AFCAT Numerical Ability · Mixture and Alligation

Profit and Loss on Mixtures

A milkman who adds water, a trader who mixes two grades or sells two parts at different rates: find the cost of one unit of the mixture, then the profit on it.

All Mixture and Alligation notes

Why this matters

One AFCAT past-year question (2019) is here: a milkman who adds water and sells at a profit. Every question first needs the cost of one unit of the mixture. The marks go when profit is worked on the selling price, or on the price of pure milk, instead of on that cost.

Concept 1 of 2: Water sold at the price of milk

Water costs nothing. A milkman who sells the mixture at the price he paid for milk gains exactly on the water: every litre of water is sold at the milk's price.

Definition

Sold at the milk's cost price: gain % $= \dfrac{\text{water}}{\text{milk}} \times 100$ , so water : milk $=$ gain : 100.
Sold at some other price: first find what one litre of the mixture costs, $\text{CP} = \text{SP} \times \dfrac{100}{100 + \text{gain}}$ .
Then use the alligation cross with water at ₹0: milk : water $= (\text{CP} - 0) : (\text{milk price} - \text{CP})$ .

Cost of one litre of the mixture

\text{CP} = \text{SP} \times \dfrac{100}{100 + \text{gain}}

Worked example

Milk costs ₹40 a litre. A milkman adds water and sells the mixture at ₹42 a litre, gaining 20%. In what ratio does he mix milk and water?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 03-202Moderate

Example 1 · Mixture and Alligation · Profit and Loss on Mixtures

Pure milk costs ₹16 per litre. After adding water the milkman sells the mixture at ₹15 per litre and thereby makes a profit of 25%. In what respective ratio does he mix milk with water?

Gain is measured on the milk, not on the mixture

1 litre of water in 6 litres of mixture is 1 litre of water to 5 of milk: a gain of 20%. Dividing by the whole 6 litres gives 16.7%.

The selling price is not the mean

With a profit in the question, the cross needs the cost of a litre of the mixture. Divide the selling price by

1 + \text{gain}

before you cross it with the price of milk.

Drill 2 more on water sold at the price of milk

Concept 2 of 2: Profit on a mixture, and mixed profit rates

A mixture bought at two prices has one cost a kg, its mean price. Profit is worked on that cost. When one stock is sold in parts at different profit rates, the overall rate is a mean too, and the cross runs it backwards.

Definition

Two grades bought and mixed: find the mean cost a kg first (the mean-price page), then the profit or loss on that cost.
Parts of one stock sold at different profit rates: the overall rate is the weighted mean of the rates, weighted by how much is sold at each.
Given the overall rate, the alligation cross on the two rates gives the ratio of the two parts.

Profit on a mixture

\text{profit} = \dfrac{\text{SP} - \text{mean cost}}{\text{mean cost}} \times 100\%

Worked example

Rice at ₹30 a kg and at ₹60 a kg is mixed in the ratio

2 : 1

and sold at ₹48 a kg. Find the profit percentage.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mixture and Alligation · Profit and Loss on Mixtures

Two varieties of rice at Rs. 35 per kg and Rs. 55 per kg are mixed in the ratio 3:2. A merchant sells the mixture at Rs. 50 per kg. What is his profit or loss percentage?

Weight the profit rates

A quarter of the stock sold at 20% and the rest at 12% gives

0.25 \times 20 + 0.75 \times 12 = 14\%

overall, not the plain average 16%.

Drill 4 more on profit on a mixture, and mixed profit rates

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)

Water sold at the price of milk
Cost of one litre of the mixture
$\text{CP} = \text{SP} \times \dfrac{100}{100 + \text{gain}}$
Profit on a mixture, and mixed profit rates
Profit on a mixture
$\text{profit} = \dfrac{\text{SP} - \text{mean cost}}{\text{mean cost}} \times 100\%$

Watch out for (3)

Gain is measured on the milk, not on the mixture
→ Water sold at the price of milk
The selling price is not the mean
→ Water sold at the price of milk
Weight the profit rates
→ Profit on a mixture, and mixed profit rates

Drill the questions on this topic

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