AFCAT Numerical Ability · Profit and Loss

False Weights, Short Measures and Adulteration

A dishonest seller gives less than he charges for; his real profit compares the money taken with what the goods actually handed over cost him.

All Profit and Loss notes

Why this matters

No AFCAT past-year question sits on this page, but more than a quarter of its rows are HARD. One idea settles all of them: the cost of what is handed over against the money taken. Marks go on dividing by the weight claimed instead of the weight given, and on adding two gains that multiply.

Concept 1 of 4: Selling at cost price with a short weight

Charging for 1000 g but handing over 900 g at cost price: the 900 g cost him the price of 900 g and fetch the price of 1000 g. His gain is 100 g worth on 900 g worth.

Definition

Gain % $= \dfrac{\text{true weight} - \text{weight given}}{\text{weight given}} \times 100$ .
The base is the weight actually given, not the kilogram claimed.
A weight ' $x\%$ less than standard' gives $(100 - x)$ g for every 100 g charged.
Short lengths and short measures (a 'metre' rod, a 'litre' can) work the same way.

Gain from a short weight

\text{Gain}\% = \dfrac{\text{error}}{\text{true value} - \text{error}} \times 100

Worked example

A shopkeeper claims to sell at cost price but uses a weight of 960 g for a kilogram. Find his gain per cent.

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The same idea in a practice question:

Practice questionModerate

Example 1 · Profit and Loss · Dishonest Dealers and False Weights

A dishonest trader claims to sell his goods at cost price but uses a false weight of 950 g for one kg weight. Find his gain per cent.

Divide by what was given

960 g for a kilogram gains 40 on 960, that is

4\tfrac 16\%

. Dividing by 1000 gives

4\%

, the option set for that slip.

Drill 4 more on selling at cost price with a short weight

Concept 2 of 4: A short weight with a profit, a loss or a discount

Take the cost of a true kilogram as 100. The price he charges comes from his stated profit, loss or discount; what he hands over costs him only the share of a kilogram he gives.

Definition

Price charged per 'kilogram' $= 100 \times \left(1 \pm \tfrac{s}{100}\right)$ , after any markup and discount.
Cost of the goods handed over $= 100 \times w$ , where $w$ is the fraction of a true kilogram given.
Actual profit % $= \left(\dfrac{1 \pm s/100}{w} - 1\right) \times 100$ .
A scale that shows $x\%$ more than the true weight hands over $\dfrac{100}{100 + x}$ of what it shows.
A stated loss can turn into a real profit once the weight is short.

Actual profit with a short weight

\text{Profit}\% = \left(\dfrac{1 \pm \frac{s}{100}}{w} - 1\right) \times 100

Worked example

A grocer sells at a

4\%

loss but gives only 800 g for every kilogram. Find his actual gain or loss per cent.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Profit and Loss · Dishonest Dealers and False Weights

A grocer sells rice at

10

% profit but uses a weight of

900

gm instead of

1

kg. Find his actual profit percent.

Two gains multiply

25\%

profit together with 800 g for a kilogram is

1.25 \times 1.25 = 1.5625

, a profit of

56.25\%

, not

50\%

The cost is of what he hands over

With 800 g given, the cost to compare against is 80, not 100. Comparing the price with a full kilogram's cost leaves only the stated profit.

Drill 11 more on a short weight with a profit, a loss or a discount

Concept 3 of 4: Different false weights for buying and selling

Cheating on both sides multiplies. Buying, he gets more than he pays for; selling, he gives less than he charges for. Follow one rupee through both.

Definition

Buying, he receives $b$ kg for each kilogram paid for ( $b > 1$ if he cheats the supplier); selling, he gives $s$ kg for each kilogram charged ( $s < 1$ ).
At cost price both ways, the profit factor is $\dfrac bs$ , so profit % $= \left(\dfrac bs - 1\right) \times 100$ .
A balance that cheats the same way both times cancels: what it gains him when buying it loses him when selling.

Two false weights

\text{Profit}\% = \left(\dfrac{b}{s} - 1\right) \times 100

Worked example

A dealer gets 1.2 kg for every kilogram he pays for when buying, and gives 0.8 kg for every kilogram he charges for when selling, both at cost price. Find his profit per cent.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Profit and Loss · Dishonest Dealers and False Weights

A shopkeeper uses a balance that gives

10

% more weight while buying and

10

% less weight while selling. If he sells at cost price, what is his profit percent?

Do not add the percentages

20\%

more when buying and

20\%

less when selling is

\tfrac{1.2}{0.8} = 1.5

, a

50\%

profit, not

40\%

One balance used both ways cancels

If the same faulty balance is used for buying and selling, the gain on one side is lost on the other: no profit, no loss.

Drill 4 more on different false weights for buying and selling

Concept 4 of 4: Mixing in water, sand or a cheaper grade

Water or sand costs nothing but is sold as the real thing. The cost is the real part only, while the money comes in on the whole mixture.

Definition

Sold at the cost price of the pure goods: gain % $= \dfrac{\text{filler}}{\text{real part}} \times 100$ . Water : milk $= 1 : 3$ gives $33\tfrac 13\%$ .
For a gain of $g\%$ at cost price, water is $\dfrac{g}{100 + g}$ of the mixture.
A filler that costs something: add both costs and divide by the total quantity.
An alloy sold at the price of the pure metal: the cost is the alloy's price and the selling price the pure metal's.
A short measure on top of the water: cost only the milk actually handed over.

Gain from a free filler at cost price

\text{Gain}\% = \dfrac{\text{filler}}{\text{real part}} \times 100

Worked example

A milkman adds 1 litre of water to every 5 litres of milk and sells the mixture at the cost price of milk. Find his gain per cent and the percentage of water in the mixture.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Profit and Loss · Dishonest Dealers and False Weights

A dishonest milkman mixes water with milk in the ratio

1:4

. If he sells the mixture at the cost price of pure milk, find his profit percent.

Water's share is of the mixture

20\%

gain at cost price means 1 part water to 5 parts milk, which is

\tfrac 16

of the mixture, not

20\%

of it.

Free filler adds weight, not cost

Sand mixed in raises the quantity sold but not the cost, so divide the target takings by the whole weight, sand included.

Drill 9 more on mixing in water, sand or a cheaper grade

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 (4)

Selling at cost price with a short weight
Gain from a short weight
$\text{Gain}\% = \dfrac{\text{error}}{\text{true value} - \text{error}} \times 100$
A short weight with a profit, a loss or a discount
Actual profit with a short weight
$\text{Profit}\% = \left(\dfrac{1 \pm \frac{s}{100}}{w} - 1\right) \times 100$
Different false weights for buying and selling
Two false weights
$\text{Profit}\% = \left(\dfrac{b}{s} - 1\right) \times 100$
Mixing in water, sand or a cheaper grade
Gain from a free filler at cost price
$\text{Gain}\% = \dfrac{\text{filler}}{\text{real part}} \times 100$

Watch out for (7)

Divide by what was given
→ Selling at cost price with a short weight
Two gains multiply
→ A short weight with a profit, a loss or a discount
The cost is of what he hands over
→ A short weight with a profit, a loss or a discount
Do not add the percentages
→ Different false weights for buying and selling
One balance used both ways cancels
→ Different false weights for buying and selling
Water's share is of the mixture
→ Mixing in water, sand or a cheaper grade
Free filler adds weight, not cost
→ Mixing in water, sand or a cheaper grade

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