2 Discount-related arithmetic problems focused on discounts, marked prices, selling prices, and their impact on profit or loss

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 2  Discount-related arithmetic problems focused on discounts, marked prices, selling prices, and their impact on profit or loss. Each problem includes a clear problem statement, step-by-step solution, and a detailed explanation of the concepts involved. These problems are designed to cover various aspects of discounts, including single discounts, successive discounts, profit/loss calculations, and more complex scenarios. The solutions use straightforward arithmetic calculations and formulas to ensure clarity.
Problem 1: Single Discount Calculation
Problem: A shirt is marked at $100, and a 20% discount is offered. Find the selling price after the discount.
Solution:
  1. Identify Marked Price (MP) and Discount %:
    • MP = $100
    • Discount % = 20%
  2. Calculate Discount Amount:
    • Discount = (Discount % × MP) / 100
    • Discount = (20 × 100) / 100 = $20
  3. Calculate Selling Price (SP):
    • SP = MP - Discount
    • SP = 100 - 20 = $80
Answer:
  • Selling Price = $80
Explanation:
  • A single discount reduces the marked price by a percentage of the MP.
  • The formula for SP after a single discount is: SP = MP × (100 - Discount %) / 100, or SP = MP - Discount.
Problem 2: Successive Discounts
Problem: A laptop is marked at $1000. Two successive discounts of 10% and 20% are offered. Find the final selling price.
Solution:
  1. Identify Marked Price and Discounts:
    • MP = $1000
    • First discount = 10%
    • Second discount = 20%
  2. Calculate Price after First Discount:
    • First discount = (10 × 1000) / 100 = $100
    • Price after first discount = 1000 - 100 = $900
  3. Calculate Price after Second Discount:
    • Second discount = (20 × 900) / 100 = $180
    • SP = 900 - 180 = $720
Alternative Method:
  • For successive discounts, use: SP = MP × (100 - D1%) / 100 × (100 - D2%) / 100
  • SP = 1000 × (100 - 10) / 100 × (100 - 20) / 100
  • SP = 1000 × 0.9 × 0.8 = 1000 × 0.72 = $720
Answer:
  • Selling Price = $720
Explanation:
  • Successive discounts are applied one after another on the reduced price, not added together.
  • The alternative method uses a single formula to combine the effects of both discounts.
Problem 3: Discount and Profit
Problem: A retailer buys a watch for $200 and marks it at $300. He offers a 15% discount and still makes a profit. Find the profit percentage.
Solution:
  1. Identify Cost Price (CP), Marked Price (MP), and Discount %:
    • CP = $200
    • MP = $300
    • Discount % = 15%
  2. Calculate Selling Price (SP):
    • Discount = (15 × 300) / 100 = $45
    • SP = MP - Discount = 300 - 45 = $255
  3. Calculate Profit:
    • Profit = SP - CP = 255 - 200 = $55
  4. Calculate Profit Percentage:
    • Profit % = (Profit / CP) × 100
    • Profit % = (55 / 200) × 100 = 27.5%
Answer:
  • Profit % = 27.5%
Explanation:
  • The selling price after the discount is compared to the cost price to determine profit.
  • Profit percentage is calculated relative to the cost price.
Problem 4: Finding Marked Price Given SP and Discount
Problem: A book is sold for $120 after a 20% discount. Find the marked price.
Solution:
  1. Identify Selling Price (SP) and Discount %:
    • SP = $120
    • Discount % = 20%
  2. Relate SP to MP:
    • SP = MP × (100 - Discount %) / 100
    • 120 = MP × (100 - 20) / 100
    • 120 = MP × 0.8
  3. Solve for MP:
    • MP = 120 / 0.8
    • MP = $150
Answer:
  • Marked Price = $150
Explanation:
  • The selling price is the marked price minus the discount. Rearrange the formula SP = MP × (1 - Discount%/100) to find MP.
Problem 5: Discount to Achieve No Profit, No Loss
Problem: A trader buys an item for $80 and wants to sell it at no profit, no loss after offering a discount. If the marked price is $100, find the discount percentage.
Solution:
  1. Identify CP, MP, and Desired SP:
    • CP = $80
    • MP = $100
    • For no profit, no loss, SP = CP = $80
  2. Calculate Discount Amount:
    • Discount = MP - SP
    • Discount = 100 - 80 = $20
  3. Calculate Discount Percentage:
    • Discount % = (Discount / MP) × 100
    • Discount % = (20 / 100) × 100 = 20%
Answer:
  • Discount % = 20%
Explanation:
  • For no profit, no loss, SP equals CP. The discount is the difference between MP and SP, expressed as a percentage of MP.
Problem 6: Equivalent Single Discount for Successive Discounts
Problem: A product is offered with two successive discounts of 15% and 10%. Find the single discount equivalent to these two discounts.
Solution:
  1. Identify the Discounts:
    • First discount = 15%
    • Second discount = 10%
  2. Use the Successive Discount Formula:
    • Equivalent single discount % = D1 + D2 - (D1 × D2) / 100
    • D1 = 15%, D2 = 10%
    • Equivalent discount = 15 + 10 - (15 × 10) / 100
    • = 25 - 1.5 = 23.5%
Verification:
  • Assume MP = $100
  • After 15% discount: Price = 100 × (100 - 15) / 100 = $85
  • After 10% discount: Price = 85 × (100 - 10) / 100 = $76.5
  • Single discount: Discount = 100 - 76.5 = $23.5
  • Discount % = (23.5 / 100) × 100 = 23.5%
Answer:
  • Equivalent Single Discount = 23.5%
Explanation:
  • The equivalent single discount combines the effect of two successive discounts into one, calculated using the formula above.
Problem 7: Discount and Loss
Problem: A bicycle is bought for $500 and marked at $600. After offering a 10% discount, the seller incurs a loss. Find the loss percentage.
Solution:
  1. Identify CP, MP, and Discount %:
    • CP = $500
    • MP = $600
    • Discount % = 10%
  2. Calculate Selling Price (SP):
    • Discount = (10 × 600) / 100 = $60
    • SP = MP - Discount = 600 - 60 = $540
  3. Determine Profit or Loss:
    • Since SP ($540) > CP ($500), there is a profit, not a loss.
    • Profit = SP - CP = 540 - 500 = $40
  4. Calculate Profit Percentage:
    • Profit % = (Profit / CP) × 100
    • Profit % = (40 / 500) × 100 = 8%
Correction:
  • The problem states a loss, but calculations show a profit. If a loss is intended, the SP must be less than CP. Let’s assume the problem meant a higher discount to cause a loss. Let’s try a 20% discount:
    • Discount = (20 × 600) / 100 = $120
    • SP = 600 - 120 = $480
    • Loss = CP - SP = 500 - 480 = $20
    • Loss % = (Loss / CP) × 100 = (20 / 500) × 100 = 4%
Answer (assuming 20% discount for loss):
  • Loss % = 4%
Explanation:
  • The original discount (10%) resulted in a profit. Adjusting to a 20% discount creates a loss, aligning with the problem’s intent. Loss % is calculated relative to CP.
Problem 8: Finding Discount % Given CP, MP, and Profit
Problem: A phone is bought for $400 and marked at $600. The seller wants a 10% profit after offering a discount. Find the discount percentage.
Solution:
  1. Identify CP, MP, and Desired Profit %:
    • CP = $400
    • MP = $600
    • Profit % = 10%
  2. Calculate Desired Selling Price (SP):
    • Profit = (Profit % × CP) / 100 = (10 × 400) / 100 = $40
    • SP = CP + Profit = 400 + 40 = $440
  3. Calculate Discount:
    • Discount = MP - SP = 600 - 440 = $160
  4. Calculate Discount Percentage:
    • Discount % = (Discount / MP) × 100
    • Discount % = (160 / 600) × 100 = 26.67%
Answer:
  • Discount % = 26.67%
Explanation:
  • The SP is determined based on the desired profit. The discount is the difference between MP and SP, expressed as a percentage of MP.
Problem 9: Discount to Clear Stock
Problem: A shopkeeper has 50 items, each bought for $20 and marked at $30. To clear stock, he offers a discount such that the total selling price of all items covers the total cost price. Find the discount percentage.
Solution:
  1. Calculate Total CP and Desired Total SP:
    • CP per item = $20
    • Total CP = 50 × 20 = $1000
    • For no profit, no loss, Total SP = Total CP = $1000
  2. Calculate Total MP:
    • MP per item = $30
    • Total MP = 50 × 30 = $1500
  3. Calculate Total Discount:
    • Total Discount = Total MP - Total SP = 1500 - 1000 = $500
  4. Calculate Discount Percentage:
    • Discount % = (Total Discount / Total MP) × 100
    • Discount % = (500 / 1500) × 100 = 33.33%
Answer:
  • Discount % = 33.33%
Explanation:
  • To break even (no profit, no loss), the total SP must equal the total CP. The discount % is calculated based on the total MP.
Problem 10: Discount and False Advertising
Problem: A retailer marks an item at $200, claiming a 25% discount, but the actual selling price is $160. Find the actual discount percentage.
Solution:
  1. Identify MP and Actual SP:
    • MP = $200
    • Actual SP = $160
  2. Calculate Actual Discount:
    • Discount = MP - SP = 200 - 160 = $40
  3. Calculate Actual Discount Percentage:
    • Discount % = (Discount / MP) × 100
    • Discount % = (40 / 200) × 100 = 20%
  4. Compare with Claimed Discount:
    • Claimed discount = 25%
    • Actual discount = 20%
Answer:
  • Actual Discount % = 20%
Explanation:
  • The actual discount is calculated based on the difference between MP and SP. The retailer’s claim of a 25% discount is false, as the actual discount is lower.
Key Concepts Summary:
  • Single Discount: SP = MP × (100 - Discount %) / 100
  • Successive Discounts: SP = MP × (100 - D1%) / 100 × (100 - D2%) / 100
  • Equivalent Single Discount: D1 + D2 - (D1 × D2) / 100
  • Profit/Loss with Discount: Compare SP (after discount) with CP to determine profit or loss.
  • Discount Percentage: (Discount / MP) × 100
  • No Profit, No Loss: SP = CP, discount adjusts MP to achieve this.
These problems cover a range of discount-related scenarios, with step-by-step solutions ensuring clarity in calculations and concepts. If you need further clarification or additional problems, please let me know!

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