Mixtures and Alligation: Solve Ratio Problems Easily

A mixture blends two or more ingredients, and its average quality lies somewhere between the qualities of the parts. Mixing equal parts of 40% and 70% acid solution gives 55% — the simple middle. Unequal quantities pull the mixture's strength towards the larger part.

Alligation is the shortcut that reverses this: given two ingredients and a target strength, it tells you the ratio in which to mix them. Shopkeepers blending tea, pharmacists preparing solutions, and bankers computing average interest on mixed loans all use this rule.

Key Concepts

• Mixture: a combination of two or more ingredients, each with a measurable quality (price, concentration, purity).
• Mean value of a mixture: the weighted average of the ingredients' qualities.
• Alligation rule: to get mean value m from qualities a (cheaper) and b (dearer), mix them in the ratio (b − m) : (m − a).
• Cheaper and dearer: a must be below m and b above m, or the ratio makes no sense.
• Alligation diagram: write a and b in a column, m between them, and take diagonal differences.

Formulas

• Mean of a mixture = (q₁×v₁ + q₂×v₂) ÷ (q₁ + q₂), where q are quantities and v are values.
• Alligation ratio (cheaper : dearer) = (dearer − mean) : (mean − cheaper).
• Quantity needed: once the ratio is known, split the total required quantity in that ratio.
• Assumes the qualities blend linearly (true for price, concentration and purity).

Worked Example

In what ratio must 40% acid solution be mixed with 70% acid solution to get a 55% solution? How much of each is needed for 6 litres?

1. Identify: cheaper a = 40, dearer b = 70, mean m = 55.
2. Apply alligation: (70 − 55) : (55 − 40) = 15 : 15 = 1 : 1.
3. Split 6 litres in ratio 1:1: each part = 6 ÷ 2 = 3 litres.

Mix 3 litres of 40% solution with 3 litres of 70% solution. Check: acid content = 0.40×3 + 0.70×3 = 1.2 + 2.1 = 3.3 litres; 3.3 ÷ 6 = 0.55 = 55%. Correct. For a harder target like 60%, the ratio would be (70−60):(60−40) = 10:20 = 1:2, needing 2 litres of 40% and 4 litres of 70%.

Common Misconceptions

• "Mixing equal parts of 40% and 70% gives 110%." Percentages never add. Equal parts give the simple average, 55%; the total quantity doubles but the strength stays between the two ingredients.
• "The ratio is (mean − cheaper) : (dearer − mean)." Reversed. The cheaper ingredient's share is the diagonal difference at the dearer end: (b − m) belongs to a. Remember: the part you need more of is the one closer to the target — the differences are cross-matched.
• "Alligation works for any target." Only for targets between the two ingredient values. You cannot make an 80% solution from 40% and 70% stock — the ratio would go negative, which is the mathematics telling you it is impossible.

Real-World Applications

• Tea and coffee traders blend grades to hit a target price and taste.
• Pharmacists dilute concentrated stock solutions to the prescribed strength.
• Jewellers mix gold of different purities to make standard carat alloys.
• Dairies standardise milk to a fixed fat percentage by mixing cream and skimmed milk.

Alligation is weighted averaging run backwards: the mixture's mean is fixed, and the diagonal differences hand you the mixing ratio. Keep cheaper below the mean and dearer above it, cross-match the differences, and always sanity-check that the target lies between the two ingredients.