1 Profit and Loss problems as a part of Arithmetic problems useful for competitive exams

1. Profit and Loss problems as a part of Arithmetic problems useful for competitive exams

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Below are 10 Profit and Loss problems with detailed steps and clear explanations for each. These problems cover a variety of scenarios, including basic calculations, discounts, marked prices, and more complex cases. Each problem includes the problem statement, step-by-step solution, and an explanation of the concepts involved.

Problem 1: Basic Profit Calculation

Problem: A shopkeeper buys a book for Rs.50 and sells it for Rs.60. Find the profit and profit percentage.

Solution:

• Identify the Cost Price (CP) and Selling Price (SP):
• CP = Rs.50
• SP = Rs.60
• Calculate Profit:
• Profit = SP - CP
• Profit = 60 - 50 = Rs.10
• Calculate Profit Percentage:
• Profit % = (Profit / CP) × 100
• Profit % = (10 / 50) × 100 = 20%

Answer = Profit = Rs.10      -  Profit % = 20%

Explanation:

• Profit is the difference between the selling price and cost price when SP > CP.
• Profit percentage is calculated relative to the cost price, expressed as a percentage.

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Problem 2: Basic Loss Calculation

 

Problem: A trader buys a phone for Rs.300 and sells it for Rs.270. Find the loss and loss percentage.

 

Solution:

• Identify CP and SP:
• CP = Rs.300
• SP = Rs.270
• Calculate Loss:
• Loss = CP - SP
• Loss = 300 - 270 = Rs.30
• Calculate Loss Percentage:
• Loss % = (Loss / CP) × 100
• Loss % = (30 / 300) × 100 = 10%

Answer:

• Loss = Rs.30
• Loss % = 10%

Explanation:

• Loss occurs when SP < CP.
• Loss percentage is calculated as a percentage of the cost price.

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Problem 3: Selling Price to Achieve Desired Profit

Problem: A retailer buys a watch for Rs.200 and wants to earn a 25% profit. What should be the selling price?

Solution:

• Identify CP and Profit %:
• CP = Rs.200
• Profit % = 25%
• Calculate Profit:
• Profit = (Profit % × CP) / 100
• Profit = (25 × 200) / 100 = Rs.50
• Calculate Selling Price:
• SP = CP + Profit
• SP = 200 + 50 = Rs.250

Alternative Method:

• SP = CP × (100 + Profit %) / 100
• SP = 200 × (100 + 25) / 100 = 200 × 1.25 = Rs.250

Answer:

• Selling Price = Rs.250

Explanation:

• To achieve a desired profit percentage, calculate the profit amount and add it to CP, or use the formula SP = CP × (1 + Profit%/100).

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Problem 4: Cost Price Given SP and Profit %

Problem: A laptop is sold for Rs.1200, earning a 20% profit. Find the cost price.

Solution:

• Identify SP and Profit %:
• SP = Rs.1200
• Profit % = 20%
• Relate SP and CP:
• SP = CP × (100 + Profit %) / 100
• 1200 = CP × (100 + 20) / 100
• 1200 = CP × 1.2
• Solve for CP:
• CP = 1200 / 1.2
• CP = Rs.1000

Answer = Cost Price = Rs.1000

Explanation:

• When SP and profit % are given, use the formula CP = SP / (1 + Profit%/100) to find the cost price.

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Problem 5: Selling at a Loss

Problem: A bicycle is bought for Rs.500 and sold at a 15% loss. Find the selling price.

Solution:

• Identify CP and Loss %:
• CP = Rs.500
• Loss % = 15%
• Calculate Loss:
• Loss = (Loss % × CP) / 100
• Loss = (15 × 500) / 100 = Rs.75
• Calculate Selling Price:
• SP = CP - Loss
• SP = 500 - 75 = Rs.425

Alternative Method:

• SP = CP × (100 - Loss %) / 100
• SP = 500 × (100 - 15) / 100 = 500 × 0.85 = Rs.425

Answer = Selling Price = Rs.425

Explanation:

• For a loss, subtract the loss amount from CP or use SP = CP × (1 - Loss%/100).

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Problem 6: Profit on Multiple Items

Problem: A shopkeeper buys 12 pens for Rs.60 and sells them at Rs.6 each. Find the total profit and profit percentage.

Solution:

• Identify CP and SP for all pens:
• Total CP = Rs.60
• SP per pen = Rs.6
• Total SP = 12 × 6 = Rs.72
• Calculate Total Profit:
• Profit = Total SP - Total CP
• Profit = 72 - 60 = Rs.12
• Calculate Profit Percentage:
• Profit % = (Profit / Total CP) × 100
• Profit % = (12 / 60) × 100 = 20%

Answer:

• Total Profit = Rs.12
• Profit % = 20%

Explanation:

• For multiple items, calculate total CP and total SP, then find profit and profit % based on these totals.

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Problem 7: Marked Price and Discount

Problem: A shirt is marked at Rs.80, and a 10% discount is offered. If the shopkeeper still makes a 20% profit, find the cost price.

Solution:

• Calculate Selling Price after Discount:
• Marked Price (MP) = Rs.80
• Discount % = 10%
• Discount = (10 × 80) / 100 = Rs.8
• SP = MP - Discount = 80 - 8 = Rs.72
• Relate SP to CP with Profit %:
• Profit % = 20%
• SP = CP × (100 + Profit %) / 100
• 72 = CP × (100 + 20) / 100
• 72 = CP × 1.2
• Solve for CP:
• CP = 72 / 1.2 = Rs.60

Answer:

• Cost Price = Rs.60

Explanation:

• The selling price is calculated after applying the discount to the marked price.
• Use the profit % to relate SP to CP and solve for CP.

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Problem 8: Two Articles with Profit and Loss

Problem: A trader sells two items. Item A is bought for Rs.400 and sold at a 10% profit. Item B is bought for Rs.600 and sold at a 10% loss. Find the overall profit or loss.

Solution:

• Calculate for Item A:
• CP of A = Rs.400
• Profit % = 10%
• SP of A = CP × (100 + Profit %) / 100
• SP of A = 400 × (100 + 10) / 100 = 400 × 1.1 = Rs.440
• Calculate for Item B:
• CP of B = Rs.600
• Loss % = 10%
• SP of B = CP × (100 - Loss %) / 100
• SP of B = 600 × (100 - 10) / 100 = 600 × 0.9 = Rs.540
• Calculate Overall CP and SP:
• Total CP = CP of A + CP of B = 400 + 600 = Rs.1000
• Total SP = SP of A + SP of B = 440 + 540 = Rs.980
• Calculate Overall Profit or Loss:
• Loss = Total CP - Total SP = 1000 - 980 = Rs.20
• Calculate Loss Percentage:
• Loss % = (Loss / Total CP) × 100
• Loss % = (20 / 1000) × 100 = 2%

Answer:

• Overall Loss = Rs.20
• Loss % = 2%

Explanation:

• Calculate SP for each item separately, sum CP and SP, and determine if there’s an overall profit or loss.

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Problem 9: False Weights

Problem: A dishonest shopkeeper buys goods at Rs.10 per kg and uses a false weight of 900 grams instead of 1 kg while selling at Rs.12 per kg. Find his profit percentage.

 

Solution:

• Understand the False Weight:
• CP = Rs.10 per kg (1000 grams)
• SP = Rs.12 per kg, but he sells 900 grams as 1 kg
• Actual quantity sold = 900 grams
• SP for 900 grams = Rs.12
• Calculate CP for 900 grams:
• CP per gram = 10 / 1000 = Rs.0.01
• CP for 900 grams = 0.01 × 900 = Rs.9
• Calculate Profit:
• Profit = SP - CP = 12 - 9 = Rs.3
• Calculate Profit Percentage:
• Profit % = (Profit / CP) × 100
• Profit % = (3 / 9) × 100 = 33.33%

Answer =  Profit % = 33.33%

Explanation:

• The shopkeeper cheats by selling less quantity (900g instead of 1000g) while charging for 1 kg, increasing his profit.

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Problem 10: Successive Discounts

Problem: A product is marked at Rs.200. Two successive discounts of 10% and 20% are offered. If the cost price is Rs.120, find the profit percentage.

Solution:

• Calculate Selling Price after Successive Discounts:
• Marked Price = Rs.200
• First discount = 10%
• Price after first discount = 200 × (100 - 10) / 100 = 200 × 0.9 = Rs.180
• Second discount = 20% on Rs.180
• Price after second discount = 180 × (100 - 20) / 100 = 180 × 0.8 = Rs.144
• SP = Rs.144
• Calculate Profit:
• CP = Rs.120
• Profit = SP - CP = 144 - 120 = Rs.24
• Calculate Profit Percentage:
• Profit % = (Profit / CP) × 100
• Profit % = (24 / 120) × 100 = 20%

Answer = Profit % = 20%

Explanation:

• Successive discounts are applied one after the other, not added together.
• SP is calculated after both discounts, and profit % is based on CP.

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Key Concepts Summary:

• Profit: SP > CP; Profit = SP - CP; Profit % = (Profit / CP) × 100
• Loss: SP < CP; Loss = CP - SP; Loss % = (Loss / CP) × 100
• Marked Price and Discount: SP = MP × (100 - Discount %) / 100
• Successive Discounts: Apply each discount sequentially on the reduced price.
• False Weights: Adjust CP and SP based on actual quantities bought and sold.

These problems cover a range of scenarios, and the step-by-step solutions ensure clarity in understanding the calculations and underlying concepts. Let me know if you need further clarification or additional problems!