26. Cost price, selling price, marked price problems

26. Cost price, selling price, marked price problems - step by step solution procedure and detailed explanation for each problem.

Below are 10 problems related to cost price (CP), selling price (SP), and marked price (MP), each with a step-by-step solution and detailed explanation. These problems cover various scenarios involving profit, loss, discounts, and their calculations. The explanations are designed to be clear and comprehensive, suitable for understanding the concepts thoroughly.
Problem 1: Basic Profit Calculation
Question: A shopkeeper buys a book for $50 and sells it for $60. Find the profit and profit percentage.
Solution:
  • Step 1: Identify the given values.
    • Cost Price (CP) = $50
    • Selling Price (SP) = $60
  • Step 2: Calculate the profit.
    • Profit = SP - CP = 60 - 50 = $10
  • Step 3: Calculate the profit percentage.
    • Profit % = (Profit / CP) × 100 = (10 / 50) × 100 = 20%
Answer:
  • Profit = $10
  • Profit % = 20%
Explanation:
  • The profit is the difference between what the shopkeeper sells the book for (SP) and what they paid for it (CP).
  • Profit percentage is calculated relative to the cost price, expressed as a percentage, which shows how much profit is made per $100 of the cost price.
Problem 2: Basic Loss Calculation
Question: A trader buys a chair for $200 and sells it for $180. Find the loss and loss percentage.
Solution:
  • Step 1: Identify the given values.
    • CP = $200
    • SP = $180
  • Step 2: Calculate the loss.
    • Loss = CP - SP = 200 - 180 = $20
  • Step 3: Calculate the loss percentage.
    • Loss % = (Loss / CP) × 100 = (20 / 200) × 100 = 10%
Answer:
  • Loss = $20
  • Loss % = 10%
Explanation:
  • Loss occurs when the selling price is less than the cost price.
  • The loss percentage is calculated similarly to profit percentage but represents the loss relative to the cost price.
Problem 3: Finding Selling Price Given Profit %
Question: A pen is bought for $10. If the shopkeeper wants a 25% profit, find the selling price.
Solution:
  • Step 1: Identify the given values.
    • CP = $10
    • Profit % = 25%
  • Step 2: Calculate the profit.
    • Profit = (Profit % × CP) / 100 = (25 × 10) / 100 = $2.5
  • Step 3: Calculate the selling price.
    • SP = CP + Profit = 10 + 2.5 = $12.5
Answer:
  • Selling Price = $12.5
Explanation:
  • To achieve a 25% profit, the shopkeeper adds 25% of the cost price to the cost price.
  • Alternatively, SP can be calculated directly as SP = CP × (1 + Profit % / 100) = 10 × (1 + 25/100) = 10 × 1.25 = $12.5.
Problem 4: Finding Cost Price Given Loss %
Question: A watch is sold for $90 at a loss of 10%. Find the cost price.
Solution:
  • Step 1: Identify the given values.
    • SP = $90
    • Loss % = 10%
  • Step 2: Use the formula for selling price with loss.
    • SP = CP × (1 - Loss % / 100)
    • 90 = CP × (1 - 10/100) = CP × 0.9
  • Step 3: Solve for CP.
    • CP = 90 / 0.9 = $100
Answer:
  • Cost Price = $100
Explanation:
  • Since the sale results in a 10% loss, the selling price is 90% of the cost price.
  • Rearranging the formula allows us to find the original cost price by dividing the selling price by (1 - Loss % / 100).
Problem 5: Marked Price and Discount
Question: A shirt is marked at $100, and a 20% discount is offered. Find the selling price.
Solution:
  • Step 1: Identify the given values.
    • Marked Price (MP) = $100
    • Discount % = 20%
  • Step 2: Calculate the discount amount.
    • Discount = (Discount % × MP) / 100 = (20 × 100) / 100 = $20
  • Step 3: Calculate the selling price.
    • SP = MP - Discount = 100 - 20 = $80
Answer:
  • Selling Price = $80
Explanation:
  • The marked price is the price listed before any discount.
  • A 20% discount means the customer pays 80% of the marked price, so SP = MP × (1 - Discount % / 100) = 100 × (1 - 20/100) = $80.
Problem 6: Finding Marked Price Given Discount and SP
Question: A gadget is sold for $120 after a 25% discount. Find the marked price.
Solution:
  • Step 1: Identify the given values.
    • SP = $120
    • Discount % = 25%
  • Step 2: Use the formula for selling price after discount.
    • SP = MP × (1 - Discount % / 100)
    • 120 = MP × (1 - 25/100) = MP × 0.75
  • Step 3: Solve for MP.
    • MP = 120 / 0.75 = $160
Answer:
  • Marked Price = $160
Explanation:
  • The selling price is what the customer pays after the discount. Since a 25% discount means the SP is 75% of the MP, we divide the SP by 0.75 to find the original marked price.
Problem 7: Profit % with Marked Price and Discount
Question: A table is bought for $400 and marked at $500. A 10% discount is given. Find the profit percentage.
Solution:
  • Step 1: Identify the given values.
    • CP = $400
    • MP = $500
    • Discount % = 10%
  • Step 2: Calculate the selling price.
    • Discount = (10 × 500) / 100 = $50
    • SP = MP - Discount = 500 - 50 = $450
  • Step 3: Calculate the profit.
    • Profit = SP - CP = 450 - 400 = $50
  • Step 4: Calculate the profit percentage.
    • Profit % = (Profit / CP) × 100 = (50 / 400) × 100 = 12.5%
Answer:
  • Profit % = 12.5%
Explanation:
  • The marked price is reduced by the discount to get the selling price.
  • Profit is then calculated based on the difference between SP and CP, and the percentage is relative to the CP.
Problem 8: Finding Discount % Given CP, SP, and MP
Question: A phone is bought for $600 and marked at $800. It is sold for $720. Find the discount percentage.
Solution:
  • Step 1: Identify the given values.
    • CP = $600
    • MP = $800
    • SP = $720
  • Step 2: Calculate the discount amount.
    • Discount = MP - SP = 800 - 720 = $80
  • Step 3: Calculate the discount percentage.
    • Discount % = (Discount / MP) × 100 = (80 / 800) × 100 = 10%
Answer:
  • Discount % = 10%
Explanation:
  • The discount is the reduction from the marked price to the selling price.
  • The discount percentage is calculated relative to the marked price, showing the percentage reduction offered.
Problem 9: No Profit, No Loss
Question: A bicycle is bought for $300 and sold at a price that results in no profit or loss. Find the selling price.
Solution:
  • Step 1: Identify the given values.
    • CP = $300
    • No profit, no loss means SP = CP
  • Step 2: Determine the selling price.
    • SP = CP = $300
Answer:
  • Selling Price = $300
Explanation:
  • When there is no profit or loss, the selling price equals the cost price, as the trader neither gains nor loses money on the transaction.
Problem 10: Complex Problem with Multiple Conditions
Question: A laptop is bought for $1000 and marked up by 50%. A discount of 20% is offered, but the shopkeeper still makes a 20% profit. Verify if this is possible and find the selling price.
Solution:
  • Step 1: Identify the given values.
    • CP = $1000
    • Markup % = 50%
    • Discount % = 20%
    • Desired Profit % = 20%
  • Step 2: Calculate the marked price.
    • Markup = (50 × 1000) / 100 = $500
    • MP = CP + Markup = 1000 + 500 = $1500
  • Step 3: Calculate the selling price after discount.
    • Discount = (20 × 1500) / 100 = $300
    • SP = MP - Discount = 1500 - 300 = $1200
  • Step 4: Calculate the actual profit and profit percentage.
    • Profit = SP - CP = 1200 - 1000 = $200
    • Profit % = (Profit / CP) × 100 = (200 / 1000) × 100 = 20%
  • Step 5: Verify the condition.
    • The profit percentage matches the desired 20%, so the scenario is possible.
Answer:
  • Selling Price = $1200
  • The scenario is possible as the profit percentage is 20%.
Explanation:
  • The markup increases the price above the cost price, and the discount reduces the marked price to determine the selling price.
  • The profit is calculated based on the final SP and CP, and the percentage confirms whether the desired profit is achieved. Here, the calculations align perfectly with the given conditions.
Key Formulas Used
  • Profit = SP - CP (if SP > CP)
  • Loss = CP - SP (if CP > SP)
  • Profit % = (Profit / CP) × 100
  • Loss % = (Loss / CP) × 100
  • SP with Profit = CP × (1 + Profit % / 100)
  • SP with Loss = CP × (1 - Loss % / 100)
  • Discount = MP - SP
  • Discount % = (Discount / MP) × 100
  • SP after Discount = MP × (1 - Discount % / 100)
General Tips
  • Always identify whether the problem involves profit, loss, or discounts, and use the appropriate formula.
  • Marked price is relevant when discounts are involved, while cost price and selling price are key for profit/loss calculations.
  • Double-check calculations, especially when percentages are involved, to avoid errors.
These problems and solutions cover a range of scenarios to build a strong understanding of cost price, selling price, and marked price concepts. 

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