How this instrument works
A proportion is a statement that two ratios are equal: a ⁄ b = c ⁄ x. Whenever three of the four terms are known, the fourth can be recovered by cross-multiplication — multiplying each numerator by the other fraction's denominator, giving a × x = c × b, then dividing to isolate x: x = (c × b) ⁄ a. This is one of the most common problem-solving tools in early algebra, and it underlies scaling recipes, map distances, unit conversions, and similar-figure geometry alike.
Cross-multiplication works because multiplying both sides of an equation by the same nonzero quantity never changes its truth — multiplying both sides of a ⁄ b = c ⁄ x by b and by x simultaneously clears both denominators at once, leaving the simpler equation a × x = c × b behind.
A proportion is really just a special, simplified case of a broader linear equation, but it's worth learning as its own pattern because it appears so often in exactly this shape: two known quantities forming one ratio, and a third known quantity paired with an unknown fourth forming a second, equal ratio.
- Enter the first ratio's numerator into the a field.
- Enter the first ratio's denominator into the b field.
- Enter the second ratio's known numerator into the c field.
- Read x: the sheet cross-multiplies to solve for the missing fourth term.
Worked example — 2 ⁄ 10 = 3 ⁄ x
Given the proportion 2 ⁄ 10 = 3 ⁄ x, cross-multiplying gives 2 × x = 3 × 10 = 30, so x = 15 — check the result directly: 2 ⁄ 10 = 0.2 and 3 ⁄ 15 = 0.2, the same ratio confirmed on both sides.
Given 4 ⁄ 8 = 6 ⁄ x, the first ratio simplifies to 1⁄2, so x must be exactly double c: x = 12, confirmed by cross-multiplication as 4 × 12 = 48 and 6 × 8 = 48, the two products matching exactly.
Questions
What is cross-multiplication?
Given a proportion a ⁄ b = c ⁄ x, cross-multiplication multiplies each numerator by the other side's denominator, producing a × x = c × b. This clears both fractions at once, leaving a simple equation solvable by ordinary division.
How do I know which term is the unknown in a proportion problem?
Set up the proportion so the two known, related quantities form one ratio (a and b), and the third known quantity pairs with the missing one to form the second, equal ratio (c and x) — as long as the two ratios genuinely represent the same relationship, cross-multiplication solves for whichever term is missing.
Can a be zero?
No — a sits in the denominator of the rearranged formula x = (c × b) ⁄ a, so a must be nonzero for the proportion to have a defined solution.
Where are proportions used practically?
Scaling a recipe up or down, converting between units, reading a map's distance scale, and solving similar-triangle side-length problems all reduce to exactly this same 'two equal ratios, one term missing' structure.
Is a proportion the same as a fraction?
Not quite — a fraction is a single ratio, like 3⁄4. A proportion is a STATEMENT that two such ratios are equal to each other, like 3⁄4 = 6⁄8, and solving one usually means finding a missing term that makes that equality hold true.