Mixtures & Alligation

high importance~1 Q in Tier 120 formulas⚡ 15 shortcuts5 subtopics

Overview Learn (5)Formulas (20)Examples (5)Practice (66)

Mixing Two Mixtures

⏱ 5 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

Two ready mixtures, each with its own ratio, are combined. Reduce each vessel to the fraction of one chosen ingredient — milk, say — never juggle both ratios at once.

$\text{blend fraction} = \dfrac{V_1 f_1 + V_2 f_2}{V_1 + V_2}$ .

For a target fraction f, mix the volumes in the ratio $(f_2 - f) : (f - f_1)$ . A withdrawal from a uniform mixture takes both liquids in proportion, so the inside ratio survives it.

Fractions first

Two vessels, each already a milk-water blend, are poured together. Give each vessel ONE number: the fraction of milk it holds.

\text{milk in the blend} = V_1 f_1 + V_2 f_2

Vessel A at 3 : 1 holds milk fraction $\frac{3}{4}$ ; vessel B at 5 : 3 holds $\frac{5}{8}$ .

Rule: Pick one ingredient (milk or water) and convert every vessel to ITS fraction. Stay with that ingredient to the end.

Mixing two mixtures

Vessel A (3 : 1, 20 L) gives 15 L of milk. Vessel B (5 : 3, 32 L) gives 20 L. Poured together: milk $15 + 20 = 35$ L, water $5 + 12 = 17$ L → $35 : 17$ .

Equal volumes make it lighter work: 4 : 1 and 3 : 2 in equal volumes give milk fraction $\frac{1}{2}\left(\frac{4}{5} + \frac{3}{5}\right) = \frac{7}{10}$ → $7 : 3$ .

Example: Two vessels contain milk and water in the ratios 3 : 1 and 5 : 3. Equal volumes are mixed. Milk fraction $= \frac{1}{2}\left(\frac{3}{4} + \frac{5}{8}\right) = \frac{11}{16}$ → milk : water $= 11 : 5$ .

Hitting a target strength

To blend A and B into a target milk fraction t, 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 the two fractions. Mixtures at $\frac{2}{5}$ and $\frac{4}{9}$ milk blended to $\frac{5}{12}$ : $\frac{q_A}{q_B} = \frac{5/12 - 4/9}{2/5 - 5/12} = \frac{1/36}{1/60} = \frac{5}{3}$ . Check: $5 \times \frac{2}{5} + 3 \times \frac{4}{9} = \frac{10}{3}$ milk in 8 parts → $\frac{5}{12}$ .

Tip: Verify every target blend by rebuilding the fraction from your volumes.

Unequal volumes: weight by volume

A 60 L vessel at 2 : 1 and a 40 L vessel at 1 : 1 are poured together. Milk = $60 \times \frac{2}{3} + 40 \times \frac{1}{2} = 40 + 20 = 60$ L in 100 L → 60% milk. Equal volumes alone would have given 58.3% — the volumes matter.

Watch: Never average two fractions when the volumes differ. Weight each fraction by its volume.

Withdrawals keep the inside ratio

Drawing from a uniform mixture removes both liquids in the vessel's ratio. 60 L of 2 : 1 (40, 20); draw 12 L → removes 8 milk, 4 water → still 2 : 1 in 48 L. Add 8 L of water → $32 : 24 = 4 : 3$ .

Only the topping-up changes a ratio. Removing 20% and topping with water multiplies the strength by 0.8 each round: 100 L at 80% milk, two rounds → $100 \times 0.8^2 = 64$ L of milk.

Transfers between vessels

Track each vessel separately.

A 40 L can of pure milk: 5 L move to an empty vessel; the can is topped with water → can at $35 : 5$ . Then 5 L of the can's mixture move to the second vessel, carrying $5 \times \frac{35}{40} = 4.375$ milk and $0.625$ water. Second vessel: $5 + 4.375 = 9.375$ milk, $0.625$ water → $15 : 1$ .

Watch: Recompute each vessel's total after every move. The components must add back to that total.

Three or more vessels

Equal capacities → the plain mean of the fractions. Vessels at $\frac{1}{2}$ , $\frac{2}{5}$ , $\frac{3}{8}$ milk → mean $\frac{17}{40}$ → milk : water $17 : 23$ . For unequal capacities, weight by volume first.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Two mixtures combined

How to spot it:

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

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

Method

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

Why it works:

Components add linearly across vessels.

Try this

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

Show solution

Fractions: $\frac{4}{5}$ and $\frac{3}{5}$ ; equal volumes → mean $\frac{7}{10}$ .
Milk : water = $7 : 3$ .

Answer

7 : 3

Type 2common2 practice Q

Ratio to hit a target strength

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}

Method

Write each vessel's milk fraction and the target fraction.
Alligate on the fractions — the two differences must share a sign.
Verify by rebuilding the blend's strength.

Why it works:

A target between the two source strengths fixes the weights.

Try this

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

Show solution

Fractions $\frac{2}{5}$ , $\frac{4}{9}$ ; target $\frac{5}{12}$ .
$\frac{q_A}{q_B} = \frac{5/12 - 4/9}{2/5 - 5/12} = \frac{1/36}{1/60} = \frac{5}{3}$ .

Answer

5 : 3

Type 3common3 practice Q

Transfer and top-up between vessels

How to spot it:

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

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

Method

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

Why it works:

A uniform mixture leaves in its own proportions.

Try this

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

Show solution

After top-up the can holds $35 : 5$ .
5 L of it carries $4.375$ milk and $0.625$ water.
Second vessel: milk $5 + 4.375 = 9.375$ , water $0.625$ → $15 : 1$ .

Answer

15 : 1

Type 4common2 practice Q

Strength of a blended solution

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}

Method

Compute the solute from each solution.
Divide the pooled solute by the pooled volume.
State the result as a percentage.

Why it works:

Concentration is volume-weighted; a plain average only works for equal volumes.

Try this

Solutions of 60% and 30% acid are mixed in the ratio 2 : 1. Find the strength of the resulting solution.

Show solution

Solute = $2 \times 60 + 1 \times 30 = 150$ parts.
Volume = $3$ parts.
Strength = $150 \div 3 = 50\%$ .

Answer

50%

Type 5occasional2 practice Q

Three vessels or equal capacities

How to spot it:

Three or more blends combine, often in equal volumes.

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

Method

Write each vessel's milk fraction.
Equal volumes → average the fractions; otherwise weight by volume.
Convert the mean fraction into the asked ratio.

Why it works:

With equal capacities all the weights are 1.

Try this

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

Show solution

Fractions: $\frac{1}{2}, \frac{2}{5}, \frac{3}{8}$ .
Mean = $\frac{17}{40}$ .
Milk : water = $17 : 23$ .

Answer

17 : 23

Formula sheet

Blend of two mixtures

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

Volumes for a target fraction

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

Fraction from ratio

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

Equal volumes

\bar{f} = \frac{f_1 + f_2 + f_3}{3}

Only when every vessel holds the same volume.

Shortcuts that save time

⚡ 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. Find the ratio of milk to water in the new mixture.

Show solution

Fractions: $\frac{3}{4}$ and $\frac{5}{8}$ .
Mean = $\frac{1}{2}\left(\frac{3}{4} + \frac{5}{8}\right) = \frac{11}{16}$ .
Milk : water = $11 : 5$ .

Answer

11 : 5

⚡ Alligate the fractions

The target sits between the two vessel fractions; the distances give the volumes.

Example

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

Show solution

$A : B = (50 - 30) : (70 - 50)$ .
= $20 : 20 = 1 : 1$ .

Answer

1 : 1

⚡ Weight by volume when sizes differ

Multiply each vessel's 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 has them in the ratio 1 : 1. They are poured together. Find the milk fraction of the blend.

Show solution

Milk = $60 \times \frac{2}{3} + 40 \times \frac{1}{2} = 60$ L.
Total = $100$ L.
Milk fraction = $60\%$ .

Answer

60% milk

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Averaging the two ratios directly (3:1 and 5:3 'average' to 4:2).

Convert each ratio to a fraction of one ingredient, then weight by volume.

Mistake 02

Mixing milk fractions with water fractions in one calculation.

Choose one ingredient at the start and stay with it.

Mistake 03

Ignoring unequal volumes when blending.

Weight each fraction by its own vessel's volume.

Mistake 04

Forgetting to reduce each ratio to a fraction of the total.

3 : 4 is 3/7 of the mixture, not 3/4.

Mistake 05

Losing track of vessel totals in transfer questions.

After every move, check that components add back to the vessel's volume.

Quick revision

Read this the night before the exam.

Convert every vessel to one ingredient's fraction before pooling.
Target blend: alligate the fractions; target must sit between them.
Unequal volumes → weight by volume.
Equal volumes → plain mean of the fractions.
A withdrawal keeps the inside ratio; the refill changes it.
Transfers: keep a separate column per vessel; totals must balance.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

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

Continue learning