💹 Profit & Loss - Complete Theory

Master Profit & Loss - essential for business math and banking exams!

🎯 Basic Concepts

Key Terms

Cost Price (CP):

• Price at which an article is purchased
• Also called “buying price”

Selling Price (SP):

• Price at which an article is sold
• Also called “selling price”

Profit:

• When SP > CP
• Profit = SP - CP

Loss:

• When CP > SP
• Loss = CP - SP

Marked Price (MP):

• Price printed on the article
• Also called “List Price” or “Tag Price”

Discount:

• Reduction given on Marked Price
• Discount = MP - SP

📐 Basic Formulas

1. Profit Formulas

Profit = SP - CP

Profit % = (Profit/CP) × 100
         = [(SP - CP)/CP] × 100

SP = CP + Profit
   = CP × (100 + Profit%)/100
   = CP × [1 + (Profit%/100)]

CP = SP × 100/(100 + Profit%)

2. Loss Formulas

Loss = CP - SP

Loss % = (Loss/CP) × 100
       = [(CP - SP)/CP] × 100

SP = CP - Loss
   = CP × (100 - Loss%)/100
   = CP × [1 - (Loss%/100)]

CP = SP × 100/(100 - Loss%)

3. Discount Formulas

Discount = MP - SP

Discount % = (Discount/MP) × 100

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

MP = SP × 100/(100 - Discount%)

💡 Solved Examples

Example 1: Basic Profit

Q: CP = ₹500, SP = ₹600. Find profit%.

Solution:

Profit = SP - CP = 600 - 500 = ₹100

Profit% = (Profit/CP) × 100
        = (100/500) × 100
        = 20%

Answer: 20% profit

Example 2: Finding SP from Profit%

Q: CP = ₹800, Profit% = 25%. Find SP.

Solution:

SP = CP × (100 + Profit%)/100
   = 800 × (100 + 25)/100
   = 800 × 125/100
   = 800 × 5/4
   = ₹1,000

Answer: SP = ₹1,000

Example 3: Finding CP from SP and Profit%

Q: SP = ₹1,200, Profit = 20%. Find CP.

Solution:

CP = SP × 100/(100 + Profit%)
   = 1200 × 100/(100 + 20)
   = 1200 × 100/120
   = 1200 × 5/6
   = ₹1,000

Answer: CP = ₹1,000

Example 4: Discount Problem

Q: MP = ₹2,500, Discount = 20%. If CP = ₹1,500, find profit%.

Solution:

Step 1: Find SP
SP = MP × (100 - Discount%)/100
   = 2500 × 80/100
   = ₹2,000

Step 2: Find Profit%
Profit = SP - CP = 2000 - 1500 = ₹500
Profit% = (500/1500) × 100 = 33.33%

Answer: 33.33% profit

🔄 Advanced Concepts

1. Successive Discounts

If two discounts are a% and b%:
Net discount = a + b - (ab/100)%

Single equivalent discount formula

Example: 20% and 10% successive discounts

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

Note: NOT 30%!

2. Dishonest Dealer (False Weight)

If dealer uses weight 'w' gm instead of 1000 gm:
Profit% = [(1000 - w)/w] × 100

If dealer claims profit of p% but uses false weight w:
Actual Profit% = [(1000 - w + wp/100)/w] × 100

Example: Uses 900g instead of 1kg

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

3. Marked Price & Discount

If MP is x% above CP and discount is d%:
Profit/Loss% = x - d - (xd/100)

If result is +ve → Profit
If result is -ve → Loss

Example: MP is 40% above CP, Discount = 20%

Result = 40 - 20 - (40×20)/100
       = 40 - 20 - 8
       = 12% profit

📊 Important Variations

Variation 1: Break-Even Point

When SP = CP:
No profit, no loss
Profit% = Loss% = 0%

Variation 2: Selling at CP After Discount

If discount = d% and SP = CP:
MP = CP × 100/(100-d)

Example: 20% discount, SP = CP = ₹800

MP = 800 × 100/80 = ₹1,000

Variation 3: Two Articles - One Profit, One Loss

If both sold at same SP:
Loss% = [(Profit% × Loss%)/(100 + Profit%)] - required for no overall loss

⚡ Quick Shortcuts

Shortcut 1: When CP = ₹100

If CP = ₹100:
Profit% = SP - 100 (directly!)

Example: CP = ₹100, SP = ₹125
Profit% = 125 - 100 = 25%

Shortcut 2: Common Profit% Multipliers

10% profit → SP = CP × 11/10
20% profit → SP = CP × 6/5
25% profit → SP = CP × 5/4
33.33% profit → SP = CP × 4/3
50% profit → SP = CP × 3/2
100% profit → SP = CP × 2

Shortcut 3: Loss% Multipliers

10% loss → SP = CP × 9/10
20% loss → SP = CP × 4/5
25% loss → SP = CP × 3/4
50% loss → SP = CP × 1/2

Shortcut 4: Discount Calculation

If MP and discount% both are given:
SP = MP × (100 - Discount%)/100

For 20% discount → multiply by 0.8 or 4/5
For 25% discount → multiply by 0.75 or 3/4

💼 Real-Life Applications

1. Retail Business: Calculating selling prices, discounts
2. E-commerce: Flash sales, coupon codes
3. Stock Market: Profit/loss on investments
4. Banking: Loan processing fees, interest calculations
5. Shopkeeping: Daily transactions, inventory valuation

🎯 Important Patterns

Pattern 1: Same SP, Different CP

If two items sold at same SP:
- One at x% profit
- Other at x% loss

Result: Always overall loss!
Loss% = (Common profit-loss%)²/100

Example: 20% profit and 20% loss
Loss% = (20)²/100 = 4%

Pattern 2: Two Different SP

If CP same for both:
Profit on first = x%
Loss on second = y%

Overall Profit/Loss% depends on actual values

Pattern 3: Successive Transactions

If buying at one price and selling at another in chain:
Overall Profit% = Product of all multipliers - 1

Example: Buy at ₹100, sell at 20% profit, buy again, sell at 10% profit
Overall = 1.2 × 1.1 - 1 = 1.32 - 1 = 32% profit

⚠️ Common Mistakes

❌ Mistake 1: Wrong Base for %

Wrong: Profit% = (Profit/SP) × 100 ✗
Right: Profit% = (Profit/CP) × 100 ✓

Always use CP as base for profit/loss%!

❌ Mistake 2: Discount Base

Wrong: Discount% = (Discount/CP) × 100 ✗
Right: Discount% = (Discount/MP) × 100 ✓

Always use MP as base for discount%!

❌ Mistake 3: Successive Discounts

Wrong: 20% + 10% = 30% total discount ✗
Right: 20 + 10 - (20×10)/100 = 28% ✓

📝 Practice Problems

Level 1:

1. CP = ₹600, SP = ₹750. Find profit%.
2. CP = ₹800, Loss = 20%. Find SP.
3. MP = ₹500, Discount = 10%. Find SP.

Level 2:

4. SP = ₹900, Profit = 20%. Find CP.
5. MP is 30% above CP. After 10% discount, profit is what%?
6. Two successive discounts of 15% and 20%. Find single equivalent discount.

Level 3:

7. A shopkeeper uses 800g weight instead of 1kg. Find profit%.
8. Two articles sold at ₹1,200 each. One at 20% profit, other at 20% loss. Overall profit/loss?
9. MP = ₹5,000, CP = ₹3,000. What discount% will give 20% profit?

🔗 Related Topics

Prerequisites:

• Percentage - Foundation for profit/loss calculations

Related:

• Discount - Part of P&L
• Simple Interest - Uses similar percentage concepts

Practice:

• Profit & Loss Questions
• Formula Sheet

Remember: Profit & Loss is all about percentages - master Percentage first! 💪