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AFCAT Numerical Ability · Mixture and Alligation

Removal and Replacement

Some of a mixture is drawn out and the same amount of a liquid is poured back, once or several times: track what each draw takes and what each top-up adds.

All Mixture and Alligation notes

Why this matters

Two AFCAT past-year questions (both 2016) are here, and every question on the page is MODERATE or HARD. A draw takes every liquid in proportion, and a top-up with water leaves the milk a fixed fraction of what it was. Miscounting the rounds is the commonest way to lose the marks.

Concept 1 of 3: One draw and one top-up: track each liquid

Drawing out some of a mixture is like pouring out a small copy of it: the litres removed are in the mixture's own ratio. Take that fraction off each liquid, then add what goes back in.

Definition

  • Drawing xx litres out of TT removes the fraction xT\dfrac{x}{T} of every liquid in it.
  • Then add whatever is poured back to its own liquid: water to the water, milk to the milk, the pure part of a solution to the pure part.
  • If the total is unknown, write each liquid in terms of TT and turn the final ratio into an equation.

What a draw leaves

left=(1−xT)×amount before\text{left} = \left(1 - \dfrac{x}{T}\right) \times \text{amount before}

Worked example

A 64-litre mixture has milk and water in the ratio 3:13 : 1. 16 litres are drawn out and replaced with water. Find the new ratio.
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 03-205Moderate

Example 1 · Mixture and Alligation · Removal and Replacement

20 L of a mixture contains milk and water in the ratio 5 : 3. If 4 L of this mixture is replaced by 4 L of milk, the ratio of milk to water in the new mixture would be

A draw takes some of every liquid

Removing 10 litres of a 4:14 : 1 milk and water mixture removes 8 litres of milk and 2 of water. Taking all 10 from the milk, or all from the water, changes the ratio by the wrong amount.
Drill 3 more on one draw and one top-up: track each liquid

Concept 2 of 3: Topping up with water: the milk keeps a fixed fraction

When the drawn mixture is replaced with water, the milk is the only thing lost. It keeps T−xT\dfrac{T - x}{T} of itself, and since the total is back to TT, its percentage shrinks by the same fraction.

Definition

  • Drawn mixture replaced with water: the milk keeps the fraction T−xT\dfrac{T - x}{T}.
  • The total is back to TT, so the milk's percentage is multiplied by T−xT\dfrac{T - x}{T} as well.
  • So milk share aftermilk share before=T−xT\dfrac{\text{milk share after}}{\text{milk share before}} = \dfrac{T - x}{T}. Solve it for TT or for xx.
  • Ratios work too: turn each into the milk's fraction first (3:23 : 2 is 35\tfrac 35 milk).

Fraction the milk keeps

share aftershare before=T−xT\dfrac{\text{share after}}{\text{share before}} = \dfrac{T - x}{T}

Worked example

A vessel holds a mixture that is 80% milk. 6 litres are drawn out and replaced with water, and the milk is now 70%. How much does the vessel hold?
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mixture and Alligation · Removal and Replacement

A mixture of milk and water in a container has milk 75% by weight. If 50 kg of this mixture is withdrawn and replaced with 50 kg of water, the percentage of milk becomes 60%. What was the original quantity?

The percentage is multiplied, not reduced by a subtraction

Drawing a quarter of a 90% milk mixture and topping it up with water leaves 90%×34=67.5%90\% \times \tfrac 34 = 67.5\% milk. Subtracting 25 points gives 65%, which is wrong.
Drill 2 more on topping up with water: the milk keeps a fixed fraction

Concept 3 of 3: Repeated replacement

Each round keeps the same fraction of the milk that is there at the time, so the fraction multiplies round after round, like compound interest in reverse.

Definition

  • Each round keeps the fraction V−xV\dfrac{V - x}{V} of the milk; after nn rounds the milk is V(1−xV)nV\left(1 - \dfrac xV\right)^n.
  • Count the rounds carefully: 'done twice in all' is 2 rounds, 'repeated two more times' is 3.
  • The water at the end is VV minus the milk left, and the ratio follows from the two.

Milk after n rounds

milk left=V(1−xV)n\text{milk left} = V\left(1 - \dfrac{x}{V}\right)^n

Worked example

A can holds 60 litres of milk. 12 litres are drawn out and replaced with water, and this is done twice in all. How much milk is left?
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 03-207Moderate

Example 3 · Mixture and Alligation · Removal and Replacement

A container contains 40 L of milk. From this container 4 L of milk was taken out and replaced by water. This process was repeated two more times. How much milk does the container now contain?

Count the rounds

'Done once and then repeated twice more' is three rounds. 'Repeated once more' after the first is two. Write the number of rounds before raising the fraction to a power.

Do not subtract the same amount each round

Two rounds of drawing 10 litres from 50 litres of milk leave 50×0.82=3250 \times 0.8^2 = 32 litres, not 50−20=3050 - 20 = 30. The second draw takes milk and water, so it removes less than 10 litres of milk.
Drill 3 more on repeated replacement

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Watch out for (4)

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