Solving Cost Price and Selling Price Profit Problems

This problem involves understanding the relationship between Cost Price (CP), Selling Price (SP), and Profit Percentage. We are given a scenario where the total cost of a certain number of articles is equal to the total selling price of a different number of articles, along with the profit percentage earned.

Understanding the Problem Statement

We are told two key things:

1. The cost price of 15 articles is the same as the selling price of A articles.
2. The profit earned is 25%.

Our goal is to find the value of A, which is the number of articles whose selling price equals the cost price of 15 articles.

Setting up the Equations

Let's assume the Cost Price (CP) of one article is \(C\), and the Selling Price (SP) of one article is \(S\).

According to the first condition:

Cost Price of 15 articles = \(15 \times C\)

Selling Price of A articles = \(A \times S\)

Since these are equal, we have our first main equation:

\(15C = AS \quad (Equation \; 1)\)

Using the Profit Percentage

The profit percentage is given as 25%. The formula for profit percentage is:

Profit% = \(\frac{\text{Profit}}{\text{Cost Price}} \times 100\)

Where Profit = Selling Price - Cost Price.

So, Profit% = \(\frac{SP - CP}{CP} \times 100\)

We can also express the relationship between SP and CP directly from the profit percentage. If there is a profit of 25%, it means the Selling Price is 25% more than the Cost Price. So, SP is 100% + 25% = 125% of the CP.

SP = 125% of CP

SP = \(\frac{125}{100} \times CP\)

SP = \(\frac{5}{4} \times CP\)

Applying this to the price of a single article, we get:

\(S = \frac{5}{4}C \quad (Equation \; 2)\)

Solving for the Value of A

Now we have two equations:

1. \(15C = AS\)
2. \(S = \frac{5}{4}C\)

We can substitute the value of \(S\) from Equation 2 into Equation 1:

\(15C = A \times \left(\frac{5}{4}C\right)\)

We can cancel \(C\) from both sides of the equation, assuming \(C \neq 0\) (the cost of an article cannot be zero):

\(15 = A \times \frac{5}{4}\)

To find the value of A, we rearrange the equation:

\(A = 15 \times \frac{4}{5}\)

\(A = \frac{15 \times 4}{5}\)

\(A = \frac{60}{5}\)

\(A = 12\)

Conclusion on Finding A

The value of A is 12. This means the cost price of 15 articles is the same as the selling price of 12 articles when there is a 25% profit.

Verification (Optional)

Let's verify this with an example. Assume the CP of 1 article is Rs. 100.

CP of 15 articles = \(15 \times 100 = \text{Rs. } 1500\).

With a 25% profit, the SP of 1 article is \(100 + 25\% \text{ of } 100 = 100 + 25 = \text{Rs. } 125\).

SP of A articles = SP of 12 articles (since A=12) = \(12 \times 125\).

\(12 \times 125 = 12 \times (100 + 25) = 1200 + 12 \times 25 = 1200 + 300 = \text{Rs. } 1500\).

Since CP of 15 articles (Rs. 1500) equals SP of 12 articles (Rs. 1500), our value of A=12 is correct.

Revision Table: Cost Price Profit Calculation

Additional Information: Profit and Loss Concepts

Profit and Loss are fundamental concepts in business and quantitative aptitude. Here are some related points:

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Profit: Occurs when SP > CP. Profit = SP - CP.
• Loss: Occurs when SP < CP. Loss = CP - SP.
• Profit Percentage: Calculated on CP. \(\frac{\text{Profit}}{CP} \times 100\).
• Loss Percentage: Calculated on CP. \(\frac{\text{Loss}}{CP} \times 100\).

When the CP of 'x' articles is equal to the SP of 'y' articles:

\(x \times CP = y \times SP\)

\(\frac{SP}{CP} = \frac{x}{y}\)

If \(x > y\), then SP > CP, resulting in a profit. Profit% = \(\left(\frac{x}{y} - 1\right) \times 100 = \left(\frac{x-y}{y}\right) \times 100\).

If \(x < y\), then SP < CP, resulting in a loss. Loss% = \(\left(1 - \frac{x}{y}\right) \times 100 = \left(\frac{y-x}{y}\right) \times 100\).

In our problem, \(x = 15\) and \(y = A\). Profit is 25%. We used the relationship \(SP = \frac{5}{4}CP\). Substituting this into \(15C = AS\) gave \(15C = A (\frac{5}{4}C)\), leading to \(15 = A \frac{5}{4}\) and \(A = 12\).

Using the direct formula \(\left(\frac{x-y}{y}\right) \times 100\) for profit%:

\(25 = \left(\frac{15-A}{A}\right) \times 100\)

\(\frac{25}{100} = \frac{15-A}{A}\)

\(\frac{1}{4} = \frac{15-A}{A}\)

\(A = 4 \times (15-A)\)

\(A = 60 - 4A\)

\(A + 4A = 60\)

\(5A = 60\)

\(A = \frac{60}{5}\)

\(A = 12\)

Both methods yield the same result, confirming the value of A is 12.