Ratios and proportions: how cross multiplication works
Learn what a ratio is, how a proportion links two ratios, and why cross multiplication solves for any missing term every time.
What a ratio really tells you
A ratio is a comparison between two amounts of the same kind, written with a colon such as 3:2. It says nothing about the total, only about relative size, so 3:2, 6:4 and 30:20 all describe the same relationship. Because of that, any ratio has one canonical form found by dividing both terms by their greatest common divisor. Reducing to lowest terms makes two ratios easy to compare at a glance, which is why 1920:1080 is far more recognizable once it becomes 16:9.
When two ratios form a proportion
A proportion is the claim that two ratios are equal, like 1:2 = 5:10. It is the workhorse of everyday scaling, from doubling a recipe to reading a map to converting units. The power of a proportion is that if you know three of the four terms, the fourth is fixed and can be recovered exactly. That is precisely the job of this calculator's Solve mode, where you leave one box blank and let it fill in the rest.
Why cross multiplication always works
For a proportion A:B = C:D, the two ratios are equal fractions, A over B and C over D. Multiplying both sides of that equation by B and by D removes the denominators and leaves A times D equals B times C. This single identity holds for any valid proportion, so isolating the unknown is just a matter of dividing the known product by the remaining known term. That is why the tool can solve for A, B, C or D with the same underlying rule rather than four separate formulas.
Common places proportions show up
Aspect ratios in photography and video, gear ratios in machinery, scale factors on blueprints, and dilution ratios in cooking and chemistry are all proportions in disguise. Mixing concrete at 1:2:3 or diluting a cleaner at 1:10 both rely on holding a ratio constant while the batch size changes. Whenever you catch yourself saying for every X we need Y, a proportion is waiting to be solved, and leaving the unknown blank here gives the answer instantly.