Profit, Loss & Discount – Complete Short Notes for Competitive Exams

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1. Basic Concepts

These concepts are part of Percentage and Arithmetic and are based on comparing Cost Price (C.P.), Selling Price (S.P.), and Marked Price (M.P.).

Term	Meaning
Cost Price (C.P.)	Price at which an article is purchased
Selling Price (S.P.)	Price at which an article is sold
Marked Price (M.P.)	Price printed on the article before discount
Profit / Gain	When S.P. > C.P.
Loss	When C.P. > S.P.

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2. Fundamental Formulas

Concept	Formula
Profit (Gain)	S.P. – C.P.
Loss	C.P. – S.P.
Profit %	(Profit / C.P.) × 100
Loss %	(Loss / C.P.) × 100
S.P. (when Profit)	C.P. × (100 + Profit%) / 100
S.P. (when Loss)	C.P. × (100 - Loss%) / 100
C.P. (when Profit)	S.P. × 100 / (100 + Profit%)
C.P. (when Loss)	S.P. × 100 / (100 - Loss%)

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3. Key Relationships

Relationship	Formula
If Profit % = Loss %	Net Loss = (Common %²) / 100
If C.P. and S.P. are interchanged	Gain% and Loss% are same numerically
If article is sold at gain = loss%	Loss% = Gain% = x → Loss = (x²)/100

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4. Example Calculations

Example 1:
C.P. = ₹200, S.P. = ₹250
→ Profit = 250 − 200 = ₹50
→ Profit% = (50/200)×100 = 25%

Example 2:
C.P. = ₹150, S.P. = ₹120
→ Loss = 150 − 120 = ₹30
→ Loss% = (30/150)×100 = 20%

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5. Successive Transactions (Profit or Loss Twice)

If an article is sold twice with x% and y% profit (or loss):

$$\text{Net Change \%}=x+y+\frac{xy}{\mathrm{100}}$$

Example:
Gain of 10% and then 20%
→ Net Gain = 10 + 20 + (10×20)/100 = 32%

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6. Discount

• Discount = Marked Price − Selling Price

• Discount % = (Discount / Marked Price) × 

• 100

$$S\mathrm{.}P\mathrm{.}=M\mathrm{.}P\mathrm{.}\times \frac{100-Discount\mathrm{\%}}{\mathrm{100}}$$

Example:
M.P. = ₹500, Discount = 20%
→ S.P. = 500 × (80/100) = ₹400

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7. Successive Discounts

If two discounts x% and y% are given successively,

$$\text{Net Discount \%}=x+y-\frac{xy}{\mathrm{100}}$$

Example:
Discounts of 10% and 20%
→ Net Discount = 10 + 20 − (10×20)/100 = 28%

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8. Relation Between Marked Price, Cost Price, and Selling Price

$$\text{S.P.} = \text{M.P.} × \frac{(100 - \text{Discount\%})}{100}$$ $$\text{Profit \%}=\left(\frac{\text{S.P.}-\text{C.P.}}{\text{C.P.}}\right)\times 100$$

Thus,

$$\text{S.P.}=\text{C.P.}\times \frac{(100+\text{Profit\%})}{\mathrm{100}}$$

Combine both when given M.P., Discount, and Profit% to find C.P. or S.P.

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9. Important Variations

Case	Formula / Concept
Discount allowed on M.P.	S.P. = M.P. × (100 − Discount%) / 100
Profit based on S.P.	Profit% = (Profit / S.P.) × 100
Loss based on S.P.	Loss% = (Loss / S.P.) × 100
Required M.P. for desired Profit%	M.P. = C.P. × 100 × 100 / ((100 − Discount%) × (100 − Profit%))

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10. Successive Profit and Loss (Mixed Cases)

When an article is sold twice with gain and loss:

$$\text{Net \% Change}=x-y-\frac{xy}{\mathrm{100}}$$

Example:
Gain 20%, Loss 10% → Net = 20 − 10 − (20×10)/100 = 8% gain

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11. Cost Price & Selling Price Interchange Trick

If an article is sold at gain equal to loss%,
then overall Loss = (Common %²) / 100

Example:
If gain = loss = 10%,
→ Net Loss = (10²)/100 = 1%

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12. Multiple Items – Equal Selling Price

If two articles are sold at the same S.P., one at x% gain and other at x% loss:
→ Net Loss % = (x²) / 100

Example:
One article at 20% gain, another at 20% loss → Net Loss = (20²)/100 = 4%

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13. False Weight / Dishonest Dealer

If a dealer cheats by giving less weight but sells at cost price:

$$\text{Gain \%}=\frac{\text{Error in weight}}{\text{True weight}-\text{Error}}\times 100$$

Example:
Dealer uses 950g instead of 1kg.
→ Gain% = (50/950)×100 ≈ 5.26%

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14. Articles Bought and Sold Together

If n articles are bought for Rs. x and sold for Rs. y,

$$\text{Profit or Loss \%}=\frac{(y-x)}{x}\times 100$$

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15. Shortcut Tricks

✅ When S.P. and Profit% are given:
C.P. = (100×S.P.) / (100 + Profit%)

✅ When S.P. and Loss% are given:
C.P. = (100×S.P.) / (100 − Loss%)

✅ When C.P. and Discount% are given:
M.P. = (100×S.P.) / (100 − Discount%)

✅ When two successive discounts:
Use Net Discount Formula: 

$x + y - \frac{xy}{100}$

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16. Important One-Liners

• Profit% and Loss% are always calculated on C.P. unless stated otherwise.

• Discount% is always on M.P.

• Successive % changes compound, not simply add.

• If M.P. = S.P., no discount is given.

• Profit and discount can occur simultaneously — always apply formulas sequentially.

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17. Quick Practice Table

Case	Formula
Profit %	(Profit / C.P.)×100
Loss %	(Loss / C.P.)×100
Discount %	(Discount / M.P.)×100
S.P. (Profit)	C.P.×(100+P)/100
S.P. (Loss)	C.P.×(100−L)/100
C.P. (Profit)	S.P.×100/(100+P)
C.P. (Loss)	S.P.×100/(100−L)
Net Discount	x + y − (xy/100)
Net Profit (2 gains)	x + y + (xy/100)
Equal Gain & Loss	Net Loss = x²/100

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18. Typical Exam Questions

Q1. A shopkeeper buys an item for ₹400 and sells it for ₹480. Find profit %.
→ Profit = 80, Profit% = (80/400)×100 = 20%

Q2. After a 10% discount, a shirt is sold for ₹450. Find M.P.
→ M.P. = (450×100)/(100−10) = ₹500

Q3. Two successive discounts of 10% and 15%. Find net discount.
→ = 10 + 15 − (10×15)/100 = 23.5%

Q4. A dishonest dealer uses 900g instead of 1kg. Find gain %.
→ Gain% = (1000−900)/900 ×100 = 11.11%

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19. Common Fractions for Fast Calculation

Fraction	Percentage
1/10	10%
1/5	20%
1/4	25%
1/3	33.33%
1/2	50%

Use them for mental math in profit/loss and discount questions.

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20. Quick Revision Summary

• Profit = S.P. − C.P.

• Loss = C.P. − S.P.

• Discount = M.P. − S.P.

• Net Change (Two Changes) → 

  $x ± y ± \frac{xy}{100}$

• Equal Profit & Loss → 

  $x^2/100$ % Loss

• Discount chain → 

  $x + y − \frac{xy}{100}$

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✅ In One Line:

“Profit, Loss, and Discount questions are all about comparing C.P., S.P., and M.P. — learn the core formulas and successive % change trick.”