — the means have traded places, and the proportion is still true.
A ratio must compare like with like. You can form 6 cm : 4 cm (both lengths), but 3 km : 2 hours is not a length ratio — those are different kinds of quantity. And remember the cross product: a : b = c : d means a·d = b·c, never a·b = c·d.
Here is one proportion so beautiful it earned a name. Cut a segment AB at a point P so that the whole is to the long part as the long part is to the short part:
AB : AP = AP : PB.
Let the whole length be AB = 1 and the long part be AP = x, so the short part is PB = 1 − x. The proportion says 1x = x1 − x. Cross-multiply (extremes = means): 1·(1 − x) = x·x, that is
x² + x − 1 = 0 ⟹ x = √5 − 12 ≈ 0.618.
So the long part is about 0.618 of the whole and the short part about 0.382. The long-to-short ratio comes out to 1.618 : 1 — the golden ratio, written with the Greek letter φ (phi). The cut point P is the golden section. The same number φ appears in the regular pentagon, in the spiral of a sunflower's seeds, and in centuries of art and architecture.
A 12 cm segment is cut at its golden point. The longer piece is 12 × 0.618 ≈ 7.4 cm, and the shorter piece is 12 − 7.4 ≈ 4.6 cm. Check the magic: 7.4 : 4.6 ≈ 1.61, and 12 : 7.4 ≈ 1.62 — the whole-to-long ratio equals the long-to-short ratio. ✓
① A ratio a : b compares two like quantities by division; reduce it like a fraction, and it has no units.
② The ratio of two segments is AB : CD = AB ÷ CD — a pure number once both are measured the same way.
③ Four segments are proportional when a : b = c : d; a and d are the extremes, b and c the means; and a : b = c : d ⟺ a·d = b·c.
④ From a proportion you may invert, alternate, and add or subtract — every move is just ad = bc dressed differently.
⑤ The golden cut divides AB so that AB : AP = AP : PB, giving φ ≈ 1.618 (long-to-short) or 0.618 of the whole.
Next up: 17.2 — Parallel Lines Cut Segments Proportionally, where a bundle of parallel lines hands these proportions straight to the geometry of triangles.
Simplify the ratio 8 : 12 to lowest terms.
Divide both by their gcd, 4: 8 : 12 = 2 : 3.
If a : b = 3 : 4 and b = 12, find a.
a : 12 = 3 : 4, so cross-multiply: 4·a = 3·12 = 36, hence a = 9. (Or: b is 4 parts = 12, so 1 part = 3, and a is 3 parts = 9.)
Are 4, 6, 10, 15 proportional?
Cross-multiply the extremes and means: 4·15 = 60 and 6·10 = 60. The products are equal, so 4 : 6 = 10 : 15 ✓ — yes, they are proportional.
Given x : 6 = 4 : 3, solve for x.
Extremes = means: x·3 = 6·4 = 24, so x = 8.
The golden ratio φ ≈ 1.618 satisfies φ² = φ + 1. Check that the value works.
φ² ≈ 1.618² ≈ 2.618, and φ + 1 ≈ 1.618 + 1 = 2.618. The two agree, so φ² = φ + 1 ✓. (This is just x² + x − 1 = 0 rearranged for the long-to-short ratio.)
A 12 cm segment is cut at its golden point. How long is the longer piece?
The longer piece is 0.618 of the whole: 12 × 0.618 ≈ 7.4 cm (and the shorter piece is about 4.6 cm).
Six questions to lock it in. Tap the answer you think is right.
This lesson is the keystone of the whole stage: it quietly converts a number idea (ratio) into a geometry idea (proportional segments), and everything that follows — the side-splitter theorem, the tests for similar triangles, the k-and-k² scaling laws — is built on the one equivalence a : b = c : d ⟺ ad = bc. Spend time making that cross-product feel inevitable, not memorized: it is simply what you get when you clear the fractions.
Two misconceptions are worth catching early. The first is forming a ratio of unlike quantities — pairing a distance with a time, or centimetres with grams, and calling it a length ratio. Insist that both sides of a geometric ratio be lengths in the same unit, so the answer is unitless. The second, and more stubborn, is writing a·b = c·d instead of a·d = b·c when cross-multiplying a : b = c : d. The fix is to name the roles out loud every time — "extremes times extremes equals means times means" — and to point at the picture: the outer two multiply, the inner two multiply.
Common Core. This lesson lives at the bridge between number and geometry: 6.RP.A and 7.RP.A (understand ratio concepts and use proportional reasoning), G-SRT.B.5 (use proportional relationships among segments), with the golden-ratio derivation touching A-REI (solving the quadratic x² + x − 1 = 0). The golden section is a wonderful optional enrichment — a real quadratic arising from a real picture.