Here are concise revision notes on “Discounting” for competitive exams like JKSSB Forester, focusing on mathematical applications:
Discounting: Key Concepts and Calculations for Competitive Exams
Discounting is a fundamental concept in commercial mathematics, frequently tested in competitive exams. It involves reducing the price of an item, and understanding its various forms and calculations is crucial.
1. Basic Terminology
Before diving into calculations, let’s clarify the key terms:
- Marked Price (MP) / List Price (LP) / Tag Price: This is the price at which a product is initially listed or tagged. It’s the price before any discount is applied. Think of it as the original, advertised price.
- Analogy: The price printed on a clothing tag.
- Sale Price (SP) / Selling Price: This is the price at which the product is actually sold after a discount has been applied.
- Analogy: The amount you pay at the cash counter.
- Discount (D): This is the reduction offered on the Marked Price. It is usually expressed as a percentage of the Marked Price or as an absolute value (Rupees).
- Analogy: The “₹100 off” or “20% off” mentioned on a store sign.
Cost Price (CP): The price at which the retailer/seller bought the product. This is not* directly involved in discount calculations, but it’s important for profit/loss calculation, which often follows discounting.
- Analogy: The price the store paid to the manufacturer.
- Profit/Loss: The difference between Selling Price and Cost Price. If SP > CP, it’s a profit. If SP < CP, it's a loss.
2. Basic Discount Formulae
The core of discounting revolves around a few essential formulae:
- Discount Amount:
Discount (₹) = Marked Price (MP) - Sale Price (SP)- Mnemonic: “D = M – S” (Discount is MP minus SP)
- Discount Percentage:
Discount % = (Discount (₹) / Marked Price (MP)) × 100
Key Highlight: Discount is always* calculated on the Marked Price unless stated otherwise. Profit/Loss is calculated on Cost Price. This distinction is vital!
- Calculating Sale Price (given MP and Discount %):
SP = MP - (Discount % of MP)SP = MP × (1 - (Discount % / 100))SP = MP × ((100 - Discount %) / 100)
Mnemonic: “SP = MP (100 – D%) / 100″
- Calculating Marked Price (given SP and Discount %):
MP = SP / (1 - (Discount % / 100))MP = SP × (100 / (100 - Discount %))
3. Successive Discounts
Often, retailers offer multiple discounts on an item, one after another. This is known as successive discounts or compound discounts. It’s crucial to understand that these discounts are not simply added up. The second discount is applied to the price after the first discount has been given.
Example: If an item has MP ₹100 and a 10% discount, then another 20% discount.
- First discount: 10% of ₹100 = ₹10. Price after 1st discount = ₹100 – ₹10 = ₹90.
- Second discount: 20% of ₹90 = ₹18. Price after 2nd discount = ₹90 – ₹18 = ₹72.
- Total discount = ₹100 – ₹72 = ₹28.
Equivalent single discount = (28 / 100) 100 = 28%.
Formula for Equivalent Single Discount for two successive discounts (d1% and d2%):
Equivalent Discount % = [d1 + d2 - (d1 × d2 / 100)] %
Key Highlight: This formula is a shortcut. For three or more successive discounts, it’s generally easier to apply them one by one or use the multiplicative factor method.
Multiplicative Factor Method (Recommended for multiple discounts):
If discounts are d1%, d2%, d3%…
Final SP = MP × [(100 – d1) / 100] × [(100 – d2) / 100] × [(100 – d3) / 100] …
Example: MP = ₹100, Discounts = 10%, 20%, 5%.
SP = 100 × (90/100) × (80/100) × (95/100)
SP = 100 × 0.90 × 0.80 × 0.95
SP = 68.4
Equivalent Discount % = [(MP – SP) / MP] × 100 = [(100 – 68.4) / 100] × 100 = 31.6%
4. Discount and Profit/Loss
This is a common scenario in exams where a shopkeeper offers a discount but still makes a profit (or incurs a loss).
Relationship between CP, MP, SP, Profit/Loss, and Discount:
- Profit/Loss is calculated on CP.
SP = CP + ProfitorSP = CP - Loss - Discount is calculated on MP.
SP = MP - Discount
Combining these, if there is a profit:
CP × (100 + Profit %) / 100 = MP × (100 - Discount %) / 100
This can be rearranged to find the ratio:
MP / CP = (100 + Profit %) / (100 - Discount %)
Mnemonic: “MP/CP = (100+P%) / (100-D%)” – Remember P for Plus, D for Minus.
Example: A shopkeeper allows a 10% discount on the MP and still gains 20%. If an item costs him ₹450, what is its MP?
CP = ₹450
Profit % = 20%
Discount % = 10%
Using the ratio:
MP / 450 = (100 + 20) / (100 – 10)
MP / 450 = 120 / 90
MP / 450 = 4 / 3
MP = (4/3) × 450
MP = 4 × 150 = ₹600
5. Buy One Get One Free / “Buy X Get Y Free” Offers
These are specific types of discounts.
“Buy 1 Get 1 Free” means you get 2 items for the price of 1.
“Buy 2 Get 1 Free” means you get 3 items for the price of 2.
Calculation of Equivalent Discount % for “Buy X Get Y Free”:
Discount % = (Number of Free Items / Total Number of Items) × 100
Discount % = (Y / (X + Y)) × 100
Example 1: “Buy 1 Get 1 Free”
Y = 1 (free item)
X = 1 (items bought)
Total items = 1 + 1 = 2
Discount % = (1 / 2) × 100 = 50%
Example 2: “Buy 2 Get 1 Free”
Y = 1
X = 2
Total items = 2 + 1 = 3
Discount % = (1 / 3) × 100 = 33.33%
Example 3: “Buy 3 Get 2 Free”
Y = 2
X = 3
Total items = 3 + 2 = 5
Discount % = (2 / 5) × 100 = 40%
6. Dishonest Dealer / Faulty Weights (Revision Pointer, less common directly under ‘Discounting’ but related to commercial math)
Sometimes, questions combine discounting with a dishonest dealer’s tactics, where the dealer uses faulty weights while selling (and sometimes even while buying).
- Selling with faulty weight: If a shopkeeper claims to sell at Cost Price but uses 900g instead of 1kg, they are making a profit.
Profit = (Error / True Value – Error) 100
Profit = (100g / 900g) 100 = 11.11%
- Selling at a discount AND using faulty weight: This makes it complex.
- Approach: Assume a base CP (e.g., ₹1000 for 1kg).
- Calculate the actual cost incurred by the dealer for the quantity they are selling.
- Calculate the SP based on the Marked Price and discount.
- Compare the actual cost with the SP to find profit/loss.
7. Important Points & Common Traps
- Always calculate discount on MP: A common mistake is to calculate discount on SP or CP.
- Successive discounts are NOT additive: Use the equivalent discount formula or apply them one by one.
- Reading comprehension: Carefully distinguish between “discount offered” and “profit earned” or “loss incurred”.
- Units: Ensure consistency in units (e.g., Rupees, percentage).
- Working with fractions/decimals: Sometimes it’s easier to work with fractions for percentages (e.g., 20% = 1/5, 25% = 1/4) to simplify calculations.
- Profit margin vs. Markup:
- Markup: The percentage increased on CP to arrive at MP.
- Profit Margin: Profit as a percentage of SP. Usually, ‘profit %’ refers to profit as a percentage of CP.
8. Practice Problems & Scenario Tables (Mental Check)
| Scenario | Given | To Find | Method / Formula |
|---|---|---|---|
| Basic Discount | MP, Discount % | SP | SP = MP × (100 - D%) / 100 |
| Finding MP | SP, Discount % | MP | MP = SP × 100 / (100 - D%) |
| Finding Discount % | MP, SP | Discount % | D% = ((MP - SP) / MP) × 100 |
| Successive Discounts | Two discounts (d1, d2) | Equivalent D% | Eq D% = d1 + d2 - (d1 × d2 / 100) |
| Profit after Discount | CP, Profit %, Discount % | MP | MP / CP = (100 + P%) / (100 - D%) |
| Buy X Get Y Free | X, Y (bought, free) | Equivalent D% | D% = (Y / (X + Y)) × 100 |
| Finding CP (given MP, D%, Profit %) | MP, Discount %, Profit % | CP | CP = MP × (100 - D%) / (100 + P%) |
| MP as % above CP | Profit %, Discount % | MP above CP % | MP = CP × (100 + P%) / (100 - D%) Then calculate (MP-CP)/CP * 100 |
9. Quick Revision Drill
- What is the base for calculating discount? Marked Price.
- What is the base for calculating profit/loss (unless specified)? Cost Price.
- Are two successive discounts of 10% and 20% equal to a single 30% discount? No.
- If an item is “Buy 1 Get 1 Free”, what is the effective discount? 50%.
- How do you calculate Sale Price if you know Marked Price and Discount Percentage?
SP = MP × (100 - D%) / 100. - What formula connects MP, CP, Profit %, and Discount %?
MP/CP = (100 + P%) / (100 - D%).
Understanding these concepts and practicing various problem types will ensure you are well-prepared for any discounting question in your Forester exam or similar competitive examinations. Focus on the core relationships between MP, SP, D, CP, and P/L, and you’ll be able to solve complex problems efficiently.
