Mixtures & Alligation

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Subtopic 5 of 5·← Milk–Water Ratio & Profit by Adulteration

Mixing Two Mixtures

When two ready mixtures (each with its own ratio) are combined, work with fractions of one chosen ingredient — never with both ratios at once.

$\text{mixed fraction} = \frac{\sum (\text{volume}_i \times \text{fraction}_i)}{\sum \text{volume}_i}$

Alligation shortcut: to get a target milk fraction f, mix volumes in the ratio $(f_2 - f) : (f - f_1)$ where $f_1, f_2$ are the two vessels' milk fractions.

Convert each ratio to the fraction of the SAME ingredient first (milk or water), then mix.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 12 practice Q

Detailed notes

Mixing two finished mixtures

Two vessels, each already a milk-water (or acid-water) blend, are poured together. Reduce each vessel to the fraction of the key component, pool the columns, and read the new ratio: $\text{milk in blend} = q_1 f_1 + q_2 f_2, \qquad f = \frac{\text{milk}}{\text{total}}$ Vessel A (milk:water 3:1, 20 L) gives 15 L milk; vessel B (5:3, 32 L) gives 20 L — together 35 : 17. Nothing deeper than bookkeeping.

Choosing the mixing ratio for a target

To blend A and B into a target strength $t$ (as a milk fraction), alligate on the fractions: $\frac{q_A}{q_B} = \frac{t - f_B}{f_A - t}$ Both differences must carry the same sign — the target must lie between $f_A$ and $f_B$ . Vessels at $\frac{2}{5}$ and $\frac{4}{9}$ milk blended to $\frac{5}{12}$ : $\frac{q_A}{q_B} = \frac{\frac{5}{12}-\frac{4}{9}}{\frac{2}{5}-\frac{5}{12}} = \frac{\frac{1}{36}}{\frac{1}{60}} = \frac{5}{3}$ . Verify: $5 \times \frac{2}{5} + 3 \times \frac{4}{9} = \frac{10}{3}$ milk in 8 parts → $\frac{5}{12}$ ✓.

Transfers between vessels

Drawing from a uniform mixture takes the components in the vessel's own ratio — the ratio inside survives a withdrawal. What changes it is the topping-up. Two classics:

Proportional draw, then top up: 60 L of 2:1, remove 12 L (8 milk, 4 water — still 2:1), add 8 L water → 32:24 = 4:3.
Cross-transfer: 40 L pure milk; 5 L moved to an empty vessel, replaced with water; then 5 L of the (now 35:5) mixture moved back. The second vessel ends at $9.375 : 0.625 = 15:1$ . Track each vessel separately; volumes must balance.

Equal capacities and three vessels

For equal volumes, the blended strength is the plain mean of the fractions: vessels at $\frac{1}{2}$ , $\frac{2}{5}$ , $\frac{3}{8}$ milk → $\frac{1}{3}\left(\frac{20+16+15}{40}\right) = \frac{17}{40}$ → milk : water 17:23. For unequal capacities, weight by the volumes first.

Dilution by repeated fractions

Removing 20% of a vessel and topping with water multiplies the strength by 0.8 each round — the replacement formula in percentage clothes. 100 L at 80% milk, two rounds: $100 \times 0.8^2 = 64$ L of milk → 64%.

One ledger, many vessels

Whatever the story — pouring, drawing, topping, blending — the method never changes: maintain a per-vessel, per-component table and update it operation by operation. The only decisions are (1) how a withdrawal splits (in the source's own ratio), and (2) which column an addition swells. Exam answers are built so that a wrong split or a missed total change lands on a distractor, so recompute the pooled total as a checksum — components must always add back to the vessel's volume.

Quick revision

Convert each vessel to its key-component fraction before pooling.
Target blend: alligate the fractions; the target sits between them.
A withdrawal preserves the internal ratio; an addition shifts it.
Equal capacities → plain mean of strengths.
Cross-transfers: keep a separate ledger per vessel.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Two mixtures combinedvery common2 practice Q

How to spot it:

Two vessels with given milk-water ratios (and volumes) are mixed — find the final ratio.

\text{milk} = q_1 f_1 + q_2 f_2, \quad \text{water} = q_1(1-f_1) + q_2(1-f_2)

Convert each vessel to litres of milk (fraction × volume).
Pool milk and water columns.
Reduce the ratio to smallest terms.

Why: components add linearly across vessels.

Example: Vessels of milk:water 4:1 and 3:2 are mixed in equal volumes. Find the milk : water ratio of the blend.

Milk fraction $= \frac{1}{2}\left(\frac{4}{5} + \frac{3}{5}\right) = \frac{7}{10}$ → 7:3.

Type 2: Ratio to hit a target strengthcommon2 practice Q

How to spot it:

'In what ratio must mixture A be mixed with mixture B to get milk : water = t?'

\frac{q_A}{q_B} = \frac{t - f_B}{f_A - t}

Write each vessel's milk fraction and the target fraction.
Alligate on the fractions — same-sign differences.
Verify by recomputing the blend's strength.

Why: a target strength between the two source strengths fixes the weights.

Example: Mixtures with milk:water 2:3 and 4:5 are blended to give milk : water = 5:7. Find the mixing ratio.

Fractions $\frac{2}{5}, \frac{4}{9}$ ; target $\frac{5}{12}$ → $\frac{q_A}{q_B} = \frac{\frac{5}{12}-\frac{4}{9}}{\frac{2}{5}-\frac{5}{12}} = 5:3$ .

Type 3: Transfer and top-up between vesselscommon3 practice Q

How to spot it:

A quantity is drawn from one vessel into another (or replaced) — track both ledgers.

\text{withdrawal splits in the source's own ratio}

Convert to component litres per vessel.
Move the drawn volume split in the source's ratio.
Apply additions; confirm each vessel's total separately.

Why: a uniform mixture leaves in its own proportions.

Example: A 40-litre can of pure milk: 5 litres are moved to an empty vessel and the can is topped with water; then 5 litres of the can's mixture are moved to the second vessel. Milk : water in the second vessel?

After water: can 35, 5. The 5 L drawn carries 4.375 milk, 0.625 water → second vessel $9.375 : 0.625 = 15:1$ .

Type 4: Strength of a blended solutioncommon2 practice Q

How to spot it:

Two solutions of given concentration are mixed in a given ratio — the resulting strength is asked.

\text{strength} = \frac{q_1 s_1 + q_2 s_2}{q_1 + q_2}

Compute solute from each solution.
Divide the pooled solute by the pooled volume.
Convert to a percentage.

Why: concentration is volume-weighted, never a plain average unless volumes match.

Example: 60% and 30% acid solutions are mixed in the ratio 2:1. The strength of the resulting solution is:

$\frac{2 \times 60 + 1 \times 30}{3} = 50\%$ .

Type 5: Three vessels / equal capacitiesoccasional2 practice Q

How to spot it:

Three (or more) blends combine, often in equal volumes.

\bar{f} = \frac{f_1 + f_2 + f_3}{3}\ \ \text{(equal volumes)}

Write each vessel's milk fraction.
Equal volumes → average the fractions; else weight by volume.
Convert the mean fraction to the required ratio.

Why: with equal capacities the weights are all 1.

Example: Three equal vessels hold milk : water 1:1, 2:3 and 3:5. Find the milk : water ratio of their mixture.

Mean fraction $= \frac{1}{3}\left(\frac{1}{2}+\frac{2}{5}+\frac{3}{8}\right) = \frac{17}{40}$ → 17:23.

Formulas

Blend of two mixtures

f = \frac{V_1 f_1 + V_2 f_2}{V_1 + V_2}

Volumes for target fraction

\frac{V_1}{V_2} = \frac{f_2 - f}{f - f_1}

Fraction from ratio

f_{\text{milk}} = \frac{m}{m + w}

Shortcut tricks

⚡ Fractions first, then average

Ratio → milk fraction per vessel → weighted average.

Example: Two vessels contain milk and water in the ratios 3 : 1 and 5 : 3. Equal volumes are mixed. The ratio of milk to water in the new mixture is:

Milk fraction $= \frac{1}{2}\left(\frac{3}{4} + \frac{5}{8}\right) = \frac{11}{16}$ ⇒ milk : water $= 11 : 5$ .

⚡ Alligate the fractions

Target fraction sits between the two vessel fractions; distances give volumes.

Example: Vessel A has 70% milk, vessel B has 30% milk. In what ratio should they be mixed to get a 50% milk mixture?

$V_A : V_B = (50 - 30) : (70 - 50) = 1 : 1$ .

⚡ Unequal volumes — weight the fractions

Multiply each vessel's milk fraction by its volume before adding.

Example: A 60-litre vessel has milk and water in the ratio 2 : 1 and a 40-litre vessel in the ratio 1 : 1. They are poured together. The milk fraction of the blend is:

Milk $= 60 \times \frac{2}{3} + 40 \times \frac{1}{2} = 60$ ; total 100 L ⇒ $60\%$ milk.

Where students lose marks

Averaging the two ratios directly ( $3{:}1$ and $5{:}3$ do not average to $4{:}2$ ).
Mixing fractions of milk with fractions of water inconsistently — pick one ingredient.
Ignoring unequal volumes when blending two mixtures.
Forgetting to reduce each ratio to its fraction of the total before blending.

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.