Basic & Compound Ratio
A ratio a:b compares two like quantities and behaves exactly like the fraction a/b — so it can be scaled up or down by any non-zero multiplier without changing its value (2:3 = 4:6 = 100:150). Always reduce to lowest terms and keep both parts in the SAME unit before comparing. The CAT power move is the "common multiplier": write a:b as 2k:3k so a total or a difference becomes one linear equation. A compound ratio multiplies ratios part-wise: (a:b) compounded with (c:d) is ac:bd. The duplicate ratio is a²:b² and the triplicate ratio is a³:b³, which show up whenever areas (square of side ratio) or volumes (cube of side ratio) appear. To compare two ratios quickly, cross-multiply: a:b > c:d exactly when ad > bc.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Ratio essentials
| Ratio of a to b | a : b = a/b (b ≠ 0) |
|---|---|
| Scaling a ratio | a : b = ka : kb for any k ≠ 0 |
| Compound ratio | (a:b) × (c:d) = ac : bd |
| Duplicate / triplicate | a²:b² (duplicate), a³:b³ (triplicate) |
| Dividing N in a:b | shares = aN/(a+b) and bN/(a+b) |
Proportion & variation
| Proportion | a:b = c:d ⇒ a×d = b×c (product of extremes = product of means) |
|---|---|
| Mean proportional of a, b | √(ab) |
| Third proportional to a, b | b²/a |
| Fourth proportional to a, b, c | bc/a |
| Direct variation | y = kx (y/x constant) |
| Inverse variation | y = k/x (xy constant) |