Profit & Loss

Profit and loss problems are among the most common in placement aptitude tests. Understanding the relationships between cost price, selling price, marked price, and discount is essential.

Core Terms

Key Formulas

Basic Profit/Loss

Profit = SP - CP (when SP > CP)
Loss = CP - SP (when CP > SP)

Profit/Loss Percentage

Profit% = (Profit / CP) × 100
Loss% = (Loss / CP) × 100

Important: Percentage is ALWAYS calculated on CP unless stated otherwise.

Finding SP from CP

SP = CP × (100 + Profit%) / 100   [when profit]
SP = CP × (100 - Loss%) / 100     [when loss]

Finding CP from SP

CP = SP × 100 / (100 + Profit%)   [when profit]
CP = SP × 100 / (100 - Loss%)     [when loss]

Discount Problems

Discount on Marked Price

Discount = MP - SP
Discount% = (Discount / MP) × 100
SP = MP × (100 - Discount%) / 100

Example: MP = ₹500, Discount = 20%

SP = 500 × (100-20)/100 = 500 × 0.8 = ₹400
Discount amount = ₹100

Successive Discounts

Two successive discounts of a% and b% are NOT the same as a single discount of (a+b)%.

Formula for equivalent single discount:

Single discount = a + b - (ab/100)

Example: Successive discounts of 20% and 10%:

= 20 + 10 - (20×10)/100 = 30 - 2 = 28%

Verify: On ₹1000:

• After 20%: ₹800
• After 10% on ₹800: ₹720
• Total discount: ₹280 → 28% ✓

Three Successive Discounts

For discounts a%, b%, c%:

SP = MP × (1-a/100)(1-b/100)(1-c/100)

Example: MP = ₹1000, discounts 10%, 20%, 30%:

SP = 1000 × 0.9 × 0.8 × 0.7 = 1000 × 0.504 = ₹504
Effective discount = 49.6%

Profit/Loss with Fractions

Common Fraction-to-Percentage Mappings

Problem: A man sells an article at a profit of 25%. If he had bought it at 20% less and sold it for ₹10.50 less, he would have gained 30%. Find the CP.

Solution:

Let CP = 100x
SP₁ = 125x (25% profit)
New CP = 80x (20% less)
New SP = 104x (30% profit on 80x)
125x - 104x = 10.50
21x = 10.50
x = 0.5
CP = 100 × 0.5 = ₹50

Dishonest Dealer / False Weight Problems

Concept

A dealer uses false weights but claims to sell at cost price. His profit% is:

Profit% = [(True weight - False weight) / False weight] × 100

Example: A dealer uses 900g instead of 1kg. Find profit%.

Profit% = [(1000 - 900) / 900] × 100 = (100/900) × 100 = 11.11%

Dealer sells at x% loss using y% less weight

Actual profit% = [(100-x)/(100-y) - 1] × 100

(Use the same formula with +x in place of -x when selling at x% profit instead of loss.) If a dealer sells at a loss of x% but uses y% less weight:

Net effect = [(100-x)/(100-y) - 1] × 100

If positive → profit, if negative → loss.

Example: Sells at 10% loss but uses 20% less weight:

= [(100-10)/(100-20) - 1] × 100
= [90/80 - 1] × 100
= [1.125 - 1] × 100 = 12.5% profit

Buy X Get Y Free

Problem: A shopkeeper offers “Buy 2 Get 1 Free”. What is the discount%?

Solution:

Customer pays for 2, gets 3
Discount% = (Free / Total) × 100 = (1/3) × 100 = 33.33%

Problem: “Buy 3 Get 1 Free”:

Discount% = (1/4) × 100 = 25%

General: “Buy (n-1) Get 1 Free” → Discount = (1/n) × 100%

Markup and Profit Relationship

Finding Markup % for Desired Profit

If a shopkeeper wants x% profit after giving d% discount:

MP = CP × (100 + x) / (100 - d)
Markup% = [(100+x)/(100-d) - 1] × 100

Example: CP = ₹200, wants 20% profit, gives 10% discount:

MP = 200 × (100+20)/(100-10) = 200 × 120/90 = ₹266.67
Markup% = (266.67-200)/200 × 100 = 33.33%

Two Items Sold at Same Price

One at Profit, One at Loss (Same %)

If two items are sold at the same price, one at x% profit and the other at x% loss:

Net result = Always a LOSS
Net loss% = x²/100 %

Example: Two items at ₹1000 each, one at 20% profit, one at 20% loss:

Loss% = 20²/100 = 4%
Net loss = 4% of (total CP)

Proof:

SP₁ = SP₂ = S (same selling price)
CP₁ = S × 100/(100+x)
CP₂ = S × 100/(100-x)
Total CP = S[100/(100+x) + 100/(100-x)]
= S × [100(100-x) + 100(100+x)] / [(100+x)(100-x)]
= S × 20000 / (10000-x²)
Total SP = 2S
Loss = Total CP - Total SP = 2S × x²/(10000-x²)
Loss% (as fraction of CP) = [x²/(10000-x²)] × 100

When x is small, 10000-x² ≈ 10000, so Loss% ≈ x²/100.
Note: For the standard interview shortcut, **Loss% = x²/100** is the exact answer when "two items at same SP, one at x% profit and one at x% loss" — both items are referenced to their CP, and the calculation simplifies to x²/100 exactly (not approximately). The formula above is the corresponding expression when SP is held fixed and CP varies.

Different Percentages

If one is sold at x% profit and other at y% loss, and SP is same:

Net loss% = (x-y)² / (200+x-y)   [if x < y, it's a loss]
Or use: Net effect = 2xy/(x+y) loss if x=y, general formula is complex.

Tricks & Shortcuts

Trick 1: CP-SP-MP Relationships

CP → (+Profit%) → SP
CP → (+Loss%) → SP (with minus)
MP → (-Discount%) → SP

Trick 2: If Profit% = Loss% on Different CPs

If selling at ₹X gives profit of P%, and selling at ₹Y gives loss of P%:

CP = (X + Y) / 2

Trick 3: Finding CP from Profit and Loss Scenarios

If selling at ₹A gives profit of x%, and selling at ₹B gives loss of y%:

CP = (A×100)/(100+x) = (B×100)/(100-y)

From these two:

CP = (100×A)/(100+x) and CP = (100×B)/(100-y)

Trick 4: Equal Profit and Loss Amount

If profit% and loss% are on the same CP and equal in amount:

Profit × CP/100 = Loss × CP/100
This only happens when profit% = loss%

Trick 5: Effect of Changing CP and SP

If CP increases by x% and SP remains the same:

• Profit decreases or loss increases

If SP increases by x% and CP remains the same:

• Profit increases or loss decreases

Practice Questions

Q1: Basic Profit/Loss

A shopkeeper buys an article for ₹400 and sells it for ₹500. Find profit%.

Solution:

Profit = 500 - 400 = ₹100
Profit% = (100/400) × 100 = 25%

Q2: Successive Discounts

Find the single equivalent discount for successive discounts of 20%, 15%, and 10%.

Solution:

Let MP = 100
After 20%: 80
After 15%: 80 × 0.85 = 68
After 10%: 68 × 0.9 = 61.2
Effective discount = 100 - 61.2 = 38.8%

Q3: False Weight

A shopkeeper sells sugar at cost price but uses 800g instead of 1kg. Find profit%.

Solution:

Profit% = [(1000-800)/800] × 100 = (200/800) × 100 = 25%

Q4: Markup and Discount

A shopkeeper marks an article 40% above CP and allows a 20% discount. Find profit%.

Solution:

Let CP = 100
MP = 140
SP = 140 × 0.8 = 112
Profit% = 12%

Q5: Two Articles Same SP

Two articles are sold at ₹990 each. One at 10% profit, other at 10% loss. Find net result.

Solution:

CP₁ = 990 × 100/110 = ₹900
CP₂ = 990 × 100/90 = ₹1100
Total CP = 2000, Total SP = 1980
Loss = ₹20
Loss% = (20/2000) × 100 = 1%

Q6: Buy X Get Y Free

A shopkeeper offers “Buy 4 Get 1 Free”. What discount% is this equivalent to?

Solution:

Pay for 4, get 5
Discount = (1/5) × 100 = 20%

Q7: Complex Problem

A man buys 5 apples for ₹3 and sells 3 apples for ₹5. Find profit% on 15 apples.

Solution:

CP of 15 apples = 3 × (15/5) = ₹9
SP of 15 apples = 5 × (15/3) = ₹25
Profit = ₹16
Profit% = (16/9) × 100 = 177.78%

Q8: Mixed Problem

A shopkeeper sells an article at 15% profit. If he had bought it at 10% less and sold it for ₹45 less, he would have gained 25%. Find the original CP.

Solution:

Let CP = 100x
Original SP = 115x
New CP = 90x
New SP = 90x × 1.25 = 112.5x
115x - 112.5x = 45
2.5x = 45
x = 18
CP = ₹1800

Summary Table