At its core, percentage is just a standardised ratio — "per hundred." Everything in profit-loss and discount reduces to one question: what fraction of the reference base are you looking at? The confusion in exam problems almost always traces back to mixing up bases.
Here is the analogy that makes this concrete. Think of a textile merchant in Delhi's Chandni Chowk. He buys fabric at a cost price (CP) from the mill. He stitches a marked price (MP) tag onto it — this is his aspirational price, often inflated. When a customer haggles, he gives a discount off the MP. Whatever the customer actually pays is the selling price (SP). His profit or loss is the gap between SP and CP.
The chain is: CP → [markup] → MP → [discount] → SP → [compare with CP] → Profit/Loss
Three different bases appear in this single transaction:
Confusing these bases is the single biggest source of errors in this topic for UPSC CSAT candidates who have taken mocks but still drop marks here.
One more thing worth anchoring: when a question says "he sold at a 20% loss," that means SP = 80% of CP, not CP = 120% of SP. The percentage belongs to the base (CP), so SP = CP × (1 − loss%). This sounds obvious on paper but under exam pressure it slips.
1. Profit and Loss on CP
Profit% = [(SP − CP) / CP] × 100
Loss% = [(CP − SP) / CP] × 100
The direction of subtraction tells you whether it's profit or loss. Always divide by CP.
Rearranging these gives you the most useful operational forms:
SP = CP × (1 + P%) / 100 → written compactly as SP = CP × (100 + P%) / 100SP = CP × (100 − L%) / 100CP = SP × 100 / (100 + P%) when you know SP and profit%CP = SP × 100 / (100 − L%) when you know SP and loss%2. Markup and Discount
Marked Price = CP × (1 + markup%) / 100
SP = MP × (1 − discount%) / 100
Chaining these: SP = CP × (1 + markup/100) × (1 − discount/100)
This single chain equation is your workhorse for every "markup then discount" problem.
3. Net Profit/Loss When Both Markup and Discount Are Given
Net profit% = [(SP − CP) / CP] × 100
But using the chain: SP/CP = (1 + markup/100) × (1 − discount/100)
So Net profit% = [(1 + markup/100)(1 − discount/100) − 1] × 100
For the standard case of 75% markup and 20% discount: 1.75 × 0.80 = 1.40 → net profit = 40%.
4. Successive Discounts
Two discounts of a% and b% are NOT the same as a single discount of (a+b)%. The effective single discount is:
Effective discount = a + b − (ab/100)
This formula comes directly from (1 − a/100)(1 − b/100) = 1 − (a+b)/100 + ab/10000, comparing with 1 − d/100.
Stop thinking in percentages during intermediate steps. Use multipliers instead.
When operations are sequential, you just multiply the multipliers together. 1.30 × 0.75 = 0.975 means a 30% markup followed by a 25% discount gives a 2.5% net loss. You read the result directly: values above 1 are profit, below 1 are loss.
This eliminates algebraic errors entirely on well-structured CSAT problems.
When a merchant sells two batches with different quantities and different profit/loss rates, the net result is a weighted average, not a simple average of the rates.
Look — if someone sells 60 articles at 20% profit and 40 articles at 10% loss (all at ₹100 CP each):
60 × 120 = 720040 × 90 = 360010,800; Total cost: 10,000(800/10,000) × 100 = 8%The arithmetic weighted average: (60 × 20 + 40 × (−10)) / 100 = (1200 − 400)/100 = 8% — this shortcut works when all CPs are equal. If CPs differ, you must compute total CP and total SP separately.
Classic CSAT trap: "sold both at the same price, one at X% profit, other at X% loss." This always results in a net loss. The formula:
Net loss% = X² / 100
So selling both items at the same price with 10% profit and 10% loss respectively gives 100/100 = 1% net loss. This is a pattern, not a formula to memorise blindly — derive it once and own it.
Convert every percentage operation to a multiplier (profit% → 1 + p/100, loss% → 1 − l/100, discount% → 1 − d/100), then multiply them all together. If the final product is > 1, subtract 1 and multiply by 100 for profit%. If < 1, subtract from 1 and multiply by 100 for loss%. Standard method (setting up algebraic equations): 50–60 seconds. Multiplier chain: 15–20 seconds. Example: 40% markup, 25% discount → 1.40 × 0.75 = 1.05 → 5% profit. Done.
When you know SP and the profit/loss%, don't set up an equation. Instead, treat SP as a percentage of CP directly. "SP is 112 when there's a 40% profit" → CP = 112 × (100/140) = 80. The rule: divide SP by the "multiplier" (140/100 or 0.80 etc.) to get CP. Standard algebraic setup: 3 steps, ~40 seconds. This substitution: 1 step, ~12 seconds. Particularly fast when CP is a round number like 80 or 100.
Two successive discounts of a% and b%: single equivalent discount = a + b − ab/100. For 20% and 25%: 20 + 25 − (20×25)/100 = 45 − 5 = 40%. Never add the percentages raw — that overcounts by exactly ab/100. Standard step-by-step: apply first discount to get intermediate price, then apply second → 2 multiplications. Formula: 1 calculation. Time saved: approximately 20–25 seconds per problem.
If two items are sold at the same price, one at X% profit and one at X% loss, net loss = X²/100%. This comes from the AM-GM inequality between the two CPs. Example: both sold at ₹2,340, one at 30% gain, other at 20% loss — these rates differ, so don't apply this. But if rates are equal (say both at 15%), net loss = 225/100 = 2.25% without any computation. Standard verification: compute both CPs, find total CP vs total SP → 5 steps. Pattern application: 1 step, under 10 seconds.
When the problem has a round CP like ₹100, ₹500, or ₹1,000, anchor there immediately. "CP = ₹80" → scale to 100 first. 80 becomes the base, so ₹32 profit on ₹80 = scale up: 32/80 = 2/5 = 40%. Fractions like 1/5 (20%), 1/4 (25%), 2/5 (40%), 3/8 (37.5%) are faster to read than percentage calculations. Memorise: 1/6 = 16.67%, 1/7 = 14.28%, 1/8 = 12.5%, 1/9 = 11.11%. Each of these saves one division step — approximately 15–20 seconds on problems that otherwise require long division.
When you see a profit-loss-discount problem in the exam hall, run through this decision sequence:
Step 1 — Identify what's given and what's asked. Is CP given? Is SP given? Is MP given? Is a rate (profit%, loss%, discount%) given? Mark each explicitly.
Step 2 — Draw the chain: CP → MP → SP. Write the multipliers on each arrow. Markup goes on the CP→MP arrow. Discount goes on the MP→SP arrow.
Step 3 — Do you need CP from SP?
Use CP = SP / multiplier. Don't write an equation.
Step 4 — Is it a two-item or two-batch problem? If yes, compute Total CP and Total SP separately. Never average the rates directly unless you verify that CPs are equal.
Step 5 — Read the answer options before computing. If options are far apart (say 35% vs 40% vs 45%), estimation might eliminate two options before you do precise arithmetic. If options are close (say 15.5% vs 15.71%), you need exact computation — don't estimate.
Step 6 — Sanity check. A 75% markup followed by a 20% discount should yield a profit — if your answer is a loss, recheck your multiplier direction.
Why this question: This is the canonical markup-then-discount problem. It tests whether you keep CP as the base for profit% and whether you execute the multiplier chain correctly.
Solving path: CP = 80. Markup 75% → MP = 80 × 1.75 = 140. Discount 20% → SP = 140 × 0.80 = 112. Profit = 112 − 80 = 32. Profit% = (32/80) × 100 = 40%. Key check: profit% is on CP, not on SP.
Why this question: This tests reverse-calculation — finding CP when SP and loss% are given. The confirming detail (item A's SP matches the stated SP) is there to reassure you mid-problem, not to complicate it.
Solving path: Item B: SP = 2340, loss = 20%, so SP = 0.80 × CP → CP = 2340 / 0.80 = 2925. The item A information merely confirms the setup is consistent — don't let it distract you. Answer: ₹2,925.
Why this question: Classic weighted-average profit problem. Many candidates compute (20% + (−10%))/2 = 5% and pick a wrong option. The unequal batch sizes (60 vs 40 articles) are the key signal.
Solving path: Since CP per article is identical (₹100), the weighted average shortcut works: (60 × 20 + 40 × (−10)) / (60 + 40) = (1200 − 400) / 100 = 8% profit. If CPs had been different, you'd need full computation.
Why this question: Two-item problem with different discount rates on different items. The trap is averaging the discounts or computing profit on MP instead of CP.
Solving path: Item X: SP = 1200 × 0.75 = 900, profit = 900 − 800 = 100. Item Y: SP = 900 × 0.80 = 720, profit = 720 − 600 = 120. Total CP = 1400, Total SP = 1620, profit = 220. Profit% = (220/1400) × 100 = 15.71%. Note: 220/1400 = 11/70 — if you recognise this fraction, the decimal conversion is straightforward.
Why this question: Introduces a unit-conversion layer (per dozen vs per piece) before the discount calculation. This is a deliberate CSAT-style complexity addition — the arithmetic is simple once you standardise units.
Solving path: Work entirely in dozens. CP per dozen = 2500. SP per dozen at standard rate = 250 × 12 = 3000. With 10% discount: SP per dozen = 3000 × 0.90 = 2700. Profit per dozen = 200. Profit% = (200/2500) × 100 = 8%. The "8 dozen" detail is a red herring — profit% is the same regardless of quantity when the rate is uniform.
Using SP as the base for profit%. Profit% is always on CP. "He made ₹20 profit on a ₹120 selling price" tempts you to compute 20/120 = 16.67%, but the correct denominator is CP = 100, giving 20%. This mistake inflates your calculated profit% in every problem.
Discounting the CP instead of the MP. When a problem says "20% discount," it means 20% off the marked price. Applying the discount directly to CP skips the markup step and gives a completely wrong SP.
Adding successive discounts directly. A 20% discount followed by a 25% discount is not a 45% discount. It is 1 − (0.80 × 0.75) = 1 − 0.60 = 40%. The correct formula: 20 + 25 − (20 × 25)/100 = 40%. This error is common because the "add them up" shortcut is presented uncritically in some study materials.
Averaging profit% across unequal batches without checking if CPs are equal. If batch A has CP ₹100 and batch B has CP ₹200, you cannot use the quantity-weighted average of rates. You must compute actual totals. The quantity-weighted shortcut is only valid when CP per unit is identical across batches.
Confusing "marked up by X%" with "marked at X% of CP". "Marked up by 75%" means MP = CP + 75% of CP = 1.75 × CP. "Marked at 175% of CP" means the same thing. These are equivalent — but "marked at 75% of CP" would mean a discount below cost, which is different. Read the preposition carefully.
Treating the "equal SP, equal rate" loss formula as universal. The formula Net loss% = X²/100 only applies when both rates are equal and both are sold at the same SP. If the rates differ (e.g., 30% gain and 20% loss), you must compute both CPs independently and find actual totals.