Percentage is the language of comparison — it expresses any quantity as a fraction of 100. When a number changes from A to B, the percentage change is ((B - A) / A) × 100. Simple as that. Everything in profit-loss is an application of this single idea.
In a commercial transaction, there are three prices you must track:
If SP > CP, you made a profit. If SP < CP, you made a loss. The profit or loss is always expressed as a percentage of the cost price — not the selling price, not the marked price. This single rule is responsible for more wrong answers than any formula error.
Here is a useful analogy. Think of CP as your "investment" in a mutual fund. The profit percentage is your return on investment — it is always relative to what you put in, never relative to what you got back.
Discount, on the other hand, is always calculated on the marked price. So you have two separate percentage calculations running in parallel: profit/loss relative to CP, and discount relative to MP. Mixing these up is the most common CDS trap.
A quick mental model: if someone says "I bought at x% below MP and sold at y% above MP," your job is to link MP, CP, and SP together with two separate multipliers, then compute profit% on CP. Keep those two calculations in separate mental lanes and you will avoid most errors.
Let CP = cost price, SP = selling price, MP = marked price, r = rate.
Profit and Loss:
Working backwards from SP:
For a 15% profit: CP = SP / 1.15. For a 10% loss: SP = CP × 0.90.
Discount:
Stop working with the profit/loss amount unless you have to. Use multipliers directly.
If CP = 100 and profit is 20%, SP = 100 × 1.20 = 120. If you know SP and want CP, divide by 1.20.
If there is a 30% discount on MP, then SP = 0.70 × MP.
When both discount and profit are involved:
More practically: find SP from MP using the discount multiplier, then compute profit% using CP. Do it in two steps — never merge them into a single formula unless you use it every day.
Two discounts of a% and b% are NOT the same as a single discount of (a+b)%. The actual combined discount is:
For 20% and 10%: effective = 20 + 10 − 2 = 28%. If you said 30%, you overstated the discount.
This is a classic CDS pattern. "CP of y articles = SP of z articles."
Note carefully: profit% = (y−z)/z × 100, not (y−z)/y × 100. The denominator is z because profit% is always on CP, and here CP corresponds to z articles' worth of SP. If y = 5, z = 4: profit% = (5−4)/4 × 100 = 25%.
When a person makes a profit on one item and a loss on another, and the net result is break-even:
Profit amount = Loss amount (in absolute rupees, not percentages).
Find CP of each item. Compute actual profit and actual loss. Set them equal. This is where most candidates go wrong — they set profit% equal to loss%, which is only valid when the two CPs are identical.
Keep these in your head — they convert percentage calculations into fraction arithmetic, which is faster:
| Percentage | Fraction | |---|---| | 10% | 1/10 | | 12.5% | 1/8 | | 16.67% | 1/6 | | 20% | 1/5 | | 25% | 1/4 | | 33.33% | 1/3 |
When a question says "profit of 25%", read it as "SP = (5/4) × CP". Division and multiplication by simple fractions is faster than working with decimals.
When SP and profit% are given, the single fastest path to CP is: CP = SP ÷ (1 + profit%/100).
For SP = ₹34,500 at 15% profit: CP = 34500 ÷ 1.15 = 34500 × (20/23) = ₹30,000.
Convert 1.15 to the fraction 23/20, so dividing by 1.15 = multiplying by 20/23. This eliminates decimal long division entirely.
Standard method (working out profit amount first, then subtracting): 4 steps, ~40 seconds. This method: 2 steps, ~15 seconds.
"CP of y articles = SP of z articles" always gives profit% = (y−z)/z × 100.
You never need to assign a price per article. Just read off y and z from the ratio and plug in.
For y:z = 5:4 — profit% = (5−4)/4 × 100 = 25%. Done in one line.
Standard method (assigning CP = 1, computing SP = y/z, finding profit): 5 lines, ~50 seconds. This method: 1 line, ~10 seconds.
Whenever a problem says "bought at x/y of listed price, sold at z% above listed price," set listed price = 100.
CP = (x/y) × 100 (no calculation needed). SP = 100 + z (read directly). Gain% = (SP − CP)/CP × 100.
For the 3/4 and 50% problem: CP = 75, SP = 150, gain% = 75/75 × 100 = 100%.
This substitution converts a fully abstract problem into a one-row arithmetic problem. Without substitution: ~60 seconds. With it: ~20 seconds.
For two discounts a% and b%, the effective single discount is a + b − ab/100.
For 20% and 10%: 20 + 10 − (20×10)/100 = 30 − 2 = 28%. You never calculate the intermediate price.
Standard method (apply first discount, then second on the new price, then compute effective loss from original): 3 multiplications. This method: 2 additions and 1 division, ~half the time.
For "no net gain or loss" problems, identify the actual profit and loss in rupees. The question almost always gives you one of them indirectly through a CP or SP.
Work out CP of each item. Compute actual profit in rupees from item A. Set loss in rupees on item B equal to that amount. Compute SP of B as CP(B) − loss(B).
Trap to eliminate: setting profit% = loss%. That only works when both CPs are equal. CDS setters know you will try it. It is almost always wrong here. Eliminating that false path saves you from re-working the problem under time pressure.
In the exam hall, classify the problem in five seconds using this flow:
Is listed/marked price mentioned? Yes — keep MP, CP, and SP as three separate quantities. No — work only with CP and SP.
Given SP and profit%, want CP? Use CP = SP ÷ (1 + r/100). Convert the divisor to a fraction immediately.
Given CP and profit%, want SP? Use SP = CP × (1 + r/100). Again, fraction form.
"CP of y = SP of z" pattern? Profit% = (y−z)/z × 100 directly. Do not assign prices.
"No net gain or loss" with two items? Compute actual profit (₹) on item A first. Set actual loss (₹) on item B equal to it. Find CP of B from the loss and loss%, then SP = CP − loss.
Two successive discounts? Use a + b − ab/100 for the combined discount.
If the numbers look messy, substitute MP or listed price = 100 and work with clean integers throughout.
Why this question: This problem tests whether you can reverse-engineer CP from SP and profit%, compute actual profit in rupees, and apply the break-even condition correctly — three separate skills in one question.
Solving path:
Why this question: A pure "anchor to 100" problem. The listed price appears in both CP and SP calculations but never as a number — which is the signal to substitute it as 100.
Solving path:
Why this question: Tests the y/z article pattern. The ratio y:z = 5:4 directly gives the answer if you know the formula; without it, you are doing 90 seconds of unnecessary work.
Solving path:
z's contribution — not y. Don't invert it.Computing profit% on SP instead of CP. This is always wrong unless explicitly asked for "profit on selling price" (rare). If your profit% feels suspiciously low, check whether you used CP or SP in the denominator.
Treating two successive discounts as additive. A 20% discount followed by a 10% discount is not 30%. The actual combined discount is 28%. On MCQs, 30% is almost always a distractor option.
Equating profit% to loss% in break-even problems. Break-even means equal rupee amounts, not equal percentages. This mistake leads to a plausible-looking but wrong answer that will appear as an option.
Confusing which base the discount applies to. Discount is always on MP (marked/listed price). Profit/loss is always on CP. Mixing the two bases is the structural error behind at least one wrong answer in every profit-loss question set.
Inverting the y/z formula. When "CP of y = SP of z," profit% = (y−z)/z, not (y−z)/y. The denominator is z because the profit is calculated against CP, and CP corresponds to the z-article side of the equation. Writing (y−z)/y gives you a loss% calculation on SP, which is not what's asked.
Not converting the listed-price fraction to CP before computing gain%. In problems like "bought at 3/4 of listed price," candidates sometimes compute gain% as a fraction of listed price rather than CP. Gain% must always be referenced to what you paid — the cost price.