Profit and Loss is the language of trade. Every transaction in the real world — a shopkeeper selling rice, a dealer marking up goods — comes down to one question: did the seller make money or lose it, and by how much?
Here is how the vocabulary maps out:
SP > CP. The extra money you pocket.SP < CP. The money you bleed.Think of it like this: you are a small trader at a mela. You buy a toy for ₹100 (CP). You write ₹140 on the tag (MP — marked up 40%). A customer bargains, you give 25% off — they pay ₹105 (SP). You still walked away with ₹5 profit. That is the complete Profit-Loss picture in one story.
The Agniveer Army CEE does not ask you to understand the concept — it asks you to reach the answer in under 60 seconds. So everything that follows is built around speed, not theory.
The three core relationships you must internalize:
Note carefully: profit% and loss% are always calculated on CP, not SP. This is the single most common trap in the exam.
Forget the long subtraction route. The fastest way to move between CP and SP is through multipliers.
| Condition | Multiplier | |---|---| | 20% profit | CP × 1.20 = SP | | 25% profit | CP × 1.25 = SP | | 10% loss | CP × 0.90 = SP | | 15% loss | CP × 0.85 = SP |
And to go backward (SP → CP):
So if SP = ₹960 at 20% profit → CP = 960 ÷ 1.20 = 800. Done in one step. This is the method behind the 2024 PYQ in this set.
Certain profit percentages convert to clean fractions. Memorize these — they eliminate decimal division entirely.
| Profit% | Fraction form of SP/CP |
|---|---|
| 20% | 6/5 |
| 25% | 5/4 |
| 33.33% | 4/3 |
| 50% | 3/2 |
| Loss 20% | 4/5 |
| Loss 25% | 3/4 |
So when SP = ₹1200 and profit = 25%, write: SP/CP = 5/4 → CP = SP × 4/5 = 1200 × 4/5 = ₹960. No decimals.
When both markup and discount are involved:
For the classic "40% markup, 25% discount" problem:
So profit = 5%. You do not need to assign CP = 100 every time — the chain multiplier gives the answer directly. But assigning CP = 100 is still the clearest approach when you are under pressure.
Whenever a question mentions percentage markup + percentage discount:
This method is slower than the chain multiplier but produces zero errors. Use it when you are unsure about the chain multiplication.
These appear regularly: "CP of 12 pens = SP of 10 pens — find profit%."
The formula:
Wait — let's be precise. If CP of m articles = SP of n articles:
For "CP of 12 = SP of 10": Profit% = (12 − 10)/10 × 100 = 20%.
For "SP of 20 = CP of 25": Here SP articles = 20, CP articles = 25.
Profit% = (25 − 20)/20 × 100 = 25%.
The rule: whichever quantity is larger tells you there is profit (CP quantity > SP quantity means you are selling fewer to recover the same money — profit). If SP quantity > CP quantity, that's a loss.
This is a two-step reverse problem:
SP = CP × (1 + profit%/100)SP = MP × (1 − discount%/100) → MP = SP ÷ (1 − discount%/100)Example: CP = ₹200, profit = 19%, discount = 15%.
Whenever you see "profit% given, SP given, find CP" — divide SP by the multiplier directly.
Profit 20% → divide by 1.2. Profit 25% → divide by 1.25. Loss 10% → divide by 0.9.
Example: SP = ₹960, profit = 20%. CP = 960 ÷ 1.2 = 800.
Standard method (writing SP = CP + 20% of CP, solving algebra): ~45 seconds. This method: ~10 seconds. Saves 35 seconds per question.
Replace decimal multipliers with fractions for 25%, 20%, 33.33%, and 50%.
Example: SP = ₹1200, profit = 25%. CP = 1200 × 4/5 = 960. No calculator, no decimals.
Standard decimal method: 3 steps. Fraction method: 1 multiplication. Saves 2 steps every time.
When you see "marked up X%, discount Y%, find profit%", multiply the two multipliers directly.
Formula: Net multiplier = (1 + x/100) × (1 − y/100)
Markup 40%, Discount 25%: 1.4 × 0.75 = 1.05 → profit = 5%.
You do not need to assign CP = 100 at all. One multiplication replaces a 4-step working. Standard CP=100 method: ~50 seconds. Chain multiplier: ~15 seconds.
For "CP of m articles = SP of n articles", profit% = (m − n)/n × 100 directly.
CP of 12 pens = SP of 10 pens: (12 − 10)/10 × 100 = 20%.
SP of 20 articles = CP of 25 articles → here m = 25, n = 20: (25 − 20)/20 × 100 = 25%.
Standard route (assign unit CP, compute per-article SP, find %): 4 steps. This formula: 1 step. Saves ~30 seconds.
For loss% questions, check options from smallest to largest. Loss% = (Loss/CP) × 100. If CP and SP are both round numbers, the answer is almost always a round number (10%, 15%, 20%).
Example: CP = ₹2000, SP = ₹1800. Loss = 200. Scan options: 8%, 10%, 12%, 15%. Test 10% first: 10% of 2000 = 200. Matches instantly. Skip the division entirely.
Standard calculation route: ~40 seconds. Smart scan from options: ~8 seconds.
When you read a Profit-Loss question in the exam hall, run this decision tree:
Step 1 — Identify what is given and what is asked.
CP = SP ÷ multiplier).SP = CP × multiplier).(SP − CP)/CP × 100.Step 2 — Is there a Marked Price / Discount involved?
(1 + markup%) × (1 − discount%) for net effect. OR set CP = 100 if you prefer.Step 3 — Is it an "articles" type?
(m − n)/n × 100.(n − m)/m × 100.Step 4 — Check your answer against the options before writing. If the answer is not one of the four options, you have flipped CP and SP somewhere — recheck. Do not re-derive from scratch; just check which relationship you applied backward.
Total target time per question: 45-60 seconds.
Why this question: This is a direct SP-to-CP reversal — the most common question type in Agniveer maths. Get this template right and you get 3-4 marks per paper.
Solving path: SP = ₹960, profit = 20%. Multiplier = 1.20. CP = 960 ÷ 1.20 = ₹800. Do not write the algebraic equation unless you have extra time — the division is enough.
Why this question: Markup + discount combinations appear in almost every Agniveer paper. This tests whether you know that "mark up 40%, discount 25%" does NOT mean 15% profit.
Solving path: Chain multiplier: 1.40 × 0.75 = 1.05. Profit = 5%. Cross-check: CP = 100, MP = 140, SP = 140 × 0.75 = 105. Profit = 5. Both routes confirm 5%.
Why this question: Another SP-to-CP reversal, this time with a fraction-friendly percentage. Tests whether you can use fractions instead of decimals.
Solving path: SP = ₹1200, profit = 25%. SP/CP = 5/4. CP = 1200 × 4/5 = ₹960. No decimal division needed.
Why this question: Two-step reverse problem — find MP when CP and both profit% and discount% are given. This is slightly harder and separates average performers from scorers.
Solving path: Step 1 — SP = 200 × 1.19 = ₹238. Step 2 — SP = 85% of MP, so MP = 238 × 100/85 = 238 × 20/17 = ₹280. The key move is recognizing that "15% discount means SP = 85% of MP."
Why this question: Articles-type problems have a clean formula that most students miss. This question tests pattern recognition over calculation.
Solving path: SP of 20 = CP of 25. Using formula: profit% = (25 − 20)/20 × 100 = 5/20 × 100 = 25%. Alternatively: let CP = ₹1 each. CP of 25 = ₹25 = SP of 20. SP per article = ₹1.25. Profit% = 0.25/1 × 100 = 25%.
Calculating profit% on SP instead of CP. The formula is Profit% = (Profit/CP) × 100, always. If you use SP in the denominator, your answer will be smaller than the correct one and will match a trap option.
Forgetting that discount is on Marked Price, not Cost Price. "15% discount" means 15% off MP. Many students subtract 15% of CP. This gives a completely wrong SP.
Assuming markup% − discount% = profit%. This is wrong. A 40% markup and 25% discount does NOT give 15% profit. The chain multiplier gives 1.4 × 0.75 = 1.05, which is only 5% profit.
Confusing m and n in articles problems. "CP of 12 = SP of 10" — the larger number is always in the CP position here, which means profit. "SP of 20 = CP of 25" — the larger number is in CP again, still profit. Draw a small table if you are confused: write CP articles and SP articles side by side.
Not checking units. If the question says ₹40 per kg and 20 kg, compute total CP and total SP before finding profit%. Working per-unit and multiplying at the end are both fine, but mixing them mid-calculation causes errors.
Rounding intermediate steps. In questions involving 19% profit or 17-unit denominators (like the MP question above), keep fractions exact through the working. Rounding 238/85 to 2.8 prematurely can push you to ₹282 instead of ₹280 — exactly the kind of error that costs marks.