proportion
If two ratios describe the same relationship, we say they are in proportion. A map drawn so that 1 cm stands for 10 km keeps the same ratio everywhere; whether you measure 3 cm or 7 cm, the ratio of map distance to real distance stays equal. A proportion is simply an equation that two ratios are equal: a/b = c/d.
Proportions are solved by the cross-multiplication rule. If a/b = c/d, then a × d = b × c, which turns the proportion into an ordinary equation you can solve for an unknown. This is why proportions are the workhorse of scaling problems — recipes, maps, currency conversion, and similar triangles all rely on them.
Cross-multiplication is just clearing both denominators at once, valid only when no denominator is zero. Be careful to keep the same kinds of quantity in matching positions; if the left ratio is miles over hours, the right ratio must also be miles over hours, not hours over miles, or the proportion will be set up backwards.
If 3 apples cost 2 dollars, how much do 12 apples cost? Set 3/2 = 12/x. Cross-multiply: 3x = 24, so x = 8 dollars.
Two equal ratios form a proportion; cross-multiplication turns it into a linear equation.
Cross-multiplying a/b = c/d gives ad = bc, but only because we multiply both sides by bd. It is not a special trick; it is the same balancing move used to solve any equation with fractions.