🏷️ Discount - Complete Theory

Master Discount calculations - essential for banking and shopping math!

🎯 What is Discount?

Discount is the reduction in the Marked Price (MP) of an article.

Key Terms:

• Marked Price (MP) = List Price / Tag Price (printed on product)
• Discount = Reduction given on MP
• Selling Price (SP) = Price after discount = MP - Discount
• Cost Price (CP) = Price at which item was purchased

Relationship:

SP = MP - Discount

📐 Basic Formulas

Formula 1: Discount Calculation

Discount = MP - SP

Discount% = (Discount / MP) × 100

Example: MP = ₹1,000, SP = ₹800. Find discount%.

Discount = 1,000 - 800 = ₹200
Discount% = (200/1,000) × 100 = 20%

Formula 2: Finding SP from Discount%

SP = MP - Discount
   = MP - (Discount% of MP)
   = MP × (100 - Discount%) / 100
   = MP × (1 - Discount%/100)

Example: MP = ₹2,500, Discount = 20%. Find SP.

SP = 2,500 × (100 - 20) / 100
   = 2,500 × 80/100
   = 2,500 × 0.8
   = ₹2,000

Formula 3: Finding MP from SP and Discount%

MP = SP × 100 / (100 - Discount%)

Example: SP = ₹1,600, Discount = 20%. Find MP.

MP = 1,600 × 100 / (100 - 20)
   = 1,600 × 100/80
   = ₹2,000

🔄 Successive Discounts

When two or more discounts are given one after another.

Formula for Two Successive Discounts

If discounts are a% and b%:

Single Equivalent Discount = [a + b - (a×b/100)]%

Key Insight: Successive discounts ≠ Sum of discounts!

Example 1: Two Successive Discounts

Q: MP = ₹5,000. Successive discounts of 20% and 10%. Find SP.

Method 1: Step-by-step

After 1st discount (20%):
Price = 5,000 × 80/100 = ₹4,000

After 2nd discount (10% on ₹4,000):
SP = 4,000 × 90/100 = ₹3,600

Method 2: Single Equivalent Discount

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

SP = 5,000 × 72/100 = ₹3,600

Answer: ₹3,600

Note: 20% + 10% ≠ 30%! It’s 28% only.

Example 2: Three Successive Discounts

Q: MP = ₹10,000. Discounts: 10%, 20%, 30%. Find SP.

Solution:

After 10%: 10,000 × 0.9 = ₹9,000
After 20%: 9,000 × 0.8 = ₹7,200
After 30%: 7,200 × 0.7 = ₹5,040

Or using formula:

SP = MP × (100-d₁)/100 × (100-d₂)/100 × (100-d₃)/100
   = 10,000 × 0.9 × 0.8 × 0.7
   = ₹5,040

Answer: ₹5,040

💼 Discount with Profit/Loss

Formula: MP, CP, Discount, and Profit

SP = MP - Discount = MP × (100 - d%) / 100
SP = CP + Profit = CP × (100 + p%) / 100

Combining:
MP × (100 - d%) / 100 = CP × (100 + p%) / 100

Example 3: Discount and Profit Together

Q: MP = ₹2,000, CP = ₹1,200. After 20% discount, find profit%.

Solution:

SP = 2,000 × 80/100 = ₹1,600

Profit = SP - CP = 1,600 - 1,200 = ₹400
Profit% = (400/1,200) × 100 = 33.33%

Answer: 33.33% profit

Example 4: Finding MP from CP and Discount

Q: CP = ₹800. After 25% discount, shopkeeper makes 20% profit. Find MP.

Solution:

For 20% profit:
SP = 800 × 120/100 = ₹960

Discount = 25%, so SP = 75% of MP
960 = MP × 75/100
MP = 960 × 100/75 = ₹1,280

Answer: ₹1,280

📊 Important Patterns

Pattern 1: Same SP after Different Discounts

If two items have same SP but different MP and discounts:
Higher MP will have higher discount amount (but may have lower discount%)

Pattern 2: Discount vs Profit Relation

If MP is x% above CP and discount is d%:

Profit/Loss% = x - d - (x×d/100)

If positive → Profit
If negative → Loss

Example: MP is 50% above CP, Discount = 20%

Result = 50 - 20 - (50×20/100)
       = 50 - 20 - 10
       = 20% profit

Pattern 3: False Discount

Some shopkeepers mark price very high, then give "attractive" discount:

Example: CP = ₹100
MP = ₹200 (100% markup!)
Discount = 40%
SP = 200 × 60/100 = ₹120 (Still 20% profit!)

💡 Solved Examples

Example 5: Finding Discount%

Q: MP = ₹3,600, SP = ₹3,000. Find discount%.

Solution:

Discount = 3,600 - 3,000 = ₹600
Discount% = (600/3,600) × 100 = 16.67%

Answer: 16.67%

Example 6: Successive vs Single Discount

Q: Which is better: Single discount of 30% OR successive discounts of 20% and 15%?

Solution:

Single discount = 30%

Successive discount = 20 + 15 - (20×15/100)
                    = 35 - 3
                    = 32%

Answer: Successive discounts (32%) are better!

Example 7: Complex Problem

Q: An article is marked 60% above CP. Two successive discounts of 20% and 10% are given. Find profit%.

Solution:

Let CP = ₹100
MP = 100 + 60% of 100 = ₹160

After 20% discount:
Price = 160 × 80/100 = ₹128

After 10% discount:
SP = 128 × 90/100 = ₹115.20

Profit = 115.20 - 100 = ₹15.20
Profit% = 15.2%

Using Formula:

Profit% = x - d₁ - d₂ - (x×d₁/100) - (x×d₂/100) + (d₁×d₂/100)
        = 60 - 20 - 10 - 12 - 6 + 2
        = 14% (wait, this doesn't match)

Let me use simpler formula:
Equivalent discount = 20 + 10 - (20×10/100) = 28%
Net = 60 - 28 - (60×28/100)
    = 60 - 28 - 16.8
    = 15.2% ✓

Answer: 15.2% profit

Example 8: Finding Original Discount

Q: After giving 10% discount on discount, final SP is ₹2,430. Original MP = ₹3,000. Find original discount%.

Solution:

Let original discount = d%

After d% discount:
Price = 3,000 × (100-d)/100

After additional 10% discount on this:
2,430 = [3,000 × (100-d)/100] × 90/100

2,430 = 3,000 × (100-d) × 0.9 / 100
243,000 = 3,000 × (100-d) × 0.9
243,000 = 2,700 × (100-d)
90 = 100 - d
d = 10%

Answer: 10% original discount

⚡ Quick Shortcuts

Shortcut 1: Common Discount Multipliers

10% discount → Multiply by 0.9 or 9/10
20% discount → Multiply by 0.8 or 4/5
25% discount → Multiply by 0.75 or 3/4
30% discount → Multiply by 0.7 or 7/10
33.33% discount → Multiply by 2/3
50% discount → Multiply by 0.5 or 1/2

Shortcut 2: Successive Equal Discounts

If same discount d% is given twice:
Equivalent = 2d - d²/100

Example: 10% + 10%
= 2(10) - 100/100
= 20 - 1 = 19%

Shortcut 3: Discount for Break-even

If MP is x% above CP, for no profit/loss:
Discount% = x × 100/(100+x)

Example: MP is 25% above CP
Break-even discount = 25 × 100/125 = 20%

Shortcut 4: Quick Mental Calculation

For 20% discount on ₹500:
20% = 1/5
500/5 = 100
SP = 500 - 100 = ₹400

🛍️ Real-Life Applications

Shopping Scenarios

Scenario 1: Festival Sale

Regular Price: ₹2,000
Discount: 30% + Extra 10% on card
Actual price = 2,000 × 0.7 × 0.9 = ₹1,260
Total saving = ₹740 (37% equivalent discount!)

Scenario 2: Clearance Sale

Original: ₹5,000
First markdown: 20% → ₹4,000
Second markdown: 30% → ₹2,800
Better than single 50% discount! (which would be ₹2,500)

⚠️ Common Mistakes

❌ Mistake 1: Adding Successive Discounts

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

❌ Mistake 2: Discount Base

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

Always use MP as base!

❌ Mistake 3: Profit Calculation

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

Profit% is always on CP!

❌ Mistake 4: Second Discount Base

In successive discounts:
Second discount is on REDUCED price, not original MP!

Example: ₹1,000 → 20% off = ₹800
Then 10% off on ₹800 (not ₹1,000) = ₹80 (not ₹100)

📝 Practice Problems

Level 1:

1. MP = ₹800, Discount = 25%. Find SP.
2. MP = ₹1,500, SP = ₹1,200. Find discount%.
3. SP = ₹2,400 after 20% discount. Find MP.

Level 2:

4. Successive discounts 30% and 20%. Find single equivalent discount.
5. MP = ₹2,000, CP = ₹1,500. After 10% discount, find profit%.
6. Three successive discounts: 10%, 20%, 30% on ₹10,000. Find final price.

Level 3:

7. MP is 40% above CP. After 25% discount, find profit%.
8. Which is better: 40% single discount OR 25% + 20% successive?
9. After giving x% discount twice, MP of ₹2,500 becomes ₹1,600. Find x.

🔗 Related Topics

Prerequisites:

• Percentage - Foundation for discount calculations

Related:

• Profit & Loss - Discount is part of P&L
• Simple Interest - Similar percentage concepts

Practice:

• Discount Questions
• Formula Sheet

Master Discount - Know your shopping math and save money! 🏷️