The Dishonest Dealer: False Weights

Profit, Loss and Discount · 20 min

Aptitude · Profit, Loss and Discount · False weights

He sells at cost price and still makes eleven per cent

A dealer claims to sell at no profit, charges the honest rate, and still comes out ahead — because his kilogram weighs 900 grams. No price appears in these questions, and none is needed. Only one thing is ever in doubt: what goes underneath.

Set the claimed and delivered weights and see both candidate answers →
Gain% = error / (true value − error) × 100. The denominator is what he actually gave, because that is what the sale cost him.

01 The idea

Cheating on the weight, not on the price

A grocer advertises that he sells at cost price, and he means it — his rate per kilogram is exactly what he paid. But when he weighs out a kilogram he hands over only 900 grams. He has made a profit without ever touching the price.

Work out how much. He charged for 1000 grams and delivered 900, so he kept 100 grams back. His outlay on that sale was the 900 grams that left the shop. So his gain is 100 grams of goods on a cost of 900 grams of goods, which is 100/900 = 11.11%.

The whole model turns on that denominator. It is tempting to divide by 1000, the amount he claimed, giving 10%. That is wrong for exactly the reason profit percentage is always on cost: the cost of this sale was the goods he parted with, not the goods he invoiced.

Notice that no rupee figure entered the calculation. Because the rate per gram is the same on both sides of the comparison, it cancels — which is why these questions can be stated entirely in grams and still have a numerical answer. Both 11.11% and 10% will be in the options.

He charged for 1000 g and gave 900 g. The gain is the 100 g held back, measured against the 900 g delivered — not against the 1000 g charged.
True weightWhat the dealer charges for — the amount he claims to be selling. Usually a round figure like 1000 g or 1 kg.
ErrorThe shortfall: true weight minus what he actually delivers. This is the goods he was paid for and kept.
Gain percentageerror / (true value − error) × 100. The denominator is the delivered amount, because that is the seller's cost on the sale.

02 Worked example

A 1000 gram weight that is really 900

The standard form of the model. A dishonest dealer claims to sell his goods at cost price, but he uses a weight of 900 grams for a kilogram. Find his gain percentage.

1
Separate what he says from what he doesHe invoices a kilogram — 1000 grams — and hands over 900. The rate per gram is honest throughout.charged for 1000 g    delivered 900 g
2
Find the errorThe difference is goods he was paid for and did not supply.error = 1000 − 900 = 100 g
3
Decide what his cost wasThis is the only real decision. His outlay on the sale is the stock that left the shop, which is 900 grams.cost of the sale = 900 g of goods
4
Divide the gain by the costOne hundred grams gained on nine hundred grams spent.100 / 900 × 100 = 11.11% gain
5
See the wrong routeDividing by the 1000 he charged for gives the standard wrong answer, and it looks tidier than the right one.100 / 1000 × 100 = 10%  ← wrong base

The tidiness of 10% is exactly what makes it dangerous — a repeating decimal feels like a mistake and a round number feels like an answer. But the rule is the same one from the first lesson in this module: profit percentage is measured on cost, and his cost is what he gave away. If you can say that sentence, you never need the formula.

03 The method

The formula, and the variants it covers

One formula, and three ways examiners dress it up. The third is the one that needs care.

gain% = error / (true value − error) × 100, where true value − error is simply the amount actually delivered. Equivalently: gain% = (claimed / delivered − 1) × 100.
When he also marks the price up, the two gains compound rather than add. A dealer who uses a 900 g kilogram and charges 20% above cost multiplies his multipliers: (1000/900) × 1.2 = 1.3333, so a 33.33% gain — not 11.11 + 20. Treat the false weight as one multiplier and the markup as another.
What he doesClaimed / deliveredGain
900 g for a kg1000/90011.11%
950 g for a kg1000/9505.26%
800 g for a kg1000/80025%
750 g for a kg1000/75033.33%
Gives 1000 g, charges for 900900/100010% loss
900 g kg plus 20% markup(1000/900)×1.233.33%
Dividing by the claimed weightwronggives 10%

05 Cheat sheet

False weights on one page

One formula and the four dressings. The last row is the error the whole model exists to test.

CaseRouteOn 900 g for a kg
Gain from a false weighterror/delivered × 100100/900 = 11.11%
Same thing as a ratio(claimed/delivered − 1)×1001000/900 − 1
With a markup toomultiply the multipliers(1000/900)×1.2 → 33.33%
Faulty measure of lengthsame formula, cm instead of g—
He gives extra by mistakenegative errora loss
Find the weight from the gaindelivered = claimed/(1+gain)1000/1.1111 = 900
Divide by the claimed weightwronggives 10%
The denominator is what he gaveHis cost on the sale is the stock that left the shop. Dividing by the invoiced weight gives a rounder, wrong number that the options will contain.
No price is neededThe rate per gram cancels from both sides, which is why these questions are stated purely in weights and still have a numerical answer.
A markup compounds, it does not addA false weight and a price markup are two multipliers. Multiply them; adding the two percentages understates the gain.

06 Where & why

Where this shows up

A small, well-defined model that appears reliably because it tests one idea and punishes one specific error.

SSC CGL · CHSL · RRB
The 900 g kilogram

Set almost verbatim, year after year. Both 11.11% and 10% appear in the options, which is the entire point of the question.

Bank PO
False weight combined with a markup

“Uses a 900 g weight and marks up 20%.” Multiplying the multipliers gets it in one line; adding the percentages gets it wrong.

Reverse questions
Find the weight from a stated gain

“He gains 25% by false weight alone — what does his kilogram weigh?” Invert the formula: 1000/1.25 = 800 g.

Faulty measures generally
Cloth, milk, fuel

The same arithmetic with metres or litres instead of grams. Recognising the shape matters more than the unit.

This model is really a one-question test of whether you understood the first lesson in the module. If profit percentage is on cost, and his cost is the goods he handed over, there is nothing further to learn here.

07 Interview questions

What gets asked

Nine, and the second one is the whole model.

A dealer sells at cost price but uses a 900 g weight for a kilogram. Find his gain.
11.11%. He charges for 1000 g and delivers 900, so he keeps 100 g. That gain is measured against the 900 g that actually cost him something: 100/900 × 100 = 11.11%, or 100/9 per cent exactly.
Why divide by 900 rather than by 1000?
Because profit percentage is always measured on cost, and his cost on this sale is the stock that left the shop — 900 grams. The 1000 g is what he invoiced, which is his revenue side, not his cost. Dividing by 1000 gives 10%, which is the standard wrong answer.
State the formula.
Gain% = error/(true value − error) × 100, where the error is the shortfall and (true value − error) is what was actually delivered. It is worth reading it as “kept divided by given” rather than memorising the symbols.
How is it possible to profit while selling at cost price?
Because the cheat is in the quantity, not the rate. He charges the honest price per gram but supplies fewer grams than he charges for, so the shortfall is pure profit. The claim to sell at cost price is true about the rate and misleading about the sale.
Why does no rupee figure appear in these questions?
The rate per gram is identical on both sides of the comparison, so it cancels. Whether the goods cost ₹10 or ₹1000 a kilo, keeping 100 g out of every 900 delivered is the same percentage gain. That is why the question can be stated purely in weights.
A dealer uses a 900 g weight and also marks his goods up 20%. Find his total gain.
33.33%. Treat the two cheats as multipliers: the false weight is 1000/900 = 1.1111 and the markup is 1.2, so together 1.1111 × 1.2 = 1.3333. That is a 33.33% gain, not the 31.11% you get by adding the two percentages.
He gains 25% by false weight alone. What does his kilogram actually weigh?
Eight hundred grams. Invert the relationship: delivered = claimed/(1 + gain) = 1000/1.25 = 800. Check it forwards — keeping 200 g on 800 g delivered is exactly 25%.
What if a dealer accidentally gives 1000 g while charging for 900?
He makes a 10% loss. The error is negative: he delivers 1000 g and is paid for 900, so he loses 100 g on a cost of 1000 g, which is 10%. The same formula works with the roles reversed, and note that the loss percentage here is not 11.11%.
Does the same model cover cloth measured with a short metre?
Yes, identically. A shopkeeper using a 90 cm metre gains 100/900 × 100 = 11.11% in exactly the same way. The unit is irrelevant; what matters is the ratio of what is charged for to what is delivered.

08 Practice problems

Six on false measures

In each one, write down the two weights and say which is the cost before you divide. The arithmetic is never the difficulty.

The standard case

Easy
A shopkeeper professes to sell his goods at cost price but uses a weight of 950 g for a kilogram. Find his gain percentage.
Follow-up
Check your answer against 5%. If you got exactly 5% you divided by the wrong weight, and the true answer is slightly higher.
Show the hint
He keeps 50 g and delivers 950 — which of those is his cost?

A bigger cheat

Easy
A grocer sells at cost price but his kilogram weighs only 800 g. Find his gain percentage.
Follow-up
This one comes out to a round number, which is unusual for the model and worth noticing — the trap answer of 20% is also round, so tidiness proves nothing here.
Show the hint
Two hundred grams kept, eight hundred delivered.

Work backwards

Medium
By using a false weight, a trader gains 25% while claiming to sell at cost price. Find the weight he uses in place of a kilogram.
Follow-up
The formula runs in reverse, and the check is to run your answer forwards again. Note that a 25% gain does not correspond to a 250 g shortfall.
Show the hint
The claimed weight is 1.25 times the delivered weight.

Two cheats at once

Medium
A dealer uses a weight of 900 g for a kilogram and additionally sells his goods at 15% above cost price. Find his overall gain percentage.
Follow-up
The two gains compound rather than add. Work out both multipliers and multiply them; then compare with what simply adding the two percentages would have suggested.
Show the hint
The false weight multiplier is 1000/900 and the markup multiplier is 1.15.

A loss by accident

Medium
A careless shopkeeper charges for 900 g but actually hands over a full kilogram, selling at his cost rate. Find his loss percentage, and explain why it is not 11.11%.
Follow-up
The roles of the two weights swap, so the denominator changes and the percentage is not the mirror image of the usual answer. This asymmetry is the point of the question.
Show the hint
His cost is now the 1000 g that left the shop, and his revenue covers only 900 g of it.

Cheat both ways, then defend the base

Hard
A milkman buys milk at ₹50 per litre. He adds water so that every litre he sells contains only 800 ml of milk, he measures with a jug that delivers 900 ml when it claims a litre, and he sells at his cost rate of ₹50 per claimed litre. (a) Find how much actual milk a customer receives per claimed litre. (b) Find his gain percentage. (c) A student answers part (b) by adding the two individual gains. State the answer they would get and explain, in terms of what each percentage is a percentage of, why compounding is correct and adding is not.
Follow-up
Two independent cheats stacked on one sale, and part (c) forces the reason rather than the rule. Each cheat is a multiplier acting on the output of the other, so their effects multiply — the same reason a markup and a discount do not simply subtract.
Show the hint
For (a), the jug gives 900 ml of liquid and only 800/1000 of any liquid is milk. Work out the milk actually delivered, then compare with what the customer paid for.