Understanding the Retailer Profit Problem

This question asks us to compare the profit a retailer makes when selling an air-conditioner under two different discount scenarios: one with a standard discount and another with an additional membership discount. We are given the marked price, information about the cost price relative to the marked price, and the discount percentages.

Calculating Key Values: Marked Price, Cost Price, and Selling Prices

First, let's determine the specific values involved in the transaction.

• Marked Price (MP): The price at which the air-conditioner is listed. Given as Rs. 92,000.
• Cost Price (CP): The price at which the retailer bought the air-conditioner. It is stated to be 70% below the marked price.
  • Amount below MP = 70% of MP
  • Amount below MP \( = \frac{70}{100} \times 92,000 \)
  • Amount below MP \( = 0.70 \times 92,000 = 64,400 \)
  • Cost Price (CP) = MP - Amount below MP
  • CP \( = 92,000 - 64,400 = 27,600 \)
  So, the cost price is Rs. 27,600.

Scenario 1: Selling Without Membership Card (Standard Discount)

In this scenario, the retailer offers a 30% discount on the marked price.

• Discount 1: 30% on MP.
• Discount Amount 1 \( = \frac{30}{100} \times 92,000 \)
• Discount Amount 1 \( = 0.30 \times 92,000 = 27,600 \)
• Selling Price 1 (SP1): Price after the first discount.
• SP1 = MP - Discount Amount 1
• SP1 \( = 92,000 - 27,600 = 64,400 \)

The selling price without the membership card is Rs. 64,400.

• Profit 1 (P1): Profit without the membership card.
• P1 = SP1 - CP
• P1 \( = 64,400 - 27,600 = 36,800 \)

The profit without the membership card is Rs. 36,800.

Scenario 2: Selling With Membership Card (Additional Discount)

If the buyer presents a membership card, a further 20% discount is offered on the already discounted price (SP1).

• Discount 2: 20% on SP1.
• Discount Amount 2 \( = \frac{20}{100} \times 64,400 \)
• Discount Amount 2 \( = 0.20 \times 64,400 = 12,880 \)
• Selling Price 2 (SP2): Price after the additional membership discount.
• SP2 = SP1 - Discount Amount 2
• SP2 \( = 64,400 - 12,880 = 51,520 \)

The selling price with the membership card is Rs. 51,520.

• Profit 2 (P2): Profit with the membership card.
• P2 = SP2 - CP
• P2 \( = 51,520 - 27,600 = 23,920 \)

The profit with the membership card is Rs. 23,920.

Comparing Profits: Percentage Calculation

The question asks for the profit of the retailer with the membership card scheme (P2) as a percentage of the profit of the retailer without the membership card scheme (P1).

• Percentage \( = \left( \frac{\text{Profit with membership}}{\text{Profit without membership}} \right) \times 100 \)
• Percentage \( = \left( \frac{P2}{P1} \right) \times 100 \)
• Percentage \( = \left( \frac{23,920}{36,800} \right) \times 100 \)
• Percentage \( = \frac{2392}{3680} \times 100 \)

Let's simplify the fraction:

• Divide both numerator and denominator by common factors. Both are divisible by 10: \(\frac{2392}{368}\).
• Both are divisible by 8: \( 2392 \div 8 = 299 \) and \( 368 \div 8 = 46 \). So we have \(\frac{299}{46}\).
• Wait, the denominator was 3680. Let's restart the simplification from \(\frac{2392}{3680}\).
• Divide both by 10: \(\frac{239.2}{368}\). This is not correct. Start again from \(\frac{23920}{36800}\).
• Cancel trailing zeros: \(\frac{2392}{3680}\).
• Both are divisible by 8: \( 2392 \div 8 = 299 \) and \( 3680 \div 8 = 460 \). So we have \(\frac{299}{460}\).
• Let's check factors of 460. \( 460 = 10 \times 46 = 2 \times 5 \times 2 \times 23 = 4 \times 5 \times 23 \).
• Let's check if 299 is divisible by 23. \( 299 \div 23 \). \( 23 \times 10 = 230 \). \( 299 - 230 = 69 \). \( 23 \times 3 = 69 \). So \( 23 \times 13 = 299 \).
• Thus, \(\frac{299}{460} = \frac{13 \times 23}{20 \times 23} = \frac{13}{20}\).
• Now calculate the percentage: \( \frac{13}{20} \times 100 \)
• Percentage \( = 13 \times \frac{100}{20} = 13 \times 5 = 65 \)

The profit with the membership card scheme is 65% of the profit without the membership card scheme.

Conclusion

By calculating the profit in both scenarios, we found that the profit with the membership card scheme is 65% of the profit without the membership card scheme.

Revision Table: Retailer Profit Calculations

Additional Information: Understanding Discounts and Profit

Discounts are reductions in price given by the seller to the buyer. They are usually calculated on the marked price or sometimes on a previously discounted price, as seen in sequential discounts.

• Marked Price (MP): The price tag amount.
• Selling Price (SP): The price at which the item is actually sold after discounts.
• Cost Price (CP): The price at which the retailer purchased the item.
• Profit: Occurs when Selling Price > Cost Price. Calculated as \(Profit = SP - CP\).
• Loss: Occurs when Selling Price < Cost Price. Calculated as \(Loss = CP - SP\).
• Sequential Discounts: When multiple discounts are applied one after another. Each subsequent discount is applied to the price after the previous discount, not the original marked price. In this problem, the 20% membership discount was applied to the price after the initial 30% discount.

Understanding the base on which each percentage is calculated (marked price, selling price, or cost price) is crucial in solving problems involving discounts and profit.