Scale ratio, scale factor and percentage

The one quantity

Uniform scaling multiplies every length by the same number. That number is the scale factor, k:

k = new length ÷ original length

k = 1 means no change. k > 1 enlarges. k < 1 reduces. Everything else is a different way of writing k.

Percentage

A percentage is k × 100. A factor of 1.5 is 150%; a factor of 0.4 is 40%. Print dialogs, copiers, slicers and image editors all use this form, and they all mean “scale to” — the output length is that percentage of the input length.

The trap is the English. “Scale to 150%” multiplies by 1.5. “Increase by 150%” multiplies by 2.5, because it keeps the original and adds 150% on top. Two readings, one number, a 67% difference in the answer.

Representative scale

Drawings, maps and models use a ratio, written 1:N. Its meaning is that one unit of the representation stands for N units of reality, so:

k = 1 ÷ N    N = 1 ÷ k

1:50 is a factor of 0.02, or 2%. A larger denominator is a smaller representation, which is why 1:100 is smaller than 1:50 despite the bigger number. Enlargements are written the other way round — 2:1 or 5:1 — and are factors of 2 and 5.

Translating between them

One object, all three notations

The fastest way to internalise the system is to watch a single change described three ways. A 1:50 site plan has a 120 mm wall drawn on it, and you need the sheet reissued at 1:20.

• As ratios: from 1:50 to 1:20. The factors are 1/50 and 1/20, and the change is (1/20) ÷ (1/50) = 50/20.
• As a factor: 50/20 = 5/2 = k = 2.5×.
• As a percentage: 2.5 × 100 = 250% — the number the plotter dialog wants. Phrased as a change it is “increase by 150%”, which is the same operation and a different figure.

wall on the new sheet: 120 × 5/2 = 300 mm  (check: 300 × 20 = 6,000 mm = 120 × 50 ✓)

The real-world dimension is unchanged at 6 m throughout. Only the representation moved, and the check confirms it: measured length × denominator must give the same reality on both sheets.

The same job done in the wrong direction

The failure is not arithmetic; it is deciding which denominator goes on top. Going from 1:50 to 1:20, the tempting move is to divide the new denominator by the old one, because it reads naturally left to right:

wrong: 20 ÷ 50 = 0.4 = 40% → 120 × 0.4 = 48 mm
right: 50 ÷ 20 = 2.5 = 250% → 120 × 2.5 = 300 mm

Both are plausible-looking numbers, and they differ by a factor of 6.25. The check that separates them takes one multiplication: a measured length times its denominator must return the same reality. 48 × 20 = 960 mm, which is not a 6 m wall, so 40% is wrong. 300 × 20 = 6,000 mm ✓. Alternatively, decide the direction before dividing: 1:20 is a bigger drawing than 1:50, so the factor must exceed 1, which rules out 0.4 before any arithmetic happens.

What the denominator means on the ground

Ratios feel abstract until they are attached to a real length. One metre of reality, drawn:

• 1:20 — 50 mm on paper. Door leaves and joinery details are legible.
• 1:50 — 20 mm on paper. A 100 mm partition is 2 mm wide.
• 1:100 — 10 mm on paper. That same partition is 1 mm, about the width of the line drawing it.

So 1:100 is not “slightly smaller” than 1:50 — it is half the length, and a quarter of the paper area, which is why a plan that moves from 1:50 to 1:100 loses detail rather than just size. The number after the colon counts reality per unit of paper, so it runs backwards to the size of the drawing: bigger denominator, smaller picture.

Two kinds of scale that are easy to confuse

A representative scale describes a state: this drawing is at 1:50. A transformation scale describes a change: print this at 141%. Applying a transformation to a representation changes it — a 1:50 plan printed at 141% is now 1:35.46, which no scale rule reads.

Keeping the two apart is the difference between “what scale is this?” and “what do I type into the dialog?”, and it is why the calculators on this site always report both.

Enlargement ratios: 2:1 is not 1:2

Ratios are ordered representation : reality, and swapping the two ends inverts the meaning entirely. This catches people constantly on macro and component drawings, where enlargement is the normal case:

• 2:1 — two units of paper per unit of reality. k = 2, 200%. A 3 mm pin is drawn 6 mm.
• 1:2 — one unit of paper per two of reality. k = 0.5, 50%. A 3 mm pin is drawn 1.5 mm.
• 5:1 — k = 5, 500%. A 3 mm pin is drawn 15 mm.
• 1:5 — k = 0.2, 20%. A 3 mm pin is drawn 0.6 mm.

2:1 and 1:2 differ by a factor of four, so a transposed ratio is never a small error. The general rule that covers both directions is k = numerator ÷ denominator, which is why the calculators here read the full ratio rather than assuming a 1 on the left — and why 2:1 → 5:1 correctly yields 250% instead of a nonsensical answer.

Why exactness matters here

1/3 as a percentage is 33.333…%, and 1:87 is 1.149…%. Rounding those at each step and then multiplying is how a chain of resizes ends up visibly wrong. Keeping fractions exact until the moment of display costs nothing and removes an entire class of error.

Concretely: reduce to 1/3, then restore with a rounded 300.0% and you land on 0.99999, a 0.001% miss. Do it with the rounded pair 33.3% and 300% and you land on 0.999 — 0.1% out, or 0.8 mm along the 841 mm side of an A1 sheet, from one rounding. Held as fractions, 1/3 × 3 = 1 exactly, and the same holds for the ugliest ratio you can enter: 1/87 × 87 = 1, with no residue to accumulate over a chain of five steps.