Solve any of A, B, C, D in a proportion with GCD simplification and dual proportion bars.
Both bars should split in identical proportions when the ratios are equivalent.
A:B = C:D ⟺ A × D = B × C. The simplified ratio uses the GCD when both terms are integers; decimal-form is shown otherwise.
Ratios pop up everywhere: aspect ratios in design (16:9, 4:3, 21:9), gear ratios in cycling, mixing ratios in chemistry and concrete, screen-resolution scaling, financial ratios (debt-to-equity), recipe scaling. The fundamental algebraic identity — A:B = C:D ↔ A·D = B·C — is the cross-multiplication rule that lets you solve for any one of the four when the other three are known. The arithmetic is trivial individually but tedious to do by hand repeatedly, especially when the result needs simplification (computing GCD by hand) or conversion between equivalent representations (16:9 vs 1.778:1 vs 1920:1080). This calculator handles the four cases (solve for any one of A, B, C, D), simplifies the resulting ratio to lowest terms, and renders dual proportion bars showing the two ratios side by side.
Given the proportion A:B = C:D:
Simplification: divide both sides of the resulting ratio by GCD(A, B). Euclidean algorithm: GCD(a, 0) = a; GCD(a, b) = GCD(b, a mod b).
Decimal form = A / B (for visual ratio interpretation: 16:9 = 1.778, 4:3 = 1.333, 21:9 = 2.333).
The chart shows two horizontal proportion bars: A:B segment and C:D segment, scaled to a common reference so the proportional shapes are visually comparable.
Pick Solve for (A, B, C, or D). Enter the three known values. The calculator displays: - The solved value. - Simplified form (e.g. 16:12 → 4:3). - Decimal form (e.g. 4:3 → 1.333). - Dual proportion bar.
The simplification only applies when both sides are integer-typed; non-integer inputs are kept in decimal form.
Solve for D: A=16, B=9, C=1920, find D.
Solve for C: A=3, B=4, D=20.
Recipe scaling: A=200 g flour : B=120 g water = C=? : D=300 g water.
Cycling gear: 52 t front × 12 t rear = 4.333 ratio. To get the same ratio with 50 t front: 50 / 4.333 = 11.5, round to 11 (3 t difference, ratio drift to 4.545).
Direction of proportion. A:B = C:D means A is to B as C is to D. Reversing one side (A:B = D:C) gives the inverse proportion — different equation entirely.
GCD only on integers. The calc simplifies if all four values are integers. With decimal inputs (1.5:2.7), it leaves the decimal form unsimplified.
Zero in denominator. Solving for A or C requires non-zero denominator (B, D). The calc rejects with an error.
Sign handling. Negative ratios are mathematically valid (slope = −2:3 = downward) but usually confusing. The calc handles them; keep track of sign meaning.
Aspect-ratio conventions. "16:9" and "1.778:1" are equivalent forms. Some tools require specific format. The calc shows both.
Different bases. A 1:64 model means the model is 1/64 the real thing. Don't invert.
Ratio vs fraction. 1:3 is not 1/3 — it's 1 part to 3 parts, total 4 parts. 1:3 = 25 % of total, not 33 %.
Multi-term ratios. 1:2:3 (cement:sand:gravel for concrete) is a triplet, not a binary ratio. The calc handles binary only — for triplets, normalize each pair separately.
Cross-multiplication direction. A·D = B·C, not A·C = B·D. Easy to flip when typing fast.
Scaling preserves the ratio. Multiplying both sides by 2 keeps 16:9 = 16:9, doesn't make it 32:9. Common error in design layouts.
Display vs production resolution. 1920×1080 is 16:9 display, but pixel aspect ratio (PAR) on legacy formats can be non-square (NTSC 720×480 has PAR 0.9, displaying as 853×480 effective). Modern devices are 1:1 PAR.
Aspect ratios in screen design are the most visible everyday use: 16:9 for HDTV/YouTube, 4:3 for legacy TV and tablets, 21:9 for ultra-wide cinema and gaming, 9:16 for vertical video (TikTok, Instagram Reels, Stories). Cropping or letterboxing between them needs the ratio math: a 1920×1080 (16:9) image displayed in a 21:9 frame either gets pillarboxed (black bars on the sides) or cropped at the top and bottom, losing 11 % of image height. Cycling-gear ratios determine pedaling effort vs road speed: a 52-tooth front × 12-tooth rear gives a 4.33 ratio, meaning one pedal revolution turns the rear wheel 4.33 times — a fast gear for downhill or flat sprinting. In recipe scaling and chemistry, ratio simplification matters less than precise multiplication; in design and construction, the simplification to lowest integer terms (5:3 instead of 1920:1152) is what makes sketches and dimensions communicable.