Profit and Loss is one of the highest-scoring arithmetic chapters in the CDS and OTA written exam. Almost every paper carries two to four questions on cost price, selling price, percentage gain, discount or dishonest shopkeepers. The maths is short, the logic is fixed, and with a handful of formulas you can solve most questions in under a minute. Let us build that speed.
Why Profit and Loss Matters in CDS
The CDS Elementary Mathematics paper has 100 questions in 120 minutes, so speed decides your final score. Profit and Loss is a guaranteed scorer because the questions are formula-driven and rarely tricky once you know the base relationships.
Every problem in this chapter revolves around just two quantities — the Cost Price (CP) at which a trader buys, and the Selling Price (SP) at which he sells. If SP is more, he profits; if SP is less, he loses. That is the entire foundation, and the rest of the chapter is simply different ways of dressing up these two numbers.
Because the topic links directly to percentages, it doubles as revision for ratio, fractions and percentage problems too. A strong grip here often raises your accuracy across the whole arithmetic section of the paper. Examiners typically blend it with discount, marked price or a dishonest-dealer twist to test whether you truly understand the base on which each percentage is taken.
Profit and loss are always calculated on the Cost Price unless the question clearly states otherwise. This single habit prevents most silly errors.
Core Terms You Must Know
Before formulas, fix the vocabulary in your mind. CDS examiners love testing whether you know which price a percentage refers to.
- Cost Price (CP): the price at which an article is purchased, including overheads like transport or repair.
- Selling Price (SP): the price at which the article is finally sold.
- Marked Price (MP) / List Price: the printed or labelled price before any discount.
- Discount: a reduction given on the Marked Price, never on the Cost Price.
- Profit (Gain): SP − CP, when SP > CP.
- Loss: CP − SP, when CP > SP.
Overheads deserve special attention. If a trader buys a cycle for ₹2000 and spends ₹200 on repairs before selling it, his effective cost price is ₹2200, not ₹2000. CDS questions frequently slip in transport, packing or repair charges precisely to see whether you fold them into the cost price. Always treat every rupee spent before the sale as part of CP.
Discount is calculated on the Marked Price, while profit and loss are on the Cost Price. Mixing the two bases is the number-one error in this chapter.
The Basic Formulas
These four relations cover the majority of CDS questions. Learn them as a single block.
Profit = SP − CP | Loss = CP − SP
Profit% = (Profit ÷ CP) × 100
Loss% = (Loss ÷ CP) × 100
From these you can move in either direction between CP and SP:
SP = CP × (100 + Profit%) ÷ 100
SP = CP × (100 − Loss%) ÷ 100
CP = SP × 100 ÷ (100 + Profit%)
CP = SP × 100 ÷ (100 − Loss%)
Notice the symmetry: to go from CP to SP you multiply, and to come back from SP to CP you divide by the same bracket. Once you internalise this, you can solve forward and backward problems with equal ease. A great deal of CDS questions are nothing more than “find CP, then re-sell at a new percentage,” which is just one multiply and one divide.
It also helps to memorise a few quick conversions: a 25% profit means SP is five-fourths of CP, a 20% profit means SP is six-fifths of CP, and a 10% loss means SP is nine-tenths of CP. Recognising these ratios on sight removes the need to compute from scratch each time.
Whenever a question gives you SP and a profit/loss %, divide back to CP using the formulas above. Many students wrongly take a percentage of SP — that gives the wrong CP every time.
Marked Price and Discount
Shopkeepers mark goods above cost, then offer a discount to attract buyers while still earning profit. The chain is: CP → MP → (discount) → SP.
SP = MP × (100 − Discount%) ÷ 100
Discount = MP − SP
Discount% = (Discount ÷ MP) × 100
A trader can give a discount and still profit, because the marked price is set above the cost price. The discount only eats into the markup, not necessarily into the cost.
Successive Discounts
Two discounts of a% and b% in succession do not add up to (a + b)%. The second discount applies to the already-reduced price.
Equivalent single discount for a% then b% = (a + b − ab÷100)%
For example, 20% then 10% gives 20 + 10 − (20×10)÷100 = 28%, not 30%. The order of the two discounts does not change the final price, since multiplication is commutative — 20% then 10% lands at exactly the same selling price as 10% then 20%.
For three or more discounts, apply them one after another as multiplying factors. A price hit by 10%, 20% and 25% discounts becomes MP × 0.90 × 0.80 × 0.75 of the marked price, which equals 0.54 — an overall 46% discount. Building the answer as a product of fractions is far safer than trying to add percentages in your head.
Useful Shortcuts for Speed
These tested results save precious seconds in the exam hall.
If an article is sold at two prices giving equal profit and loss%, the overall result is a loss, and net Loss% = (common %)2 ÷ 100.
- If CP of x articles = SP of y articles, then Profit% = ((x − y) ÷ y) × 100.
- When a man sells two items at the same SP, one at p% gain and the other at p% loss, he always suffers a net loss of (p2÷100)%.
- To convert a fraction into a quick profit%: a gain that makes SP = (5÷4)CP means 25% profit.
The “same SP, equal % gain and loss” trap appears almost every alternate year. Memorise that the answer is always a small loss, never break-even.
Dishonest Dealers and False Weights
A favourite CDS theme: a shopkeeper claims to sell at cost price but cheats using a faulty weight (say, a 1000 g weight that is actually 960 g). He still makes a profit because he gives less than he charges for.
Gain% = (Error ÷ (True value − Error)) × 100
where Error = claimed weight − actual weight delivered.
So a dealer using a 960 g weight in place of 1000 g has Error = 40 g and True value = 1000 g, giving Gain% = (40 ÷ 960) × 100 ≈ 4.17%. The denominator is the actual quantity handed over (960 g), because that is what truly cost the dealer money — a point students often get wrong by dividing by 1000.
If the dealer also marks the goods up or down, combine the false-weight gain with the stated profit/loss using the standard percentage-on-CP method.
Worked Example
Let us apply several rules together in one typical CDS-level question.
A shopkeeper marks his goods 40% above cost price and then allows a discount of 25% on the marked price. Find his net profit or loss percent.
So despite a generous 25% discount, the shopkeeper still earns a 5% profit, because the markup was large enough to absorb the discount.
Always assume CP = 100 (or 100 units) when only percentages are given. It turns abstract markups into simple arithmetic.
A Second Quick Illustration
Here is the “CP of x = SP of y” shortcut in action.
The cost price of 20 articles equals the selling price of 16 articles. Find the gain percent.
The dealer makes a clean 25% gain without computing any actual rupee figures.
Do not divide by x. The denominator must be y — the number of articles whose selling price is referenced.
Previous-Year Style Question
This mirrors the difficulty and phrasing of recent CDS papers.
Q. By selling an article for ₹450, a man loses 10%. At what price should he sell it to gain 20%?
Answer: First find CP. CP = SP × 100 ÷ (100 − Loss%) = 450 × 100 ÷ 90 = ₹500. To gain 20%, new SP = 500 × 120 ÷ 100 = ₹600.
Notice the two-step pattern: recover the cost price first, then re-apply the desired profit percentage. Nearly every CDS profit-loss problem reduces to finding CP and moving forward.
You can also chain this with the equal-ratio shortcut. Since a 10% loss makes SP nine-tenths of CP, and a 20% gain makes SP six-fifths of CP, the required price equals 450 × (6÷5) ÷ (9÷10) = 450 × 12÷9 = ₹600 in a single line, confirming our answer instantly.
Where These Questions Appear
Beyond the textbook framing, CDS sets profit-loss inside real-world wrappers. Recognising the disguise is half the battle.
- Trader and middleman chains: goods pass through two sellers, each adding a profit; compute the final buyer's cost by multiplying the gain factors.
- Partnership and instalment sales: the profit is shared or the price is paid in parts, but the underlying CP-to-SP logic is unchanged.
- Comparison of two schemes: “buy one get one” versus a flat discount — convert both to an effective discount percentage and compare.
In each case, reduce the wordy situation to CP, SP and a percentage. Once the numbers are extracted, the formulas you have already learnt finish the job. The skill that separates a fast solver from a slow one is translating English into these three quantities quickly.
Underline the price each percentage is taken on as you read. A circled “CP” or “MP” beside every number stops you from choosing the wrong base under time pressure.
Common Mistakes to Avoid
Knowing the formulas is not enough — the exam punishes careless base selection. Watch for these traps.
- Calculating profit% on SP instead of CP.
- Adding two successive discounts directly instead of using the combined formula.
- Assuming equal % gain and loss on the same SP cancel out — they never do; it is always a loss.
- Forgetting to add overhead or transport charges to the cost price.
- Confusing Marked Price with Selling Price after a discount.
- Rounding off intermediate values and carrying the error into the final percentage — keep fractions exact until the last step.
When a question says “profit is 25% of selling price,” do not treat it as 25% of CP. Convert carefully: SP profit of 25% means CP = 75% of SP, giving a 33⅓% profit on cost.
Quick Revision
- Profit = SP − CP; Loss = CP − SP; both percentages are on CP.
- SP = CP × (100 ± %) ÷ 100; reverse to get CP by dividing.
- Discount is always on Marked Price: SP = MP × (100 − D%) ÷ 100.
- Two discounts a% then b% = (a + b − ab÷100)% combined.
- Equal % gain and loss on same SP → net loss = (%)2÷100.
- False weight gain% = Error ÷ (True − Error) × 100.
Practise five mixed problems daily for a week. Profit and Loss rewards pattern recognition far more than calculation power.
Frequently asked questions
Is profit always calculated on cost price or selling price?
By convention and in CDS exams, profit and loss percentages are always calculated on the cost price unless the question explicitly states they are on the selling price. Always read the wording carefully.
Why do two successive discounts not simply add up?
The second discount applies to the price already reduced by the first, not to the original marked price. Use the formula a + b − ab÷100 to find the single equivalent discount.
If I sell two articles at the same price, one at 10% gain and one at 10% loss, do I break even?
No. You always make a net loss equal to (common%)² ÷ 100, which is 1% here. Equal percentage gain and loss on the same selling price never cancel out.
How does a dishonest dealer profit while selling at cost price?
By using a false weight that delivers less than the stated quantity. The gain percent equals Error ÷ (True value − Error) × 100, so a 960 g weight sold as 1 kg yields about 4.17% profit.
What is the fastest way to handle markup-and-discount questions?
Assume the cost price is 100, build the marked price from the markup, subtract the discount to get the selling price, then read off the profit or loss directly as a percentage.
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