AFCAT Numerical Ability · Mixture and Alligation

Mean Price and Mean Concentration

The price of a mixture, or its strength, is a weighted average: total cost over total quantity, or total pure substance over total volume.

All Mixture and Alligation notes

Why this matters

None of these questions is HARD: each asks for the price or the strength of a mixture from what went into it. The one mistake that costs marks is averaging the prices or percentages without weighting them by how much of each was used. The next page runs the same average backwards.

Concept 1 of 2: Mean price of a mixture

Mixing two rices in the ratio

3 : 2

is the same as buying 3 kg of one and 2 kg of the other. The mixture's price is what the 5 kg cost, divided by 5.

Definition

Mixed in the ratio $a : b : c$ at prices $p, q, r$ a kg: take $a$ , $b$ and $c$ kg.
Mean price $= \dfrac{ap + bq + cr}{a + b + c}$ .
The mean always lies between the cheapest and the dearest price, nearer the one used more.
This is the cost price of the mixture. Profit on it comes later, on the profit page.

Mean price

\bar p = \dfrac{ap + bq + cr}{a + b + c}

Worked example

Rice at ₹36 a kg and rice at ₹51 a kg are mixed in the ratio

3 : 2

. Find the price of the mixture a kg.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Mixture and Alligation · Mean Price and Mean Concentration

A mixture contains three types of rice in the ratio 2:3:5. If the three types cost Rs. 20, Rs. 30, and Rs. 40 per kg respectively, what is the average cost of the mixture per kg?

Weight the prices; do not average them

₹45 and ₹75 a kg mixed

2 : 1

cost

\dfrac{90 + 75}{3} = 55

rupees a kg. The plain average, 60, treats the two as used equally.

Divide by the sum of the ratio

Three kinds mixed

1 : 2 : 3

make 6 kg in one round, so divide the total cost by 6, not by 3, the number of kinds.

Drill 6 more on mean price of a mixture

Concept 2 of 2: Mean strength: add up the pure part

Mixing solutions of different strengths, count the litres of pure substance each one brings, add them, and divide by the total volume. Alloys work the same way with kilograms of metal.

Definition

Pure part $=$ strength $\times$ volume. Strength of the mixture $=$ total pure part $\div$ total volume.
Given the ratio of the volumes instead, weight each strength by its term, exactly as with prices.
A mixture given as a ratio (gold : copper $5 : 3$ ) is a fraction first: gold is $\tfrac 58$ of it.
Equal amounts of two mixtures: average the two fractions, then turn the result back into a ratio.

Mean strength

c = \dfrac{c_1 V_1 + c_2 V_2}{V_1 + V_2}

Worked example

30 litres of a 15% salt solution are mixed with 15 litres of a 45% salt solution. Find the strength of the mixture.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Mixture and Alligation · Mean Price and Mean Concentration

A milkman has two types of milk. In one, the percentage of water is 10 and in the second the percentage of water is 20. If he takes 90 L of milk of type I and 60 L of milk of type II and prepares a new mixture, the percentage of water in this new mixture is

Do not add the ratio terms

Equal amounts of a

1 : 1

and a

1 : 4

mixture are not

2 : 5

. Milk is

\tfrac 12

of one and

\tfrac 15

of the other, so the mixture is

\tfrac{7}{20}

milk:

7 : 13

Drill 6 more on mean strength: add up the pure part

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)

Mean price of a mixture
Mean price
$\bar p = \dfrac{ap + bq + cr}{a + b + c}$
Mean strength: add up the pure part
Mean strength
$c = \dfrac{c_1 V_1 + c_2 V_2}{V_1 + V_2}$

Watch out for (3)

Weight the prices; do not average them
→ Mean price of a mixture
Divide by the sum of the ratio
→ Mean price of a mixture
Do not add the ratio terms
→ Mean strength: add up the pure part

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