Ratio Calculator

A Ratio Is a Comparison, Not Just a Fraction

A ratio compares two quantities by division, written as A:B. It looks similar to a fraction, but the mindset behind it is a little different — a fraction usually describes a part out of a whole, while a ratio describes how two separate quantities relate to each other, even when they don't add up to a single "whole" at all.

If a recipe calls for a 3:2 ratio of flour to sugar, that's not saying flour is 3/5 of some total mixture — it's saying that for every 3 units of flour, you need 2 units of sugar, in whatever quantity you're scaling to. That distinction is why ratios show up constantly in mixing, scaling, and comparing things that don't naturally combine into one pool.

At a glance:
• A ratio A:B simplifies just like a fraction — divide both sides by their GCD
• A proportion is two equal ratios: A:B = C:D
• Cross-multiplication (A × D = B × C) solves for a missing value in a proportion
• Ratios can compare more than two quantities at once, like A:B:C

Ratio Calculator

Simplify a Ratio

Visual Ratio Bar

Solve a Proportion (A : B = C : ?)

Simplifying Ratios & Solving Proportions

Worked Example — Simplifying

Given: 12 : 18

Step 1: Find the GCD of 12 and 18 → 6
Step 2: Divide both terms by 6 → 12 ÷ 6 = 2, and 18 ÷ 6 = 3
Result: 2 : 3

Worked Example — Solving a Proportion

Given: 2 : 3 = 10 : ?

Step 1: Cross-multiply → 2 × ? = 3 × 10
Step 2: Simplify the right side → 2 × ? = 30
Step 3: Divide both sides by 2 → ? = 15

Ratios You'll Recognize

Certain ratios appear so often in screens, recipes, and design that recognizing them saves a calculation entirely.

Reference Table

Where Ratios Actually Get Used

Cooking & Baking: Recipes are built on ratios — a basic vinaigrette is roughly 3:1 oil to vinegar, and pie dough is often 3:2:1 flour to fat to water by weight, letting cooks scale a recipe to any size while keeping it balanced.

Maps & Scale Models: A map scale of 1:50,000 means every 1 unit on the map represents 50,000 of the same unit in reality, letting cartographers and model builders represent huge or tiny things at a usable size.

Construction & Materials: Concrete mixes are commonly specified by ratio, like 1:2:3 for cement, sand, and gravel, ensuring consistent strength regardless of the total batch size being mixed.

Finance: A debt-to-income ratio, current ratio, or price-to-earnings ratio all compress two related financial figures into a single comparable number that's easier to evaluate at a glance.

Photography & Design: Aspect ratios determine how an image is cropped and displayed across formats, and design layouts often use classic ratios to guide proportions that feel visually balanced.

Gear & Mechanical Systems: A gear ratio like 3:1 tells you how many times the smaller gear rotates for every single rotation of the larger one, directly determining speed and torque trade-offs in a mechanism.

Working With Ratios Smoothly

✓ Order matters: A ratio of 3:2 is not the same as 2:3 — always keep the terms in the order the context describes them (flour:sugar is not the same relationship as sugar:flour).

✓ Scaling a ratio means multiplying both sides equally: To scale 2:3 up for a bigger batch, multiply both numbers by the same factor — 2:3 scaled by 4 becomes 8:12, which simplifies right back down to 2:3.

✓ A ratio can be converted to "parts of a whole": For a ratio A:B, the total is divided into (A + B) equal parts — useful for splitting a fixed quantity, like dividing $50 in a 2:3 ratio between two people.

✓ Three-term ratios work the same way: A:B:C simplifies by dividing all three terms by their shared GCD, and scales by multiplying all three terms by the same factor.

✓ Watch for mismatched units: A ratio comparing minutes to hours needs both sides converted to the same unit first, or the ratio will be misleading.

✓ Zero on one side breaks the ratio: A ratio like 5:0 is undefined in practical terms, since it implies an infinite comparison — check your inputs if you land on this.

Ratios Predate Modern Arithmetic Notation

A Separate Concept in Greek Mathematics: Ancient Greek mathematicians, particularly in Euclid's "Elements," treated ratio ("logos") as fundamentally distinct from number — a way of comparing magnitudes like lengths or areas directly, without necessarily reducing them to a single numeric value the way we do today.

Eudoxus and the Theory of Proportion: The mathematician Eudoxus of Cnidus developed a rigorous theory of proportion around the 4th century BCE that could handle both commensurable ratios (which reduce to whole numbers) and incommensurable ones (like the ratio between a square's side and its diagonal) — a genuinely sophisticated achievement for its time.

The Colon Notation Came Much Later: Writing ratios with a colon, as in A:B, is a comparatively modern notational convention, becoming standard only centuries after the underlying mathematical ideas were already well developed.

Merging With Fraction Notation: Over time, as algebra and fraction notation matured, the practical distinction between "ratio" and "fraction" blurred in everyday use, even though mathematicians still recognize the conceptual difference described above.

Frequently Asked Questions