Mixtures & Alligation

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high importance~1 Q in Tier 119 formulas⚡ 15 shortcuts5 subtopics

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Subtopic 1 of 5·Rule of Alligation →

Mixture Concentration & Amounts

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A mixture holds ingredients in a fixed ratio or concentration. The two working quantities are:

Amount of each ingredient from the ratio: in 40 L of milk : water $= 3:1$ , milk $= 40 \times \frac{3}{4} = 30$ L.
Concentration (per cent) of an ingredient: $\frac{\text{amount of ingredient}}{\text{total volume}} \times 100$ .

When something is added or drained, only ONE of the two quantities changes — track that one and rebuild the ratio.

Adding water dilutes; adding pure ingredient strengthens; removing mixture removes both ingredients proportionally.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 13 practice Q

Detailed notes

What a mixture is

Two or more ingredients combined by quantity or by cost. Every mixture question reduces to one bookkeeping idea: track the component amounts (litres of milk, kg of salt, rupees of cost) through each operation, keeping the total consistent.

The two-column ledger

Write each ingredient's quantity and its "value" (price per kg, % strength, litres of milk). Then:

Total quantity = sum of the quantities.
Total value = sum of quantity × value, ingredient by ingredient.
Mean value $= \frac{\text{total value}}{\text{total quantity}}$ . 40 kg of rice at ₹6 plus 60 kg at ₹7: total ₹660 over 100 kg → mean ₹6.60/kg. This one line solves every "find the average price of the mixture" question.

Ratios inside the mixture

When a mixture holds milk and water in a ratio like 7:5, the parts share the total: 12 parts = 72 L → 1 part = 6 L → milk 42 L, water 30 L. Adding water grows only the water column and the total; the milk column does not move. Adding one ingredient leaves the other ingredient's absolute quantity untouched — every ratio question turns on this.

Adding to shift a ratio

"A 40 L mixture has milk and water 3:1; how much milk makes it 4:1?" Water is fixed at 10 L; milk must become 4 × 10 = 40 L, so add 10 L. Routine:

Convert the ratio to absolute amounts (parts × total/parts).
Freeze the unchanged component.
Solve for the added quantity so the new ratio holds.

Percentage strength

Concentration questions are the same ledger in per cent. A 150 L solution at 60% acid holds 90 L of acid; adding x litres of pure acid gives $\frac{90 + x}{150 + x} = 0.75$ — one linear equation. Watch the base: after adding, the denominator grows too.

Alloys and blended units

Alloys (copper:zinc 5:3 in a 24 g piece → 15 g and 9 g) and 'mixing sugar at two rates' are identical arithmetic: fix the component quantity, adjust the total, recompute the ratio. The value can be rupees, litres, grams — the method never changes.

Common slips

Applying the new ratio to the old total instead of the grown total.
Treating a percentage of the mixture as a percentage of the other ingredient (25% water in the mixture is a $\frac{25}{75} = 33\frac{1}{3}\%$ water-to-milk ratio).
Adding quantities in litres while the ratio is in parts without converting.

Quick revision

Mean price = total value ÷ total quantity.
Parts: total ÷ (sum of ratio) = one part.
Adding X changes only X's column — freeze the other component.
New ratio → new equation in one unknown.
% of mixture vs % of the other ingredient are different fractions.

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: Mean price / average of the mixturevery common2 practice Q

How to spot it:

Quantities and unit prices are given; the mean price of the mixture is asked.

\text{mean} = \frac{\sum q_i p_i}{\sum q_i}

Multiply each quantity by its price; add to get the total value.
Divide by the total quantity.
Fractions of rupees are normal — ₹6.60 is a clean exam answer.

Why: the mean value is value-weighted, not count-weighted.

Example: 40 kg of rice at ₹6 per kg is mixed with 60 kg of rice at ₹7 per kg. The average price of the mixture is:

$\frac{40 \times 6 + 60 \times 7}{100} = ₹6.60$ per kg.

Type 2: Ratio to absolute amountsvery common2 practice Q

How to spot it:

A ratio (milk:water 7:5) with a total quantity is given — extract each part.

\text{one part} = \frac{\text{total}}{\text{sum of ratio terms}}

Divide the total by the sum of the ratio terms.
Multiply out each component.
For ratios inside the mixture, remember every operation keeps some column fixed.

Why: ratios are relative; the total converts them to litres or kg.

Example: A 72-litre mixture has milk and water in the ratio 7:5. The quantity of water is:

$\frac{72}{12} = 6$ per part → water $= 5 \times 6 = 30$ L, milk 42 L.

Type 3: Adding an ingredient to change the ratiovery common2 practice Q

How to spot it:

'How much X must be added so the ratio becomes a:b?'

\frac{\text{component}_1}{\text{component}_2 + x} = \frac{a}{b}\ \ \text{or}\ \ \frac{\text{component}_1 + x}{\text{component}_2} = \frac{a}{b}

Get absolute amounts from the original ratio and total.
Freeze the component not being added.
Set the new ratio as an equation and solve for x.

Why: only the added column grows; the other is untouched.

Example: A 40-litre mixture has milk and water in the ratio 3:1. How much milk must be added to make the ratio 4:1?

Water $= 10$ L stays; milk must reach $40$ L → add $40 - 30 = 10$ L.

Type 4: Percentage strength of a solutioncommon2 practice Q

How to spot it:

A solution's concentration is given; after adding solvent or solute, find the new strength.

\text{strength} = \frac{\text{amount of solute}}{\text{total volume}} \times 100

Compute the solute amount from the percentage.
Add to the solute (or leave it) and grow the total as required.
Write the new fraction and convert to a per cent.

Why: percentages are fractions of the current total, which changes.

Example: 150 L of a solution contains 60% acid. How much pure acid must be added to make it 75% acid?

Acid $= 90$ ; $\frac{90+x}{150+x} = \frac{3}{4}$ → $x = 90$ L.

Type 5: Alloys and multi-part blendingcommon2 practice Q

How to spot it:

Alloys with metal ratios, or a blend whose parts are used in further mixing.

\text{component} = \frac{\text{ratio term}}{\text{sum}} \times \text{total}

Convert each alloy's ratio to grams (or kg) of each metal.
Pool the columns across all sources.
Read the new ratio (or solve for the addition needed).

Why: metals from different sources simply add up column-wise.

Example: An alloy has copper and zinc in the ratio 5:3. In 24 g of it, how much copper must be added to make the ratio 3:1?

Copper 15 g, zinc 9 g; zinc fixed → copper $= 27$ g → add 12 g.

Formulas

Amount from ratio

\text{part} = \text{total} \times \frac{\text{part's share}}{\text{sum of shares}}

Concentration

C = \frac{\text{ingredient}}{\text{mixture}} \times 100\%

Adding water

C' = \frac{A}{M + w} \quad (A \text{ unchanged})

Removing mixture

\text{each ingredient} \to \text{same fraction removed}

Shortcut tricks

⚡ Track the constant ingredient

Water added ⇒ milk unchanged. Everything hinges on the unchanged part.

Example: A 40-litre mixture has milk and water in the ratio 3 : 1. How much water does it contain?

Water $= 40 \times \frac{1}{4} = 10$ litres.

⚡ Rebuild the per cent after dilution

Milk fixed, total grows — divide again.

Example: 20 litres of a mixture contains 60% milk. After adding 5 litres of water, the milk percentage is:

Milk $= 12$ L; new total $= 25$ L ⇒ $\frac{12}{25} = 48\%$ .

⚡ Ratio change ⇒ one equation

Set the unchanged quantity against the wanted ratio.

Example: A 50-litre mixture of milk and water in the ratio 4 : 1 needs how much water to become 2 : 1?

Milk 40 L must be $\frac{2}{3}$ : water total $= 20$ ⇒ add $20 - 10 = 10$ litres.

Where students lose marks

Changing both parts of a ratio when only the denominator changed (adding water leaves milk untouched).
Adding x litres of water and thinking the milk percentage drops by x%.
Taking the ratio share of the wrong ingredient ( $\frac{1}{4}$ vs $\frac{3}{4}$ ).
Forgetting the total volume changes when anything is added (even when mixture is drained first).

Practice sets — 13 questions

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

Topic test · 13 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.