Profit and Loss is one of the most formula-heavy and high-scoring topics in arithmetic. But many students find it confusing because the same article can produce different outcomes depending on cost, selling price, and percentage calculations.
If you understand these three ideas clearly:
✔ Cost Price (C.P.)
✔ Selling Price (S.P.)
✔ Profit or Loss Percentage
Then you can solve any Profit & Loss question effortlessly.
This guide covers all formulas, concepts, FAQs, and exam-style examples, making it a perfect one-stop solution for all competitive exams.
Quick Overview: Profit & Loss Formulas
| Concept / Situation | Reference Considered (Equivalent to Distance) | Variable Used (Equivalent to Speed) | Formula (Meaning of Symbols in Row) |
|---|---|---|---|
| Profit / Gain | Difference between S.P. and C.P. | S.P. – C.P. | Profit = S.P. – C.P. (S.P. = Selling Price, C.P. = Cost Price) |
| Loss | Difference between C.P. and S.P. | C.P. – S.P. | Loss = C.P. – S.P. |
| Gain % | Profit relative to C.P. | (Profit/C.P.)×100 | Gain% = (Profit/C.P.)×100 (Profit = S.P.–C.P.) |
| Loss % | Loss relative to C.P. | (Loss/C.P.)×100 | Loss% = (Loss/C.P.)×100 |
| Finding S.P. (Profit) | C.P. is base value | Gain% applied on C.P. | S.P. = (100+Gain%)/100 × C.P. |
| Finding S.P. (Loss) | C.P. is base value | Loss% deducted from C.P. | S.P. = (100–Loss%)/100 × C.P. |
| Finding C.P. (Profit) | S.P. known | Gain% used reverse | C.P. = (100×S.P.)/(100+Gain%) |
| Finding C.P. (Loss) | S.P. known | Loss% used reverse | C.P. = (100×S.P.)/(100–Loss%) |
| Equal Gain% & Loss% | Identical % on two items | x²/100 loss rule | Loss% = (x²)/100 |
| False Weights | Difference between true & fake weight | Error/(True–Error) | Gain% = (Error/(True–Error))×100 |
Value Conversion Formulas (Foundation of Profit & Loss)
Before using advanced formulas, always identify whether the percentage applies to C.P. or S.P.. In Profit & Loss, every percentage is reckoned on C.P. Using the wrong base gives incorrect results.
1. Converting C.P. to S.P. (Profit Case)
When there is a gain, Selling Price becomes more than Cost Price.
Formula:
S.P. = C.P. × (100 + Gain%) / 100
Why this formula works
Profit is calculated on C.P.
If gain is 20%, the item is sold at 120% of its cost.
Example
C.P. = ₹500, Gain% = 20%
S.P. = 500×120/100 = ₹600
2. Converting C.P. to S.P. (Loss Case)
When there is a loss, Selling Price becomes less than Cost Price.
Formula:
S.P. = C.P. × (100 – Loss%) / 100
Why this formula works
Loss reduces value below the cost, so S.P. becomes a percentage of C.P.
Example
C.P. = ₹400, Loss% = 10%
S.P. = 400×90/100 = ₹360
Basic Profit–Loss Relations (Foundation Formulas)
All Profit & Loss questions are built from three essential relationships. These are similar to the Speed–Time–Distance base formulas in train problems.
1. Profit Formula
Profit = S.P. – C.P.
2. Loss Formula
Loss = C.P. – S.P.
3. Profit Percentage
Profit% = (Profit / C.P.) × 100
4. Loss Percentage
Loss% = (Loss / C.P.) × 100
These four equations appear in almost every question involving profit, discount, marked price, and reverse-calculation.
Finding Cost Price (Reverse Formulas)
These formulas are the Profit & Loss equivalent of converting m/s → km/hr or using relative speed backward.
When S.P. and % are known, we must reverse the formula to find C.P.
1. Finding C.P. When S.P. and Profit% Are Given
Formula:
C.P. = (100 × S.P.) / (100 + Gain%)
Why this formula works
If S.P. is 120% of C.P., then C.P. = S.P. divided by 1.20.
Example
S.P. = ₹240, Profit% = 20%
C.P. = (240×100)/120 = ₹200
2. Finding C.P. When S.P. and Loss% Are Given
Formula:
C.P. = (100 × S.P.) / (100 – Loss%)
Why this formula works
If S.P. is 80% of C.P., then C.P. = S.P. divided by 0.80.
Example
S.P. = ₹400, Loss% = 20%
C.P. = (400×100)/80 = ₹500
Special Profit & Loss Formulas (Advanced Cases)
These formulas appear in high-level exam questions (SSC CGL, SBI PO, CAT).
1. Equal Gain% and Loss% on Two Items (Always Produces Net Loss)
When one item is sold at x% gain and another at x% loss, the seller always incurs a loss.
Formula:
Net Loss% = (x² / 100)
Why this formula works
Profit applies on one C.P., loss applies on another. The base is different, so the effects don’t cancel.
Example
Gain = 10%, Loss = 10%
Loss% = 10²/100 = 1% Loss
2. False Weights Formula (Cheating Questions)
This is the Profit & Loss equivalent of relative speed in trains, one formula solves all such tricky questions.
When a trader pretends to sell at cost price but gives less quantity, he makes a profit.
Formula
Gain% = (Error / (True Weight – Error)) × 100
Where:
- True Weight = actual required weight (e.g., 1 kg = 1000g)
- Error = shortage given (difference)
Example
Seller gives 900g but charges for 1kg
Error = 100g
Gain% = 100/(1000–100) ×100 = 11.11%
3. Profit or Loss Based on Percentage Value Conversion
Sometimes an article is sold at "x% of C.P." or "y% of S.P."
Convert it into multipliers.
Examples
- “S.P. = 135% of C.P.” → S.P. = 1.35×C.P.
- “Sells at 75% of C.P.” → S.P. = 0.75×C.P.
- “S.P. is 60% more than C.P.” → S.P. = 1.60×C.P.
These help in solving ratio and chain-rule based P&L questions.
Smart Tips and Practical Tricks for Solving Profit and Loss
Mastering Profit and Loss becomes simple when you understand how cost price, selling price, and percentage work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding the relationship between C.P., S.P., and profit/loss percent.
This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.
1. Always Identify Whether the Question Is Based on C.P. or S.P.
Almost every mistake in Profit & Loss happens because students use the wrong base. All percentages are always calculated on C.P., not on S.P.
Example:
C.P. = ₹200, S.P. = ₹260
Profit = ₹60
Profit% = (60/200)×100 = 30%
(Not (60/260) which gives wrong answer)
2. Convert Profit% or Loss% Into Multipliers for Fast Calculation
Instead of long formulas, convert percentages into multipliers.
Profit:
S.P. = (1 + Profit%) × C.P.
Loss:
S.P. = (1 – Loss%) × C.P.
Example:
C.P. = 500, Profit% = 20%
Multiplier = 1.20
S.P. = 500 × 1.20 = ₹600
This one method reduces calculation time drastically.
3. Break Questions Into Profit/Loss + Percent + Difference
Most questions are combinations of:
✔ Difference between S.P. & C.P.
✔ Percentage gain/loss
✔ Reverse-finding of C.P. or S.P.
Once you identify which combination is used, the problem solves itself.
Example:
S.P. = 240
Profit% = 20%
C.P. = (100×240)/(120) = ₹200
4. Understand That Equal Gain% and Loss% Always Produce Overall Loss
This is one of the most repeated exam traps. When one item is sold at x% gain and another at x% loss, there is always a loss, never a break-even.
Formula:
Loss% = x² / 100
Example:
Gain = 10%, Loss = 10%
Loss% = 10²/100 = 1% Loss
5. Carefully Read Whether the Question Gives S.P. or C.P.
Numerous exam mistakes happen because students use forward formulas for reverse situations.
Forward formula:
S.P. = (100 ± %)/100 × C.P.
Reverse formula:
C.P. = 100×S.P. / (100 ± %)
Example:
S.P. = 540, Loss% = 10%
C.P. = 540×100/90 = ₹600
6. Never Ignore Units in Mixed Questions
Sometimes S.P. is given per unit and C.P. is for total units. Always equalize units before applying formulas.
Example:
C.P. for 5 kg = ₹200
S.P. per kg = ₹50
S.P. total = 50×5 = 250
Profit = ₹50 (25% profit)
7. False Weight Questions Need Only One Formula
All false-weight questions follow the same concept:
The trader gives less quantity but charges for full.
Formula:
Gain% = (Error / (True – Error)) × 100
Example:
Seller calls 900g as 1kg
Error = 100g
Gain% = 100/(1000–100) ×100 = 11.11%
FAQs About Profit and Loss
Q1. Why is Profit or Loss always calculated on C.P.?
Because C.P. (Cost Price) is the original investment on which all percentages must be based.
Example:
C.P. = 400, S.P. = 500
Profit = 100
Profit% = 100/400 ×100 = 25%
Q2. Why do equal gain% and loss% always result in a net loss?
Because the gain and loss percentages apply to different cost prices, not to the same value.
Formula:
Loss% = (x²)/100
Example:
Gain = 20%, Loss = 20%
Loss% = 400/100 = 4% loss
Q3. Why is multiplier method recommended for Profit and Loss?
Using multipliers (1 + p%) or (1 – p%) reduces steps and avoids formula confusion.
Example:
C.P. = 300, Profit% = 25%
S.P. = 1.25 × 300 = ₹375
Q4. Why do false-weight questions generate profit even at “cost price”?
Because a seller gives less quantity but charges as if full weight is given. This increases the effective S.P. automatically.
Example:
Gives 900g but charges for 1000g
Error = 100g
Gain% = 100/900 ×100 = 11.11%
Q5. Why is reverse calculation necessary for finding C.P. from S.P.?
Because the percentage given applies to C.P., so we must backtrack the base value.
Example:
S.P. = 240, Profit% = 20%
C.P. = (240×100)/120 = ₹200
Q6. Why do students often get profit% and loss% calculations wrong?
Because they mistakenly use S.P. instead of C.P. as the base for percentages.
Example:
C.P. = 150, S.P. = 180
Profit = 30
Profit% = 30/150 ×100 = 20%
(Not 30/180)
Q7. How can diagrams or tables help in Profit and Loss?
Using a simple CP→SP flow table helps to identify what is given and what must be found.
Example:
C.P. = 100 → Profit 20% → S.P. = 120
Q8. What is the simplest way to master Profit and Loss?
Recognize patterns:
✔ Profit% → Increase C.P.
✔ Loss% → Decrease C.P.
✔ C.P. unknown → Use reverse formula
✔ Equal gain-loss → always loss
✔ False weights → always profit
Mini Example:
C.P. = 600
Loss% = 15%
S.P. = 600 × 0.85 = ₹510
Q9. How does Profit & Loss help in competitive exams?
It strengthens percentage understanding and commercial arithmetic, used across SSC, Banking, Railways, and CAT.
Example:
If profit = ₹50 and Profit% = 10%
C.P. = 50×100/10 = ₹500
Q10. Why is practice important in Profit & Loss?
Because small mistakes in percentages or reverse formulas lead to big errors. Regular practice builds automatic formula selection.
Mini Example:
Loss = 30, Loss% = 10%
C.P. = 30×100/10 = ₹300