Profit Loss & Discount

Core Formulas

Profit and Loss are fundamental concepts in commercial mathematics. Understanding them is crucial for SSC CGL.

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Profit: Occurs when SP > CP.
  • Profit = SP - CP
• Loss: Occurs when CP > SP.
  • Loss = CP - SP

Profit and Loss are always calculated on the Cost Price (CP) unless stated otherwise.

• Profit Percentage (P%): P% = (Profit / CP) * 100
• Loss Percentage (L%): L% = (Loss / CP) * 100

From these, we can derive:

• SP = CP * (100 + P%) / 100
• SP = CP * (100 - L%) / 100
• CP = SP * 100 / (100 + P%)
• CP = SP * 100 / (100 - L%)

Worked Example 1

Question: A shopkeeper buys a watch for ₹800 and sells it for ₹1000. Find the profit and profit percentage. Solution:

1. CP = ₹800, SP = ₹1000
2. Since SP > CP, there is a Profit.
3. Profit = SP - CP = ₹1000 - ₹800 = ₹200
4. Profit Percentage = (Profit / CP) * 100 = (200 / 800) * 100 = (1/4) * 100 = 25%

Worked Example 2

Question: An article is purchased for ₹1200 and sold for ₹1080. Calculate the loss and loss percentage. Solution:

1. CP = ₹1200, SP = ₹1080
2. Since CP > SP, there is a Loss.
3. Loss = CP - SP = ₹1200 - ₹1080 = ₹120
4. Loss Percentage = (Loss / CP) * 100 = (120 / 1200) * 100 = (1/10) * 100 = 10%

Shortcuts & Tricks

1. Fractional Method for Percentages: Instead of (100+P%)/100, use fractions. For example, 20% profit means SP is 120% of CP, or SP = (6/5) * CP. 10% loss means SP is 90% of CP, or SP = (9/10) * CP. This avoids large multiplications and divisions by 100.
   • Example: If CP = ₹500, 20% profit: SP = 500 * (6/5) = ₹600.
2. Direct CP/SP Calculation: If you know SP and profit/loss percentage, you can directly find CP. If an item is sold for ₹660 at a 10% profit, it means ₹660 is 110% of CP. So, CP = 660 / 1.10 = ₹600. This is faster than CP = SP * 100 / (100 + P%).

Common Mistakes

1. Calculating % on SP: Always remember that Profit% and Loss% are calculated on CP, not SP, unless explicitly stated otherwise. This is a very common trap.
2. Confusing Profit/Loss: Incorrectly applying Profit formula when there's a Loss, or vice-versa. Always check if SP > CP (Profit) or CP > SP (Loss).
3. Arithmetic Errors: Simple calculation mistakes, especially with fractions or decimals, can lead to incorrect answers. Practice mental math and quick calculations.

Understanding Discount & Marked Price

In commercial transactions, the price at which an item is listed for sale is called the Marked Price (MP) or List Price. A Discount is a reduction offered on this Marked Price. The price at which the item is actually sold after the discount is the Selling Price (SP).

Core Formulas

1. Discount Amount: Discount = Marked Price (MP) - Selling Price (SP)
2. Discount Percentage: Discount % = (Discount Amount / Marked Price (MP)) * 100
3. Selling Price (SP) after a single discount: SP = MP * (100 - Discount %) / 100
4. Successive Discounts: When multiple discounts (d1%, d2%, etc.) are offered, they are applied one after another on the reduced price. The Net/Effective Discount (D_eff) for two successive discounts d1% and d2% is: D_eff = d1 + d2 - (d1 * d2) / 100 The Selling Price (SP) after successive discounts d1%, d2%, ... dn% is: SP = MP * ((100 - d1) / 100) * ((100 - d2) / 100) * ... * ((100 - dn) / 100)

Worked Example 1

Q: A shopkeeper marks an article at ₹800 and offers a discount of 15%. What is the selling price? A: Step 1: Identify MP and Discount %. MP = ₹800, Discount % = 15% Step 2: Use the formula SP = MP * (100 - Discount %) / 100. SP = 800 * (100 - 15) / 100 SP = 800 * 85 / 100 SP = 8 * 85 SP = ₹680

Worked Example 2

Q: An item is marked at ₹1200. The shopkeeper offers successive discounts of 10% and 20%. Find the final selling price and the effective discount percentage. A: Step 1: Identify MP and successive discounts. MP = ₹1200, d1 = 10%, d2 = 20% Step 2: Calculate the final SP using the successive discount formula. SP = MP * ((100 - d1) / 100) * ((100 - d2) / 100) SP = 1200 * ((100 - 10) / 100) * ((100 - 20) / 100) SP = 1200 * (90 / 100) * (80 / 100) SP = 1200 * 0.9 * 0.8 SP = 1200 * 0.72 SP = ₹864 Step 3: Calculate the effective discount percentage. D_eff = d1 + d2 - (d1 * d2) / 100 D_eff = 10 + 20 - (10 * 20) / 100 D_eff = 30 - 200 / 100 D_eff = 30 - 2 D_eff = 28%

Shortcuts & Tricks

1. Multiplier Method for SP: For successive discounts d1%, d2%, ..., simply multiply the MP by the remaining percentage factors: SP = MP * (1 - d1/100) * (1 - d2/100) * .... This is faster than calculating intermediate prices.
2. Assume MP = 100: If only percentages are given (e.g., profit %, discount %), assume MP = 100 (or CP = 100 if starting from cost price) to simplify calculations and find the net percentage change directly.
3. Fraction Equivalents: Convert common percentages to fractions (e.g., 10% = 1/10, 20% = 1/5) for quick calculations. A 10% discount means MP * (9/10), a 20% discount means MP * (4/5).

Common Mistakes

1. Adding Successive Discounts Directly: Students often add 10% and 20% to get a 30% total discount. This is incorrect. Discounts are applied sequentially, leading to a smaller effective discount.
2. Calculating Discount on SP: Always remember that a discount is calculated on the Marked Price (MP), not the Selling Price (SP) or Cost Price (CP).
3. Confusing MP and CP: Discount is on MP, while Profit/Loss is calculated on CP. Keep these distinct in multi-step problems.

Core Formula

When a dishonest shopkeeper claims to sell goods at cost price but uses a false weight, the gain percentage is calculated as:

Gain % = ( (True Weight - False Weight) / False Weight ) * 100

Alternatively, and often more intuitively:

Gain % = ( Error / (Weight Actually Delivered) ) * 100

Where:

• True Weight is the weight the customer should receive (e.g., 1000g for 1kg).
• False Weight is the weight the customer actually receives (e.g., 900g).
• Error = True Weight - False Weight.

Worked Example 1

A shopkeeper sells goods at cost price but uses a weight of 900 grams for a kilogram. What is his gain percentage?

Solution:

1. Identify True Weight and False Weight:
   • True Weight (what he should give) = 1000 grams
   • False Weight (what he actually gives) = 900 grams
2. Calculate the Error:
   • Error = True Weight - False Weight = 1000g - 900g = 100 grams
3. Apply the formula:
   • Gain % = ( Error / (Weight Actually Delivered) ) * 100
   • Gain % = ( 100 / 900 ) * 100
   • Gain % = (1/9) * 100 = 11 1/9 %

Worked Example 2

An unscrupulous merchant marks up his goods by 20% and also uses a faulty balance that reads 800g for 1kg. What is his total profit percentage?

Solution: This problem combines markup with false weight.

1. Profit from False Weight:
   • True Weight = 1000g, False Weight = 800g
   • Error = 1000g - 800g = 200g
   • Gain % from false weight = (200 / 800) * 100 = (1/4) * 100 = 25%
2. Profit from Markup:
   • Markup = 20%
3. Combine Profits (Successive Percentage Change):
   • If P1 is profit from markup and P2 is profit from false weight, total profit = P1 + P2 + (P1 * P2 / 100)
   • Total Profit % = 20 + 25 + (20 * 25 / 100)
   • Total Profit % = 45 + (500 / 100)
   • Total Profit % = 45 + 5 = 50%

Shortcuts & Tricks

• Ratio Method: If a shopkeeper gives 'x' grams instead of 'y' grams, his profit is (y-x) grams on 'x' grams. So, profit ratio is (y-x)/x. Multiply by 100 for percentage. For example, 900g instead of 1000g: profit is 100g on 900g, so 100/900 = 1/9 = 11.11%.
• Direct Calculation for Common Errors:
  • 10% less weight (900g for 1kg) -> 100/900 = 11 1/9% gain.
  • 20% less weight (800g for 1kg) -> 200/800 = 25% gain.
  • 25% less weight (750g for 1kg) -> 250/750 = 33 1/3% gain.
• Cheating at both buying and selling: If a shopkeeper cheats by x% at buying and y% at selling (e.g., gets 10% more weight while buying and gives 10% less weight while selling), the total gain is NOT simply x+y. It's a successive percentage increase. If he cheats 10% by weight at both ends, it means he gets 10% extra weight for the price of 100 units (10% gain) and gives 10% less weight for the price of 100 units (11 1/9% gain). Use the formula A + B + (AB/100) for the combined effect of these two gains.

Common Mistakes

1. Incorrect Denominator: Students often use the True Weight (e.g., 1000g) in the denominator instead of the False Weight (e.g., 900g) or Weight Actually Delivered. Remember, profit is calculated on the cost incurred, which for the shopkeeper is the value of the weight actually given.
2. Confusing Error with False Weight: Mixing up Error (the difference) with False Weight (the actual amount delivered) in the formula.
3. Ignoring Stated CP/SP: If the problem states the shopkeeper sells at a profit/loss in addition to using false weights, these must be combined using successive percentage change formulas, not simply added.