Every commercial transaction boils down to two numbers: what you paid and what you received. The difference — and the direction of that difference — determines whether you made a profit or took a loss.
Cost Price (CP): The price at which you buy the article. This is your investment.
Selling Price (SP): The price at which you sell the article. This is your return.
If SP > CP, the surplus is profit. If CP > SP, the deficit is loss. When SP = CP, you break even.
Here is the analogy that keeps it concrete: think of a shopkeeper in a Delhi market. He buys a cricket bat at ₹400 (CP), slaps a tag of ₹600 on it (Marked Price or MP), then offers a 10% discount during a sale. The price after discount becomes his actual SP. His profit is SP minus his CP. The percentage version of that profit, relative to CP, is what every exam question is really asking.
Two additional terms you will encounter:
The entire chapter lives on two foundational relationships:
Notice: percentage profit and percentage loss are always computed on CP, not SP. This single fact causes more wrong answers than any formula confusion.
You need both the forward and backward forms of every relationship. Exam questions almost always give you SP and ask for CP, or give you a loss% and ask for the new SP at profit%.
Forward (given CP, find SP):
Backward (given SP, find CP):
Look — the multiplier form is the fastest path. When you see "sold at 20% loss for ₹480", immediately write:
No intermediate steps. This is what separates 30-second solvers from 2-minute solvers.
Where D% is the discount percentage on the marked price.
If a question involves both a markup (profit on CP to get MP) and a discount (reduction from MP to get SP), you chain the multipliers:
Example: An article is marked 25% above CP and sold at 20% discount. Net profit/loss?
Net effect: no profit, no loss. Elegant — and a question type that appears regularly.
Here is a classic trap: a person sells two articles each at ₹X, one at 20% profit and one at 20% loss. Students assume they break even. They don't.
Total CP =
Total SP =
Since (i.e., ), there is always a net loss when same SP is used with equal profit% and loss%. The loss% formula:
At 20% each: loss = . Memorise this result — it appears as a direct question.
When the question says "9 books purchased at ₹18" — the CP is for the whole batch, and the SP per book or vice versa. Find CP per unit and SP per unit, then apply the standard formula. Don't try to work with the whole-batch numbers and partial-batch numbers simultaneously.
Two discounts of a% and b% are NOT equivalent to (a+b)%. The equivalent single discount is:
Example: 20% and 10% successive discounts = single discount.
When SP and profit/loss% are given, find CP in one step using the multiplier.
Rule: CP = SP ÷ multiplier, where multiplier = (100 ± p%) / 100.
Sold at 20% loss for ₹480 → CP = 480 ÷ 0.8 = 600. Done. Sold at 25% profit for ₹750 → CP = 750 ÷ 1.25 = 600. Done.
Standard method (setting up the ratio equation): ~40 seconds. Multiplier method: ~10 seconds. You save 30 seconds per question — that is significant when you have many questions.
When two items are sold at the SAME selling price — one at x% profit, one at x% loss — there is ALWAYS a net loss.
At x = 20%: loss = 4%. At x = 10%: loss = 1%. At x = 30%: loss = 9%.
Standard method (finding both CPs, summing, comparing): 6 steps, ~90 seconds. This formula: 1 step, ~8 seconds.
Chain the two multipliers and compare to 1.
Markup 25%, Discount 20%: 1.25 × 0.80 = 1.00 → break even. Markup 30%, Discount 25%: 1.30 × 0.75 = 0.975 → 2.5% loss. Markup 20%, Discount 10%: 1.20 × 0.90 = 1.08 → 8% profit.
Standard method: convert each to fraction, find SP in terms of CP, then compare: ~70 seconds. Multiplier chain: multiply two decimals mentally: ~15 seconds.
For two discounts a% and b%, the single equivalent discount is:
Example: 30% and 20% → .
Standard method: apply first discount to MP, then second discount to new price, calculate overall reduction in 4 steps. This formula: 1 calculation, ~12 seconds vs ~50 seconds.
Standard profit/loss percentages map to clean fractions of CP:
| % | Fraction | Use | |---|----------|-----| | 25% profit | SP = 5/4 of CP | | | 20% profit | SP = 6/5 of CP | | | 33.33% profit | SP = 4/3 of CP | | | 20% loss | SP = 4/5 of CP | | | 25% loss | SP = 3/4 of CP | |
Recognising these fractions lets you skip decimal arithmetic entirely. For "sold at 25% profit for ₹750", write SP = 5/4 × CP → CP = 750 × 4/5 = 600. Three steps, no decimals, ~10 seconds vs standard ~35 seconds.
When you see a profit and loss question in the exam hall, run this decision tree:
Step 1 — Identify what's given and what's asked.
Step 2 — Check for special structures.
Step 3 — Work with multipliers, not raw differences. Avoid writing "profit = SP − CP" as your first line unless the question literally hands you both in rupees. Multipliers are faster and less error-prone.
Step 4 — Sanity check the answer. If SP > CP, your profit% must be positive. If CP was ₹600 and you're getting profit% as negative, you flipped something.
For Agniveer Vayu, these questions tend to be one-step or two-step. If your working exceeds three lines, you have probably missed a shortcut.
Why this question (PYQ 1): This tests your ability to handle batch-to-unit conversion before applying the profit% formula. A common trap is using 18 and 20 directly as if they are per-book prices.
Solving path: First, find CP per book: ₹18 for 9 books → CP = 18/9 = ₹2 per book. SP per book: ₹20 for 9 books → SP = 20/9 per book.
The answer is option (B): .
Why this question (PYQ 2): This is a two-stage problem — recover CP from a loss scenario, then apply a profit% to the same CP. The multiplier method makes it two lines.
Solving path: SP at 20% loss = ₹480. Using the multiplier:
For 20% profit on the same CP:
The answer is option (C): Rs. 720.
Why this question (PYQ 3): This is the most direct form — given both CP and SP in rupees, find profit%. Good for calibrating your baseline speed.
Solving path: Profit = 2100 − 2000 = ₹100.
The answer is option (B): 5%.
Observation: recognise that 100/2000 = 1/20 = 5% without any multiplication. If you see CP ending in round hundreds and profit is a small number, convert to fraction first.
Calculating profit% on SP instead of CP. The formula is always (Profit/CP) × 100. Not SP. If a question gives you SP and asks for profit%, you must find CP first.
Using batch price as unit price directly. In PYQ 1, students add 18 + 20 or compare 18 to 20 directly. Always divide by the number of units to get per-unit price before applying the formula.
Assuming equal profit% and loss% at same SP cancel out. They don't. There is always a net loss equal to %. Treating this as break-even is a guaranteed wrong answer.
Applying discount percentage to CP instead of MP. Discount is always on the Marked Price. If a question says "20% discount on MP of ₹500", the discount amount is ₹100, not a percentage of CP.
Successive discounts: adding percentages. A 30% discount followed by a 20% discount is NOT 50% off. Use the formula or chain the multipliers. Adding the percentages overstates the discount.
Sign error when recovering CP from a loss. For a 20% loss, the multiplier is 0.8, not 0.2 and not 1.2. Writing CP = SP × 0.8 instead of CP = SP ÷ 0.8 gives a CP smaller than SP, which means profit — contradiction. Always check direction.