Proportions and cross-multiplication
A proportion is the statement that two ratios are equal: a:b = c:d. The defining property is that the cross products match, a·d = b·c. That single equation is what lets you recover any one term when the other three are known — rearrange it to isolate the unknown and divide. Solving for d, for instance, gives d = (b·c) / a.
Cross-multiplication works because multiplying both sides of an equality by the same quantity keeps it balanced, turning a fraction equation into a simple product equation.
Simplifying a ratio
The calculator also reduces a:b to its lowest terms by dividing both sides by their greatest common divisor. A ratio is unchanged when you scale both parts by the same factor, so 8:12 and 2:3 describe the same relationship — the simplified form is just the clearest way to read it.
- Equivalent ratios share the same simplified form.
- Scaling both terms up or down by the same number preserves the ratio.
- A simplified ratio with whole numbers is easiest to compare and communicate.
Everyday uses and watch-outs
Proportions are the workhorse behind recipe scaling, map scales, unit conversions, mixing concentrations and resizing images while keeping their shape. Whenever two quantities should grow in lockstep, a proportion finds the missing piece.
One requirement to keep in mind: the term you divide by cannot be zero, so when solving for a given position the corresponding known term must be non-zero. Also make sure both ratios are written in the same order and units — comparing parts-to-parts against parts-to-whole, or mixing units, is the most common source of error.
Formula
a:b = c:d ⇒ a·d = b·cFrequently asked questions
- How is the missing term found?
- Cross-multiplication: since a·d = b·c, any one of the four terms can be solved when the other three are known.