A ratio compares two quantities, and the same relationship can be written in infinitely many equivalent ways — 1:2, 2:4, 50:100. This calculator handles the three things people actually need to do with a ratio: reduce it to its simplest form, solve for a missing term in a proportion, and scale it up or down while keeping the relationship intact.
How simplifying works
Simplifying a ratio means finding the smallest whole numbers that express the same relationship. The method is identical to reducing a fraction: find the greatest common divisor (GCD) of both terms using Euclid's algorithm, then divide both terms by it. 18:12 shares a GCD of 6, so it simplifies to 3:2. If either term has a decimal — say 0.5:0.25 — the calculator first scales both terms by a power of 10 (here, x100 -> 50:25) so the GCD step always works on whole numbers, then simplifies as usual (50:25 -> 2:1).
How solving a proportion works
A proportion states that two ratios are equal: A:B = C:D. If you know any three of the four terms, cross-multiplication finds the fourth. For example, in 3:4 = 9:x, cross-multiplying gives 3 × x = 4 × 9, so x = 36/3 = 12. The same technique solves for A, B, or C if that's the term left blank — just multiply the two terms diagonally opposite the unknown and divide by the remaining known term.
Why scaling preserves the ratio
Multiplying (or dividing) both terms of a ratio by the same non-zero factor doesn't change the underlying relationship — it just changes the size of the numbers. 2:3 scaled by 4 becomes 8:12, and 8:12 simplifies right back down to 2:3. This is exactly what happens when you double a recipe, redraw a map at a different scale, or mix a larger batch of paint or concrete in the same proportions.