Understanding and Solving Markup, Discount, and Profit Problems: A Comprehensive Guide

In the world of retail and business, understanding markup, discount, and profit percentages is crucial for financial success. This guide will walk you through the process of solving a specific problem that combines these elements, providing a step-by-step solution and a deeper understanding of the underlying concepts.

The Scenario: A Retailer's Profit Calculation

Let's consider the case of Ram, who marks up an item by Rs. 220 and sells it at a discount of 25% such that he earns a percentage profit of 33 13/14%.

Objective

The objective is to find out the cost price of the item.

Solution Steps

Step 1: Define Variables and Initial Conditions

Let the cost price (CP) of the item be x. The marked price (MP) is x 220.

Step 2: Calculate the Selling Price (SP) after Discount

The item sells at a 25% discount, so the selling price (SP) is:

Selling Price after discount (x 220) × (1 - 0.25) 0.75(x 220)

Step 3: Determine the Profit Percentage

The percentage profit is given by:

[ frac{33 frac{13}{14} 0.75(x 220) - x}{x} times 100 ]

Converting the mixed number 33 13/14 into a decimal:

[ 33 frac{13}{14} 33 frac{13}{14} 33.928571 ]

Substituting the values:

[ 33.928571 frac{0.75(x 220) - x}{x} times 100 ][ 33.928571 frac{0.75x 165 - x}{x} times 100 ][ 33.928571 frac{165 - 0.25x}{x} times 100 ][ frac{33.928571}{100} frac{165 - 0.25x}{x} ][ 0.33928571 frac{165 - 0.25x}{x} ]

Multiplying both sides by x:

[ 0.33928571x 165 - 0.25x ]

Combining like terms:

[ 0.33928571x 0.25x 165 ][ 0.58928571x 165 ]

Solving for x:

[ x frac{165}{0.58928571} approx 281.81 ]

The cost price of the item is approximately Rs. 281.81.

Alternative Method

Step 1: Using the Selling Price

If the selling price (SP) is Rs. 1476, we proceed as follows:

[ text{SP} 1476 ][ text{MP} frac{SP}{1 - text{Discount}} frac{1476}{1 - 0.25} frac{1476}{0.75} 1968 ]

Step 2: Calculate Cost Price (CP) without Discount

If the marked price (MP) is 124% of the cost price:

[ text{CP} frac{SP}{1 text{Profit}} frac{1476}{1.24} 1190.32 ]

Step 3: Using Ratios and Proportions

If the marked price (MP) is 100%, then:

[ text{124%} 1000 ][ text{100%} frac{1000}{124} times 100 806.45 ]

Therefore:

[ text{CP} frac{1476}{1.24} 1190.32 ]

This calculation shows the cost price to be approximately Rs. 1190.32.

Conclusion

By understanding and applying the concepts of markup, discount, and profit percentages, we can accurately determine the cost price of an item. This example demonstrates a comprehensive approach to solving complex retail problems, ensuring profitability and competitiveness in the market.