A ratio like 4:5 is really a set of instructions for splitting anything — money, ages, or a bucket of paint — into parts that scale together.
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Mean, third and fourth proportional each have their own fixed formula, built from the same idea: a:b = c:d.
📌 The three proportional formulas
Mean proportional between a and b: x = √(a × b). Between 4 and 9: √36 = 6.
Third proportional to a and b (a:b = b:x): x = b² ÷ a. To 4 and 8: 64 ÷ 4 = 16.
Fourth proportional to a, b, c (a:b = c:x): x = (b × c) ÷ a. To 2, 3, 4: (3×4) ÷ 2 = 6.
Compounded ratio: to compound (a:b) with (c:d), multiply straight across: (a×c) : (b×d). Compounding (2:3) with (4:5) gives (8:15).
A mixture ratio tells you what FRACTION of the whole each part is — always add the ratio numbers to find the total number of parts.
📊 Worked example
| Given | Working |
|---|---|
| Milk : Water = 3 : 2, total = 25 L | Total parts = 3+2 = 5. Each part = 25÷5 = 5 L. |
| Milk | 3 parts × 5 L = 15 L |
| Water | 2 parts × 5 L = 10 L |
Trap: "3:2" means 3 parts to 2 parts, NOT 3 litres and 2 litres unless the total happens to be exactly 5. Always divide the actual total by the SUM of the ratio parts first.
Age-ratio problems use the same "parts" method as mixtures — set each age as a multiple of one common unit, x.
Method: if two ages are in ratio 4:5 and sum to 45, call them 4x and 5x. Then 4x+5x=9x=45, so x=5, giving ages 20 and 25.
12 questions on ratio, proportion, mixtures and age problems.
