Profit as a Percentage of Selling Price - A Comprehensive Analysis

Understanding the relationship between profit, cost, and selling price is crucial for businesses aiming to optimize their financial performance. In this detailed article, we will explore a specific scenario involving a shop where the profit is 200% of the cost. If the cost increases by 25 but the selling price remains constant, we will calculate what percentage of the selling price the new profit represents.

Problem Statement

Let's address the problem: in a shop, the profit is 200% of the cost. If the cost increases by 25% but the selling price remains constant, what percentage of the selling price is the new profit?

Symbol Definitions

We define the following variables for clarity and simplicity:

S - the constant selling price C1 - the initial cost price C2 - the new cost price after the increase

Solving the Problem

Given that the profit is 200% of the cost, we can express this mathematically as:

S - C1 200/100 C1

Simplifying this equation, we get:

S 3C1

From this, we can express C1 in terms of S:

C1 S/3

The cost increases by 25%, leading to:

C2 C1 25% of C1 1.25C1 5/4 C1

Expressing C2 in terms of S, we have:

C2 5/4 * S/3 5/12 S

Now, the new profit P can be calculated as:

P S - C2 S - 5/12 S 7/12 S

Expressing this as a percentage of the selling price:

(7/12 S) / S * 100 58.33%

Geometric Interpretation

Visually, we can represent the initial and new state of the shop's financial health. Initially, the profit is 200% of the cost, indicating that the selling price is three times the cost. After the cost increases by 25%, the new profit is 58.33% of the selling price.

Conceptual Breakdown

Cost Price (CP): The initial cost price of the goods in the shop. Profit (P): The amount of money gained from selling the goods at a higher price, which is 200% of the CP. Selling Price (SP): The price at which the goods are sold.

Step-by-Step Solution

Define the variables: CP (cost price), P (profit), and SP (selling price). Express the profit in terms of CP: SP - CP 2CP. Express the selling price in terms of CP: SP 3CP. Apply the cost increase: CP increases by 25%, leading to a new CP of 1.25CP. Calculate the new profit: SP - 1.25CP. Express the new profit as a percentage of the selling price: (SP - 1.25CP) / SP * 100.

Example Calculations

Let's consider an example where:

CP 100 Profit 220 SP 320 New CP 125 SP remains 320 New profit 320 - 125 195 Percentage profit (195 / 125) * 100 156%

Another example:

SP 2.2CP CP increases by 25% New profit 2.2CP - 1.25CP 0.95CP Percentage profit (0.95CP / 2.2CP) * 100 43%

Final calculation using the original problem statement:

100 * 210 - 40 / 310 100 * 170 / 310 100 * 17 / 31 54.84%

In conclusion, the new profit is approximately 58.33% of the selling price when the cost increases by 25% and the selling price remains the same. This understanding is vital for businesses to manage and optimize their profit margins effectively.