Profit and Loss is one of those topics where the concept is completely intuitive — you've lived it every time you've bought or sold anything. The challenge in the exam hall is not understanding, it is executing the right formula fast under pressure.
Here's the core idea. Every transaction has two prices: what you paid (Cost Price, CP or लागत मूल्य) and what you received (Selling Price, SP or विक्रय मूल्य). If SP > CP, you made a profit (लाभ). If SP < CP, you took a loss (हानि).
Think of it like this: you buy mangoes at ₹10 per dozen from the wholesale market and sell them at ₹15 per dozen in your shop. The ₹5 difference is your profit. Now if a customer haggles you down to ₹8 per dozen, you're selling at a loss of ₹2. Simple as that.
Where it gets exam-worthy is when the question introduces a third price — the Marked Price (MP or अंकित मूल्य), also called the listed price or printed price. This is the price written on the tag before any discount. The shopkeeper marks goods above CP to create room for a discount while still making a profit.
The chain looks like this: CP → (add markup) → MP → (subtract discount) → SP.
A question on UP Police Constable may ask you to find any one of these four quantities (CP, SP, MP, Profit/Loss%) given the others. Sometimes it throws in a false-weight twist, which is essentially a hidden extra profit the shopkeeper earns by cheating on the weighing scale.
The good news: every Profit and Loss question on this exam reduces to one of five standard types. Learn the formula for each type, practice two questions per type, and you're done. No special talent required — just clean arithmetic and the right formula.
Profit and Loss (absolute values):
Percentage form — always on CP:
This "always on CP" rule is the single most important thing to burn into memory. Profit% and Loss% are calculated as a fraction of CP, not SP. Most wrong answers in exams come from dividing by SP instead of CP.
Finding SP when CP and percentage are given:
Finding CP when SP and percentage are given:
The discount is always calculated on MP, not on CP. This is another common trap.
The combined formula (when you know both markup above CP and discount on MP):
Let CP = 100. Then:
Always take CP = 100 as your base. This single assumption eliminates half the algebra in marked-price questions.
When a dishonest seller claims to sell at CP but uses a short weight, the gain comes purely from the shortfall in weight.
Look at the denominator — it is the false weight (what the seller actually gives), not the true weight. This trips up many students who put 1000 in the denominator instead of 960.
This is a two-step process that appears frequently:
Step 1: Find CP from the given SP and current profit/loss%.
(use + for profit, − for loss in denominator)
Step 2: Find new SP using the target profit%:
Given: markup% above CP and a target profit%, find the discount%.
Take CP = 100. Calculate MP. Calculate required SP for the target profit. Then:
This is cleaner than any algebraic approach and takes about 20 seconds.
Instead of computing profit/loss separately and then dividing, collapse it into one multiplication.
SP = CP × multiplier, where multiplier = (100 ± %)/100.
Example: CP = ₹40,000, profit = 15%. Standard method: find 15% of 40,000 = 6,000, add to get 46,000. Two steps, 30 seconds.
Shortcut: 40,000 × 1.15 = 46,000. One step, 8 seconds. Similarly, 2.5% loss on ₹18,560 → 18,560 × 0.975 = 18,096. Direct.
For finding CP from SP: flip the multiplier. SP at 15% loss = 1900, so CP = 1900 / 0.95 = 2000. No formula memorization needed beyond "multiply or divide by the decimal."
Any question involving markup% and discount% together — immediately set CP = 100. Don't use variables.
Example: marked 160% above CP, 25% discount. CP = 100, MP = 260, SP = 260 × 0.75 = 195. Profit% = 95%. Total time: 15 seconds.
Standard algebraic method (using CP = x, forming equations): 45-60 seconds, with risk of sign errors.
This substitution works because profit% is scale-independent — it doesn't matter whether CP is ₹100 or ₹10,000, the percentage comes out the same.
False weight gain% = [(True − False) / False] × 100.
Micro-example: 960 gm sold as 1 kg. Gain% = (40/960) × 100 = 25/6 = 4¹⁄₆%.
The trap: students put 1000 in the denominator, getting 4%, which is a wrong answer that actually appears as a distractor in the options. The denominator is what the seller gives (960), because that's the seller's actual cost. Standard approach with full ratio setup: 40 seconds. Pattern-recall with this rule: 10 seconds.
When told "sold at X% loss for ₹P, at what price to gain Y%?" — recover CP first, then find new SP.
CP = (P × 100) / (100 − loss%). New SP = CP × (100 + gain%) / 100.
Example: sold for ₹1,700 at 15% loss. CP = 1700 × 100 / 85 = 2,000. New SP for 20% gain = 2,000 × 1.20 = ₹2,400.
Students who skip the CP-recovery step try to directly ratio 1,700 to find the new price — that gives the wrong answer every time. Always recover CP first. This two-step structure reduces errors to near zero: 4 multiplications, no algebra.
Given: MP is X% above CP, required profit is Y%. Find discount%.
Set CP = 100. MP = (100 + X). Required SP = (100 + Y). Discount% = [(MP − SP) / MP] × 100.
Example: MP is 40% above CP, required profit = 12%. CP = 100, MP = 140, SP = 112. Discount% = 28/140 × 100 = 20%.
Standard method using simultaneous equations: 60+ seconds. This substitution method: under 20 seconds, zero equation setup required.
Read the question and identify which of the five types it is:
Type 1 — Direct profit/loss (absolute): CP and SP both given. Subtract. Done in 5 seconds.
Type 2 — Find SP from CP and %: Use SP = CP × multiplier. One multiplication.
Type 3 — Find CP from SP and %: Use CP = SP / multiplier. One division.
Type 4 — Markup + Discount → Profit%: Set CP = 100 immediately. Three multiplications.
Type 5 — False weight: Gain% = (shortfall / false weight) × 100. Identify false weight as the denominator — not 1000.
Type 6 — New SP for target profit: Step 1: recover CP. Step 2: apply new%. Two multiplications.
Decision rule for discount questions: Is the discount given on MP or on CP? It is always on MP. Is profit/loss% always on CP? Yes, always. If you're unsure which price to divide by, ask yourself "whose money are we measuring gain on?" — the buyer's cost is CP, so divide by CP.
If the arithmetic looks messy (e.g., 18,560 × 0.975), break it: 18,560 × 1 = 18,560, minus 18,560 × 0.025. 18,560 × 0.025 = 464. 18,560 − 464 = 18,096. Clean.
Why this question: Tests the absolute basics — total CP vs total SP across multiple units. If you can't do this in under 10 seconds, fix your foundation first.
Solving path: Total CP = 150 × 10 = ₹1,500. Total SP = 150 × 5 = ₹750. Since SP < CP, it's a loss. Loss = 1,500 − 750 = ₹750. No formula needed — pure arithmetic.
Why this question: Tests the multiplier method for loss%. The number 18,560 is deliberately uncomfortable — the question is checking whether you use the clean multiplier or get stuck in long division.
Solving path: SP = CP × (1 − 0.025) = 18,560 × 0.975. Break it: 18,560 − (18,560 × 0.025) = 18,560 − 464 = ₹18,096. Match to option C.
Why this question: Classic markup-discount-profit% chain. The most common three-variable question type on this exam. Tests whether you know to set CP = 100.
Solving path: CP = 100. MP = 100 + 160 = 260. Discount = 25%, so SP = 260 × 0.75 = 195. Profit% = (195 − 100)/100 × 100 = 95%. Answer: option C.
Why this question: Straightforward profit% to SP. Tests the multiplier formula. Should take under 10 seconds if the method is internalized.
Solving path: SP = 40,000 × 1.15 = 46,000. That's 40,000 + 6,000 = ₹46,000. Answer: option D.
Why this question: The false-weight trap question. Four wrong-looking answer choices, and the specific distractor of 4% (using 1000 as denominator) is very likely to catch you if you haven't practiced this type.
Solving path: True weight = 1000 gm, false weight = 960 gm. Shortfall = 40 gm. Gain% = (40/960) × 100 = 4000/960 = 25/6 = 4¹⁄₆%. Key: denominator is 960 (false weight), not 1000. Answer: option B (4¹⁄₆%).
Why this question: Two-step CP recovery question. Tests whether you can work backwards from SP to CP and then forward to a new SP.
Solving path: Sold at ₹1,900 with 5% loss. CP = 1900 × 100/95 = ₹2,000. New SP = ₹2,080. Profit = 2,080 − 2,000 = ₹80. Profit% = (80/2,000) × 100 = 4%. Answer: option D.
Why this question: Tests finding discount% when both markup and target profit are given. The CP = 100 substitution makes this trivial.
Solving path: CP = 100, MP = 140 (40% above CP). For 12% profit, SP = 112. Discount = 140 − 112 = 28. Discount% = 28/140 × 100 = 20%. Answer: option D.
Why this question: Two-step question with 15% loss going to 20% profit. Tests clean CP recovery and forward calculation.
Solving path: SP = ₹1,700 at 15% loss. CP = 1700/0.85 = 1700 × 100/85 = ₹2,000. For 20% gain, new SP = 2,000 × 1.20 = ₹2,400. Answer: option C.
Calculating profit% on SP instead of CP. The formula is always (Profit / CP) × 100. If you divide by SP, your answer will be lower than the correct answer and will usually match a wrong option planted specifically for this mistake.
Using 1000 as the denominator in false-weight problems. The gain% formula uses the false weight in the denominator (what the seller actually gives), not the standard 1 kg. Always ask: what does the seller actually hand over?
Calculating discount on CP instead of MP. Discount is always a percentage of the marked price, not the cost price. When the question says "25% discount," it means 25% off the MP.
Not recovering CP before finding a new selling price. In questions of the type "sold at X% loss, find SP for Y% profit," students try to scale directly from old SP to new SP. This gives the wrong answer. Always go back to CP first — it is the anchor of every calculation.
Getting confused by "marks goods at X% above CP." This means MP = CP + X% of CP = CP × (1 + X/100). Students sometimes read "above CP" as "above SP" and set MP = SP × (1 + X/100), which is wrong.
Misreading mixed fraction answers. Options like 4¹⁄₆% and 5³⁄₁₇% look similar. Compute your fraction fully (e.g., 25/6) and convert it correctly before matching. 25/6 = 4.1666... = 4¹⁄₆, not 5-something.