the properties of proportions
Once you know that two ratios are equal, you can rearrange the proportion in several legal ways and it stays true. These rearrangements are handy because the form you start with is often not the form a particular problem needs. They all follow from the single fact a/b = c/d, but each has a classical name worth recognising.
Start from a/b = c/d (all terms nonzero). Alternation swaps the means: a/c = b/d. Inversion flips both sides: b/a = d/c. The addition property (componendo) adds 1 to each side and recombines: (a + b)/b = (c + d)/d. The subtraction property (dividendo) gives (a - b)/b = (c - d)/d. Combining the last two gives (a + b)/(a - b) = (c + d)/(c - d). Each is just algebra on a true equation — for instance, alternation is what you get by cross-multiplying a/b = c/d to a·d = b·c and then dividing by c·d.
These properties shine in geometry proofs. When a line splits two sides of a triangle proportionally you often get a ratio like AD/DB = AE/EC; the addition property converts that into AD/AB = AE/AC, which compares each piece to the whole side instead of to the other piece — exactly the form the side-splitter theorem wants. Knowing the moves lets you slide between 'part to part' and 'part to whole' without re-deriving everything.
Given AD/DB = 2/3, the addition property gives AD/AB = AD/(AD + DB) = 2/(2 + 3) = 2/5. So the marked part is two-fifths of the whole side, even though it is two-thirds of the other part.
The addition property converts a part-to-part ratio into a part-to-whole ratio.
These are equalities you may apply, not new ratios you may invent. A frequent error is to 'add across' a proportion as a/b = c/d implies (a + c)/(b + d) — that one happens to be valid, but learners often misremember which combinations are legal; when in doubt, fall back on cross-multiplication, which never lies.