Understanding the Profit and Loss Problem

This problem involves calculating the original cost price of articles based on their marked price, different discounts applied during sales, and the total profit earned. We need to use concepts of cost price, marked price, selling price, discount percentage, and profit.

Step-by-Step Solution to Find the Cost Price

Let's denote the cost price (CP) of each article by \( \text{Rs. } x \).

There are a total of 30 articles, and each has the same cost price \( x \).

• Total Cost Price = \( 30 \times x = 30x \)

The marked price (MP) of each article is set at 30% above its cost price.

• Marked Price (MP) of one article = \( \text{CP} + 30\% \text{ of CP} \)
• MP of one article = \( x + 0.30x = 1.30x \)

Akshay sells the 30 articles in two lots:

1. 15 articles are sold at a discount of 20% each.
2. The remaining 15 articles are sold at a discount of 10% each.

Calculating Selling Price for the First Lot (15 articles)

Discount on each article is 20% of the Marked Price.

• Discount amount = \( 20\% \text{ of MP} = 0.20 \times 1.30x = 0.26x \)
• Selling Price (SP) of one article in this lot = \( \text{MP} - \text{Discount Amount} \)
• SP of one article = \( 1.30x - 0.26x = 1.04x \)
• Total SP for the first 15 articles = \( 15 \times 1.04x = 15.6x \)

Calculating Selling Price for the Second Lot (15 articles)

Discount on each article is 10% of the Marked Price.

• Discount amount = \( 10\% \text{ of MP} = 0.10 \times 1.30x = 0.13x \)
• Selling Price (SP) of one article in this lot = \( \text{MP} - \text{Discount Amount} \)
• SP of one article = \( 1.30x - 0.13x = 1.17x \)
• Total SP for the remaining 15 articles = \( 15 \times 1.17x = 17.55x \)

Calculating Total Selling Price and Total Profit

• Total Selling Price (SP) for all 30 articles = \( \text{SP of first 15 articles} + \text{SP of remaining 15 articles} \)
• Total SP = \( 15.6x + 17.55x = 33.15x \)

The total profit is given as Rs. 630.

• Total Profit = Total Selling Price - Total Cost Price
• \( 630 = 33.15x - 30x \)
• \( 630 = 3.15x \)

Solving for the Cost Price (x)

We need to solve the equation \( 3.15x = 630 \) for \( x \).

• \( x = \frac{630}{3.15} \)

To make the division easier, we can multiply both the numerator and the denominator by 100:

• \( x = \frac{630 \times 100}{3.15 \times 100} = \frac{63000}{315} \)

Now, perform the division:

• \( 63000 \div 315 \)
• We know that \( 315 \times 2 = 630 \).
• So, \( 315 \times 200 = 63000 \).
• Therefore, \( x = 200 \).

The cost price of each article is Rs. 200.

Verification

Let's verify the answer by calculating the total profit with \( x = 200 \).

• CP of one article = Rs. 200
• MP of one article = \( 1.30 \times 200 = \text{Rs. } 260 \)
• SP of one article (20% discount) = \( 1.04 \times 200 = \text{Rs. } 208 \)
• SP of one article (10% discount) = \( 1.17 \times 200 = \text{Rs. } 234 \)
• Total SP for first 15 articles = \( 15 \times 208 = \text{Rs. } 3120 \)
• Total SP for remaining 15 articles = \( 15 \times 234 = \text{Rs. } 3510 \)
• Total SP for all 30 articles = \( 3120 + 3510 = \text{Rs. } 6630 \)
• Total CP for 30 articles = \( 30 \times 200 = \text{Rs. } 6000 \)
• Total Profit = Total SP - Total CP = \( 6630 - 6000 = \text{Rs. } 630 \)

The calculated total profit matches the given total profit, confirming our answer.

The cost price of each article is Rs. 200.

Revision Table: Profit and Loss Concepts

Additional Information on Pricing and Profit Calculations

Understanding the relationships between Cost Price, Marked Price, Selling Price, Discount, and Profit is crucial for solving problems like this. Here are some additional points:

• The marked price is often higher than the cost price to allow for discounts and still make a profit.
• Discount is always calculated on the marked price, unless stated otherwise.
• Profit or Loss is always calculated based on the cost price, unless stated otherwise.
• Different articles or batches of articles can be sold at different selling prices, depending on the discounts offered.
• The total profit (or loss) is the difference between the total selling price of all items sold and the total cost price of those items.