Profit, Loss and Discount

Profit, loss and discount is the most commercially intuitive — and the most error-prone — chapter in the quantitative aptitude paper of judiciary and CLAT-PG examinations. Almost every candidate knows that profit means selling above cost; far fewer reliably remember that profit and loss are reckoned on cost price while discount is reckoned on marked price, and that conflating the two bases is the single largest source of avoidable error. This chapter builds the topic from first principles — cost price, selling price, marked price and discount — and then climbs to the higher-order traps that examiners love: successive discounts, marking goods up so that a discount still leaves a profit, the equal-selling-price illusion, and the dishonest dealer who hides his gain inside a faulty weight. Every formula below is stated, derived in words, and then exercised on a worked numerical example, so that you can reproduce the method under time pressure rather than memorise an opaque shortcut.

The four quantities: cost price, selling price, marked price and discount

Every problem in this chapter is a relationship among four quantities, and your first task on reading any question is to label each number with the correct one. The cost price (CP) is what the seller paid to acquire the article — including, where the question says so, transport, repair or overhead, all of which are added to CP before any profit calculation. The selling price (SP) is what the buyer actually pays. The marked price (MP) — also called the list price, printed price or labelled price — is the price tagged on the article before any reduction. The discount is the reduction the seller offers off the marked price, so that SP = MP − Discount.

The governing principle, which you must internalise before anything else, is this: profit and loss are always calculated on the cost price, but discount is always calculated on the marked price. When SP exceeds CP the difference is profit (gain); when CP exceeds SP the difference is loss. Discount lives in a separate world — it is the gap between MP and SP and has nothing directly to do with CP. Many candidates lose marks by computing a discount percentage on the cost price or a profit percentage on the marked price; the bases are different and must never be swapped. We develop the percentage machinery you need for these conversions in our companion chapter on percentage, ratio and proportion, and the underlying numerical fluency in number system and arithmetic foundations.

A worked illustration fixes the vocabulary. A shopkeeper buys a fan for Rs 1,200, marks it at Rs 1,600, and sells it after a 10% discount. The discount is 10% of the marked price, i.e. Rs 160, so SP = 1,600 − 160 = Rs 1,440. The profit is SP − CP = 1,440 − 1,200 = Rs 240, reckoned on CP. Thus profit percentage = (240 / 1,200) × 100 = 20%, while the discount percentage = (160 / 1,600) × 100 = 10%. Two different percentages on two different bases, both correct, and neither interchangeable.

Profit per cent and loss per cent — the cost-price base

The two foundational formulas are simply percentage statements built on cost price. Profit% = (Profit / CP) × 100 = ((SP − CP) / CP) × 100, and Loss% = (Loss / CP) × 100 = ((CP − SP) / CP) × 100. Both denominators are CP, never SP. From these, the two most useful working forms are derived by rearrangement: SP = CP × (100 + Profit%)/100 when there is a gain, and SP = CP × (100 − Loss%)/100 when there is a loss. The mirror forms recover cost price from selling price — CP = SP × 100/(100 + Profit%) or CP = SP × 100/(100 − Loss%) — and these matter because a frequent examiner trick is to give you SP and a profit percentage and ask for CP, tempting candidates to wrongly take the percentage of SP instead of dividing by the multiplier.

Consider: an article sold for Rs 1,955 yields a 15% profit; find the cost price. The wrong instinct is to subtract 15% of 1,955. The correct route is CP = 1,955 × 100/115 = Rs 1,700, and you can verify that 15% of 1,700 is Rs 255, with 1,700 + 255 = 1,955. The asymmetry between "per cent of SP" and "per cent of CP" is precisely what the question is testing.

A standard variation supplies two selling prices and asks you to relate the percentages. If selling at Rs 900 gives the same percentage loss as selling at Rs 1,260 gives percentage profit, find CP. Equal magnitudes of loss and gain mean CP is the arithmetic mean: CP = (900 + 1,260)/2 = Rs 1,080. The symmetry holds only because the loss and gain percentages are equal and both measured on the same CP.

Marked price and discount: marking up to discount down

Real traders rarely sell at the marked price; the marked price exists so that a visible discount can be offered while the seller still earns the intended profit. The mechanics are: discount = MP × (discount%)/100, and SP = MP − discount = MP × (100 − discount%)/100. Because SP simultaneously equals CP × (100 + profit%)/100, equating the two expressions gives the chapter's most commercially important formula: MP = CP × (100 + profit%)/(100 − discount%). This single relation answers the perennial question — "at what price must I mark goods so that after offering a stated discount I still make a stated profit?"

Worked example: a trader buys goods at Rs 800 and wishes to make a 25% profit even after allowing a 20% discount on the marked price. Then MP = 800 × 125/80 = Rs 1,250. Check: a 20% discount on 1,250 is Rs 250, so SP = Rs 1,000, which is exactly 25% above the CP of 800. The trader advertises a generous-looking 20% discount yet pockets a 25% margin — the everyday economics this chapter models.

A subtler version asks for the relationship between marked price and cost price as percentages. If goods are marked 40% above cost and a 25% discount is given, the net effect is MP = 1.40 × CP and SP = 0.75 × MP = 0.75 × 1.40 × CP = 1.05 × CP, a 5% profit. Notice that the 40% markup and the 25% discount do not simply subtract to 15%; they compose multiplicatively because they sit on different bases. That multiplicative composition is the gateway to successive discounts.

Successive discounts and the equivalent single discount

When two discounts are applied one after another — say a festival discount followed by a loyalty discount — they are not added. The second discount is computed on the price that already survived the first. If two successive discounts of a% and b% are applied, the price retained is (100 − a)/100 × (100 − b)/100 of the marked price, and the equivalent single discount = a + b − (ab/100), expressed as a percentage. The subtracted term ab/100 is exactly the amount by which naive addition over-counts, because the second discount acts on an already-reduced base.

Worked example: successive discounts of 20% and 10%. The equivalent single discount = 20 + 10 − (20 × 10)/100 = 30 − 2 = 28%, not 30%. On a marked price of Rs 500, the retained fraction is 0.80 × 0.90 = 0.72, giving SP = Rs 360, i.e. Rs 140 off, which is indeed 28% of 500. The formula extends to three discounts by chaining the multipliers — for discounts of a%, b% and c%, SP = MP × (1 − a/100)(1 − b/100)(1 − c/100) — and it is almost always faster to multiply the surviving fractions than to apply the two-term shortcut repeatedly.

Examiners exploit this in a comparison frame: which is better for the buyer, a single discount of 28% or successive discounts of 20% and 10%? They are identical, as we just saw. But successive discounts of 15% and 15% give an equivalent single discount of 30 − 2.25 = 27.75%, which is worse for the buyer than a flat 28% — a counter-intuitive result that rewards the candidate who computes rather than guesses. This compounding logic is the same multiplicative structure that governs simple and compound interest, and recognising the parallel makes both chapters easier.

Two articles at the same selling price: the hidden loss

A classic trap presents two articles sold at the same selling price, one at a profit of x% and the other at a loss of x%, and asks for the net result. Intuition says break-even, because the gain and loss percentages are equal. Intuition is wrong, and reliably so. There is always an overall loss, and its magnitude is fixed at (x/10)% squared — that is, (x/10)²% of the combined cost. The loss arises because the x% gain is measured on a smaller cost price while the x% loss is measured on a larger cost price, so the rupee loss outweighs the rupee gain even though the percentages match.

Worked example: two watches are each sold for Rs 1,200, one at 20% profit and the other at 20% loss. The first has CP = 1,200 × 100/120 = Rs 1,000; the second has CP = 1,200 × 100/80 = Rs 1,500. Total CP = Rs 2,500, total SP = Rs 2,400, so the dealer loses Rs 100 on an outlay of Rs 2,500 — a 4% loss. The formula confirms it: (20/10)² = 2² = 4%. Memorise both the qualitative result (always a loss) and the quantitative shortcut, because the temptation to mark break-even is strong and the formula is fast.

The result depends on the selling prices being equal. If instead the cost prices are equal and the percentages are equal and opposite, the transaction genuinely breaks even — so read whether the question equalises SP or CP before reaching for the (x/10)² shortcut.

The dishonest dealer and the faulty weight

A merchant who professes to sell at cost price yet gives less than the stated weight earns a concealed profit. He charges for one kilogram but hands over, say, 800 grams; his revenue corresponds to 1,000 grams while his cost corresponds to only the 800 grams actually delivered. The gain comes entirely from the shortfall. The formula is Gain% = (Error / (True Weight − Error)) × 100, where Error = True Weight − False Weight, equivalently Gain% = ((True − False)/False) × 100.

Worked example: a grocer sells at cost price but uses an 800-gram weight in place of a kilogram. Error = 1,000 − 800 = 200 g, so Gain% = (200 / (1,000 − 200)) × 100 = (200/800) × 100 = 25%. Reading the formula in reverse is a common exam demand: a dealer professes cost-price selling yet gains 12.5%; what weight does he use for a kilogram? Setting (1,000 − F)/F = 0.125 gives F = 1,000/1.125 ≈ 888.9 g. Note that 25% gain corresponds to an 800-gram weight, not a 750-gram weight — the gain is measured against the delivered quantity in the denominator, which is the step candidates most often botch.

The numerical discipline here — converting between fractions and percentages cleanly, and keeping the denominator straight — is exactly the fluency drilled in number system and arithmetic foundations. Treat the faulty-weight gain as a special case of profit-on-cost where the "cost" is the cost of what was actually given away.

Combining a markup with a false weight

The most demanding version of the dishonest-dealer problem layers two independent profits: the dealer both marks the goods up (or claims a discount) and cheats on the weight. Because the two effects sit on independent bases, they compose multiplicatively, giving the composite rule Overall Gain% = P + Q + (PQ/100), where P is the percentage gain from pricing and Q is the percentage gain from the false weight. This is the same composition law that produced the equivalent single discount, with the sign of each term set by whether it is a gain or a reduction.

Worked example: a dealer marks his goods 20% above cost and additionally uses a weight that by itself yields a 25% gain. Overall gain = 20 + 25 + (20 × 25)/100 = 45 + 5 = 50%. The two 5-percentage-point intuition (just add to 45%) under-counts by the cross term, exactly as it does with discounts. If instead the dealer offers a genuine 10% loss on price but still cheats the weight for a 25% weight-gain, set P = −10 and Q = +25: overall = −10 + 25 + (−10 × 25)/100 = 15 − 2.5 = 12.5% gain — so the customer who thinks he is getting a 10% loss-leader is in fact funding a 12.5% profit.

The safest method in the exam is to track multipliers directly rather than memorise the signed formula: convert each effect to a multiplier (a 20% markup is ×1.20, a 25% weight gain is ×1.25), multiply them (1.20 × 1.25 = 1.50), and read off the overall gain (50%). Multiplier-tracking removes sign errors and generalises to any number of layered effects.

When the gain is stated in articles, not rupees

A distinctive question type expresses profit not as a percentage but as a number of articles: "the cost price of 20 articles equals the selling price of 16 articles; find the profit per cent." Here the trick is that equal total money buys 20 at cost but only 16 at selling, so the per-article SP exceeds per-article CP. Let each article cost Rs 1; then total money = Rs 20 is the SP of 16 articles, so SP per article = 20/16 = Rs 1.25, giving a profit of Re 0.25 on Re 1, i.e. 25%.

The general shortcut: if CP of m articles = SP of n articles, then Gain% = ((m − n)/n) × 100 when m > n (a profit), and the same expression yields a negative number, i.e. a loss, when m < n. In the example, (20 − 16)/16 × 100 = 25%. A symmetric variant — "by selling 33 metres of cloth a trader loses the selling price of 11 metres; find the loss per cent" — is handled identically by setting up the per-unit prices and solving; here the loss works out to 25% as well. These problems reward the candidate who assigns a convenient unit price (Re 1 per article) and computes directly, a tactic rooted in the ratio reasoning developed in percentage, ratio and proportion.

Chains of buying and selling

Goods often pass through several hands, each taking a margin, and the question asks for the final price or the original cost. Each transaction multiplies the running price by its own (100 ± percentage)/100 factor, and the chain is solved by multiplying these factors in sequence. If A sells to B at 20% profit, B sells to C at 25% profit, and C pays Rs 225, then C's price = original CP × 1.20 × 1.25 = original CP × 1.50, so the original cost = 225/1.50 = Rs 150.

The reverse chain — given the first cost, find the last selling price — simply multiplies forward. Where a loss intervenes, its factor is below 1: a 20% profit followed by a 10% loss gives an overall factor of 1.20 × 0.90 = 1.08, an 8% net gain, again illustrating that percentages on shifting bases compound rather than add. This is structurally identical to the multiplier method for successive discounts, and treating both with the same multiplier discipline keeps your method uniform across the whole chapter. For broader practice in reading the multi-row data that examiners attach to such chains, see data interpretation.

The discount-versus-profit confusion examiners exploit

The richest source of marks — and of errors — is the deliberate entanglement of discount and profit. A typical statement reads: "a trader marks his goods 50% above cost and allows a discount of 20%; find his profit per cent." The naive subtraction (50 − 20 = 30%) is wrong because the 50% sits on CP and the 20% sits on MP. Correctly: MP = 1.50 × CP, SP = 0.80 × MP = 0.80 × 1.50 × CP = 1.20 × CP, a 20% profit. The 30% "answer" is the trap option that catches the candidate who adds across bases.

A second pattern reverses the unknown: "after a 10% discount a trader still makes a 17% profit; by what per cent above cost were the goods marked?" Using MP = CP × (100 + 17)/(100 − 10) = CP × 117/90 = 1.30 × CP, the markup is 30%. A third pattern hides a loss inside a discount: if the marked price is below an honest level, a large discount can convert an apparent gain into a real loss, so never assume that the presence of a discount implies a profit. The disciplined habit is to write SP in two ways — once from CP and profit, once from MP and discount — and equate them; this single technique dissolves almost every discount-versus-profit problem in the paper.

Shortcuts worth memorising and pitfalls worth fearing

A handful of results recur often enough to memorise. First, a profit of x% means SP/CP = (100 + x)/100, so a 25% profit is the fraction 5/4 and a 20% loss is 4/5 — converting percentages to such fractions speeds mental arithmetic enormously. Second, two equal selling prices at ±x% always lose (x/10)²%. Third, the equivalent single discount of a% and b% is a + b − ab/100. Fourth, MP = CP × (100 + profit%)/(100 − discount%). Fifth, false-weight gain = error/(true − error) × 100.

The pitfalls mirror the shortcuts. Never compute profit on selling price or discount on cost price — the bases are CP and MP respectively and are not interchangeable. Never add markup and discount percentages directly; compose them as multipliers. Never assume equal ± percentages break even; they lose. Never forget to fold transport, repair or overhead into CP when the question mentions them, because that overhead silently raises the cost base and shrinks the true profit. And always re-read whether the question equalises selling prices or cost prices before invoking a symmetry shortcut. The underlying speed in switching between fractions, percentages and ratios is the same competence rewarded across the aptitude paper, from time and work to time, distance and speed.

Exam strategy for the judiciary and CLAT-PG aptitude paper

In a timed paper, profit-loss-discount questions reward a fixed routine over inspiration. Step one: label every number as CP, SP, MP or discount before doing any arithmetic — mislabelling is the costliest error. Step two: decide which base each percentage sits on, and convert percentages to multipliers (or to fractions like 5/4 for 25%) so that composition is multiplication, never addition. Step three: where two routes to SP exist — one from CP and profit, one from MP and discount — write both and equate; this single move cracks the densest discount problems. Step four: sanity-check the answer against the trap options, because examiners plant the result of the most common mistake (such as 30% for a markup-minus-discount that should be 20%) as a distractor.

Allocate roughly a minute to a straightforward percentage conversion and up to two minutes to a layered markup-discount-or-false-weight problem; if a question runs longer, it usually means you have added across bases somewhere and should restart with multipliers. Build fluency by working a mixed set drawn from the linked sibling chapters — particularly percentage, ratio and proportion and simple and compound interest — and return to the full aptitude and reasoning hub to revise the whole quantitative module before the examination. The arithmetic of trade is small in volume but high in yield: a candidate who never confuses the cost-price base with the marked-price base will bank these marks every time.

Frequently asked questions

Are profit and loss calculated on the cost price or the selling price?

Profit and loss are always calculated on the cost price. Profit% = (Profit/CP) × 100 and Loss% = (Loss/CP) × 100. Discount, by contrast, is always calculated on the marked price. Computing profit on the selling price, or discount on the cost price, is the single most common error in this chapter and is frequently the planted distractor in the options.

Why do two successive discounts of 20% and 10% not equal a 30% discount?

Because the second discount acts on the price that already survived the first, not on the original marked price. The equivalent single discount is a + b − ab/100 = 20 + 10 − (200/100) = 28%, not 30%. The subtracted term ab/100 corrects for the over-counting in naive addition; equivalently, you multiply the surviving fractions 0.80 × 0.90 = 0.72, a 28% reduction.

If I sell two articles at the same price, one at 20% profit and one at 20% loss, do I break even?

No. When two articles are sold at the same selling price with equal profit and loss percentages, there is always a net loss of (x/10)²%. For x = 20 the loss is (20/10)² = 4%. The loss arises because the gain is measured on a smaller cost price and the loss on a larger one. Break-even occurs only if the cost prices are equal, not the selling prices.

How does a dishonest dealer who uses a false weight make a profit while claiming to sell at cost price?

He charges for the stated weight but delivers less, so his revenue corresponds to the full weight while his cost corresponds only to what he actually hands over. The gain is Error/(True Weight − Error) × 100. Using 800 g for a kilogram gives 200/(1000 − 200) × 100 = 25% profit, even though every rupee is nominally charged at cost price.

How do I find the marked price needed to make a profit after giving a discount?

Use MP = CP × (100 + desired profit%)/(100 − discount%). For a cost of Rs 800 with a target 25% profit after a 20% discount, MP = 800 × 125/80 = Rs 1,250. A 20% discount on Rs 1,250 leaves Rs 1,000, which is exactly 25% above the Rs 800 cost. This is why traders mark goods well above cost before advertising a discount.

What happens when a dealer both marks goods up and uses a false weight?

The two gains compose multiplicatively: Overall Gain% = P + Q + PQ/100, where P is the pricing gain and Q the weight gain. A 20% markup combined with a 25% weight gain yields 20 + 25 + (20×25)/100 = 50%, not 45%. The cleanest method is to multiply the multipliers: 1.20 × 1.25 = 1.50, i.e. a 50% overall gain, which avoids sign and cross-term errors.