AFCAT Numerical Ability · Profit and Loss

Lots Sold at Several Prices, and Mixtures

For a lot bought or sold at more than one price, compare the total cost with the total takings; the cost price of a mixture is the weighted average of its parts.

All Profit and Loss notes

Why this matters

Most rows here are a two-line total, and one AFCAT past-year question, from 2018, sells parts of a single purchase at different prices. Marks are lost by averaging the prices instead of adding the totals, and by taking the profit on the takings instead of the cost.

Concept 1 of 3: Total cost against total takings

However many prices appear, the question asks one thing: what did the whole lot cost, and what did it fetch? Add each side, then take the difference on the cost.

Definition

Total cost $= \sum (\text{quantity} \times \text{buying price})$ ; total takings $= \sum (\text{quantity} \times \text{selling price})$ .
Profit % $= \dfrac{\text{takings} - \text{cost}}{\text{cost}} \times 100$ .
Parts given as fractions ('a quarter of them', 'the rest'): turn each into a count first.
An average of the selling prices works only when the quantities are equal.

Profit on a lot

\text{Profit}\% = \dfrac{\text{total takings} - \text{total cost}}{\text{total cost}} \times 100

Worked example

A trader buys 50 kg of dal at ₹90 a kg. He sells 20 kg at ₹105 a kg and the rest at ₹85 a kg. Find his profit or loss per cent.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 02-207Moderate

Example 1 · Profit and Loss · Mixed Lots and Mixtures

I purchased 120 exercise books at the rate of ₹3 each and sold

\frac{1}{3}

of them at the rate of ₹4 each,

\frac{1}{2}

of them at the rate of ₹5 each and the rest at the cost price. My profit per cent was

Add the totals, do not average the prices

Selling 20 kg at ₹105 and 30 kg at ₹85 is not an average price of ₹95 a kg; it is

4650 \div 50 = 93

The base is the total cost

A profit of ₹150 on a lot costing ₹4,500 is

3\tfrac 13\%

. Dividing by the takings of ₹4,650 gives a smaller, wrong figure.

Drill 33 more on total cost against total takings

Concept 2 of 3: Cost price of a mixture

Mixing two grades bought at different prices gives one grade whose cost per kg is the weighted average. Find that average cost, then treat the mixture as a single article.

Definition

Cost per kg of a mixture $= \dfrac{q_1 c_1 + q_2 c_2}{q_1 + q_2}$ .
Mixed in the ratio $m : n$ : cost per kg $= \dfrac{m c_1 + n c_2}{m + n}$ .
To reach a target cost $c$ , mix the cheap and the dear grade in the ratio $(\text{dear} - c) : (c - \text{cheap})$ . This is alligation.
Then the selling price for a profit of $p\%$ is the mean cost times $1 + \tfrac{p}{100}$ .

Mean cost of a mixture

\bar c = \dfrac{q_1 c_1 + q_2 c_2}{q_1 + q_2}

Worked example

Tea at ₹180 a kg and ₹240 a kg is mixed in the ratio

2 : 1

. At what price per kg should the mixture be sold to gain

25\%

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Profit and Loss · Mixed Lots and Mixtures

A shopkeeper mixes two varieties of rice costing

\text{Rs. } 40

per kg and

\text{Rs. } 60

per kg in the ratio

3:2

. If he sells the mixture at

\text{Rs. } 55

per kg, find his profit percentage.

Weight the average

Mixed

2 : 1

, tea at ₹180 and ₹240 averages ₹200 a kg, not ₹210. The plain average is right only for equal quantities.

Alligation pairs each grade with the other's gap

The share of the cheap grade is (dear minus mean), not (mean minus cheap). Swapping them mixes the grades the wrong way round.

Drill 22 more on cost price of a mixture

Concept 3 of 3: Part at one profit, the rest at another

When parts of a stock earn different profit percentages, the overall percentage is their average weighted by cost. Read it backwards to find how much went at each rate.

Definition

Overall profit % $= \dfrac{w_1 p_1 + w_2 p_2}{w_1 + w_2}$ , where the weights are the costs of the parts (or the quantities, when every unit cost the same).
To find the split, use alligation on the percentages: the parts at $p_1$ and $p_2$ are in the ratio $(p_2 - P) : (P - p_1)$ .
Parts that cost different amounts: work out each part's profit in rupees and add.
A missing rate for a target: the total selling price needed, minus what the known part fetched, is what the rest must fetch.

Splitting by two rates

\dfrac{\text{part at } p_1}{\text{part at } p_2} = \dfrac{p_2 - P}{P - p_1}

Worked example

A dealer has 240 kg of wheat. He sells part at

6\%

profit and the rest at

16\%

profit, gaining

12\%

on the whole. How much does he sell at

16\%

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Profit and Loss · Mixed Lots and Mixtures

A merchant has

1000

kg of sugar, part of which he sells at

8\%

profit and the rest at

18\%

profit. He gains

14\%

on the whole. The quantity sold at

18\%

profit is:

Weight by cost

Three quarters at

20\%

and a quarter at a

12\%

loss is not the plain average of

4\%

; the bigger part counts three times, giving

12\%

Each rate is on its own part

A part's profit percentage is on that part's cost, not on the whole stock. Work part by part in rupees when in doubt.

Drill 7 more on part at one profit, the rest at another

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)

Total cost against total takings
Profit on a lot
$\text{Profit}\% = \dfrac{\text{total takings} - \text{total cost}}{\text{total cost}} \times 100$
Cost price of a mixture
Mean cost of a mixture
$\bar c = \dfrac{q_1 c_1 + q_2 c_2}{q_1 + q_2}$
Part at one profit, the rest at another
Splitting by two rates
$\dfrac{\text{part at } p_1}{\text{part at } p_2} = \dfrac{p_2 - P}{P - p_1}$

Watch out for (6)

Add the totals, do not average the prices
→ Total cost against total takings
The base is the total cost
→ Total cost against total takings
Weight the average
→ Cost price of a mixture
Alligation pairs each grade with the other's gap
→ Cost price of a mixture
Weight by cost
→ Part at one profit, the rest at another
Each rate is on its own part
→ Part at one profit, the rest at another

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