AFCAT Numerical Ability · Mixture and Alligation

The Rule of Alligation

Given two prices or strengths and the mean of their mixture, the alligation cross gives the ratio in which they were mixed, and from it any quantity or missing price.

All Mixture and Alligation notes

Why this matters

The cross is the fastest tool in the chapter, and the profit page leans on it. It also settles questions that are not about mixtures at all: heads and legs, wages of men and women. Marks go when the two differences are attached to the wrong sides, or when the ratio is scaled against the wrong known quantity.

Concept 1 of 2: The alligation cross

Alligation is the mean price run backwards. Given the two prices and the mean, it gives the ratio of the quantities. Each kind gets the other kind's distance from the mean: the further the dear one is above the mean, the more of the cheap one you need.

Definition

Cheaper price $c$ , dearer price $d$ , mean $m$ , with $c < m < d$ .
Cheaper : dearer $= (d - m) : (m - c)$ .
Draw it as a cross: the two values on top, the mean in the middle, and each difference written under the opposite value.
It works for anything averaged by quantity: prices, percentage strengths, and mixtures given as ratios once they are turned into fractions (alcohol : water $2 : 3$ is 40% alcohol).

Rule of alligation

\dfrac{\text{cheaper}}{\text{dearer}} = \dfrac{d - m}{m - c}

Worked example

In what ratio must tea at ₹120 a kg be mixed with tea at ₹180 a kg to make a mixture worth ₹140 a kg?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Mixture and Alligation · Rule of Alligation

Two alloys have gold content 30% and 50% respectively. In what ratio should they be mixed to get an alloy with 44% gold content?

The cheaper kind gets the dearer kind's distance

Mixing ₹10 and ₹20 to make ₹12: the cheaper gets

20 - 12 = 8

and the dearer

12 - 10 = 2

, so cheaper : dearer

= 4 : 1

. Pairing each price with its own distance gives

1 : 4

, the ratio turned round.

The mean must lie between the two

A 40% and a 60% solution can never make a 70% one. If the target is outside the two values, a difference comes out negative: re-read the stem.

Drill 6 more on the alligation cross

Concept 2 of 2: From the ratio to a quantity, a price, or any average

The cross gives a ratio. A question usually wants litres, kilograms or a price, so scale the ratio to the one quantity you are given.

Definition

Known amount of one kind: divide it by its own term to get one part, then multiply by the other term.
Known total: split it in the ratio, as in dividing a sum.
Unknown price: write the given ratio as $(d - m) : (m - c)$ and solve for the missing price.
Any two-group average: heads and legs (2 and 4 legs, the mean is legs per head), wages of men and women, two parts of a stock. Put the two values and their average in the cross.

Scaling the ratio

\text{cheaper} = \dfrac{d - m}{m - c} \times \text{dearer}

Worked example

How much of a 25% acid solution must be added to 12 litres of a 55% solution to make a 35% solution?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mixture and Alligation · Rule of Alligation

A variety of rice at ₹14 per kg is mixed with 15 kg of another variety of rice at ₹20 per kg, so as to make a mixture worth ₹16.50 per kg. The quantity of rice at ₹14 per kg must be

Scale from the right side

If cheaper : dearer is

3 : 2

and 12 kg of the dearer kind is used, one part is 6 kg and the cheaper kind is 18 kg. Multiplying 12 by

\tfrac 23

attaches the known amount to the wrong term.

Drill 4 more on from the ratio to a quantity, a price, or any average

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 (2)

The alligation cross
Rule of alligation
$\dfrac{\text{cheaper}}{\text{dearer}} = \dfrac{d - m}{m - c}$
From the ratio to a quantity, a price, or any average
Scaling the ratio
$\text{cheaper} = \dfrac{d - m}{m - c} \times \text{dearer}$

Watch out for (3)

The cheaper kind gets the dearer kind's distance
→ The alligation cross
The mean must lie between the two
→ The alligation cross
Scale from the right side
→ From the ratio to a quantity, a price, or any average

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