Mixtures and Alligation

Module 10 — Lesson 0010 · In Depth · 35 min

Alligation is the cross method for weighted averages — two ingredients at prices C and D mixed to mean price M. The ratio falls out as diagonal differences: (D−M):(M−C). No system of equations needed.

1. Basic Definitions

Mixture: combining two or more ingredients.
Mean price (M): cost/concentration of the final mixture.
Cheaper (C): lower value ingredient.
Dearer (D): higher value ingredient.

Ordering: C < M < D — mean always lies between the two components. If M is outside, you can't make it by mixing (negative ratio).

2. The Rule of Alligation (Cross Method)

Ratio in which to mix two ingredients to hit mean price M:

Quantity of Cheaper : Quantity of Dearer = (D − M) : (M − C)

Always subtract smaller from larger (diagonal differences stay positive).

      C            D−M

         \     /

            M

         /     \

      D            M−C

      Cheaper quantity ∝ (D−M), Dearer ∝ (M−C) — the farther M is from an ingredient, the less of
        that ingredient you need.

Cross method: cheaper quantity ∝ (D−M), dearer ∝ (M−C). Farther M is from an ingredient, less of it you need.

When to use: two concentrations (20% and 50% acid), two prices ($50/kg and $80/kg), average speed with time as weight, profit% blends.

3. Solution Concepts

a) Concentration

Pure liquid: 100% concentration.
Water: 0% (unless specified otherwise).

Worked: Pure milk (100%) mixed with water (0%) to get 80% milk.
Ratio milk:water = (80−0):(100−80)=80:20=4:1. So 4 parts milk +1 part water.
Algebra check: (4×100 +1×0)/5=80% ✓ — alligation is just weighted average solved by cross differences.

Adding water: total solute unchanged, volume grows — C₁V₁ = C₂V₂. Milk-water 4:1 to 80% is the same idea.

b) Adding Water to Reduce Concentration

Use dilution: C₁V₁ = C₂V₂ — total solute stays same, volume changes.

C₁, V₁: initial concentration and volume
C₂, V₂: final concentration and volume (V₂ = V₁ + added water)

Worked: 20L of 50% acid, add water to get 20% acid. 0.50×20=0.20×V₂ → V₂=50L → add 30L water.

Each removal leaves 1−x/V of the solute. After n cycles: Initial × (1−x/V)ⁿ.

c) Replacement Formula (Repeated Dilution)

Remove quantity x from capacity V, replace with water, repeat n times:

Final Quantity = Initial Quantity × [1 − x/V]ⁿ

Initial = pure substance present at start; final = that substance left after n operations.

Worked: 40L pure milk, remove 4L, replace with water, repeat once (n=2): Final milk =40×(1−4/40)²=40×0.9²=32.4L. So after 2 cycles, 32.4L milk, 7.6L water.
Key: x is relative to V, not to current amount — the formula uses constant x/V each time.

d) Algebraic Method (Alternative to Alligation)

Sometimes one line of algebra is faster, especially for simple mixes:

Mix x liters at p% and y liters at q% → final % = (x·p + y·q)/(x+y) — weighted average directly.

Alligation is this equation solved for the ratio x:y when final % is known. Use alligation when ratio is unknown; use algebra when amounts are known and you need final %.

4. Tips for Solving

Check units: all prices must be Cost Price (CP) — don't mix CP and SP in one alligation.
Ratios are quantity ratios (weight, volume), not price ratios.
Water is free: CP of water = 0 in profit mixing problems.
Mean must be between C and D — if M < C or M > D, ratio would be negative → impossible mix or you swapped C/D.

5. Summary of Variables

Symbol	Meaning
`C, D, M`	Cheaper, Dearer, Mean price
`Qc, Qd`	Quantity of cheaper / dearer
`x`	Quantity removed/replaced
`V`	Total volume / capacity
`n`	Number of replacement operations

Alligation ratio (Cheaper : Dearer) = ?

Pure milk 100% + water 0% → 80% milk. Milk:Water?

C₁V₁=C₂V₂ is used for?

40L milk, replace 4L with water twice → milk left?

Mean price must be?

In mixtures, water CP is?

Mix x at p% and y at q% → final %?

Notes

Mixture :- Basics :-

Mixture ⇒ combine ingredients ; M ⇒ mean price (C < M < D)

Alligation (Cross) :-

Cheaper : Dearer ⇒ (D−M) : (M−C) — diagonal differences

C \ / D−M ; M ; D / \ M−C

(eg 100%+0%→80% : 80:20=4:1)

Concepts :-

a) Concentration ⇒ pure 100%, water 0%

b) Dilution ⇒ C₁V₁=C₂V₂ (V₂=V₁+added)

c) Replacement ⇒ Final=Initial×[1−x/V]ⁿ (x removed, V capacity)

(eg 40L,4L×2: 40×0.9²=32.4)

d) Algebraic ⇒ final % = (xp+yq)/(x+y) (weighted avg)

Tips :-

All CP (not SP) ; ratio is quantity, not price ; water CP=0 ; M between C and D

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