Solving Profit and Discount Percentages: A Practical Guide

When dealing with financial calculations in business, it is essential to understand the relationships between cost prices (CP), marked prices (MP), profit, and discount percentages. This article provides a detailed step-by-step guide to solving a specific problem involving these elements using given ratios. We will break down the problem and offer a clear solution.

Ratios and Initial Definitions

Given the problem, we are provided with the following ratios:

The ratio of CP to MP is 8:15. The ratio of profit to discount is 25:24.

We need to determine the difference between the profit percentage and the discount percentage. To start, let's define our variables and set up the equations based on the given ratios.

Defining Variables

From the ratios, we can write:

Let the CP be ( 8x ). Let the MP be ( 15x ). Let the profit be ( 25y ). Let the discount be ( 24y ).

Calculating Profit and Discount

First, we need to calculate the actual profit and discount amounts using the defined variables:

The profit is calculated as: Profit MP - CP ( 15x - 8x 7x ).

Converting this to a percentage of CP:

Profit Percentage ( left(frac{7x}{8x}right) times 100 87.5% )

Next, we calculate the discount:

The selling price (SP) can be derived from the discount percentage: SP MP - Discount

The discount amount is calculated as:

Discount ( frac{24y}{100} times MP frac{24y}{100} times 15x 3.6xy )

So the selling price becomes:

SP MP - Discount ( 15x - 3.6xy )

Finding Selling Price in Terms of CP

To express SP in terms of CP, we relate SP to the CP and profit:

SP CP Profit ( 8x 7x 15x )

Thus:

15x - 3.6xy 15x

This implies:

3.6xy 0

This means the discount must be based on the value of y in such a way that the ratios hold true.

Difference Between Profit and Discount

The profit percentage is ( 25y ) and the discount percentage is ( 24y ).

The difference is:

Difference Profit - Discount ( 25y - 24y y )

To find the actual difference in percentage terms, we note that ( y ) is a common multiplier and represents the ratio of the profit and discount percentages. Thus:

Difference 25 - 24 1

Therefore, the difference between the profit percentage and the discount percentage is 1 percentage point.

Conclusion

This step-by-step guide demonstrates how to solve a problem involving the ratios of CP and MP as well as the ratio of profit and discount percentages. By defining the variables and using the given ratios, we were able to calculate the profit percentage and discount percentage and ultimately determine the difference between them.