Understanding the Cost Price Problem

This problem involves finding the cost price (CP) of an article. We are given two different selling prices (SP) for the same article. The key information is that the profit made when selling at the higher price is exactly equal to the loss incurred when selling at the lower price.

Defining Profit and Loss

In business transactions, understanding profit and loss is crucial. Here’s how they are defined in relation to cost price (CP) and selling price (SP):

• Profit: Occurs when the selling price is greater than the cost price. $$ \text{Profit} = \text{SP} - \text{CP} $$
• Loss: Occurs when the selling price is less than the cost price. $$ \text{Loss} = \text{CP} - \text{SP} $$

Setting Up the Equation

Let the cost price of the article be denoted by '$ \text{CP} $'.

We are given two scenarios:

1. Selling Price 1 ($ \text{SP}_1 $) = ₹6,250. In this case, a profit is earned. $$ \text{Profit} = \text{SP}_1 - \text{CP} = ₹6,250 - \text{CP} $$
2. Selling Price 2 ($ \text{SP}_2 $) = ₹3,750. In this case, a loss is incurred. $$ \text{Loss} = \text{CP} - \text{SP}_2 = \text{CP} - ₹3,750 $$

The problem states that the profit earned is the same as the loss incurred. Therefore, we can set the two expressions equal to each other:

Step-by-Step Calculation

Now, we need to solve the equation for '$ \text{CP} $'.

1. Rearrange the equation: Move all terms involving '$ \text{CP} $' to one side and the constant values to the other side. $$ ₹6,250 + ₹3,750 = \text{CP} + \text{CP} $$
2. Combine terms: Add the constant values and the '$ \text{CP} $' terms. $$ ₹10,000 = 2 \times \text{CP} $$
3. Isolate CP: Divide both sides of the equation by 2 to find the value of the cost price. $$ \text{CP} = \frac{₹10,000}{2} $$ $$ \text{CP} = ₹5,000 $$

Conclusion

By setting the profit equal to the loss and solving the resulting algebraic equation, we find that the cost price of the article is ₹5,000.

Let's verify this:

• If CP = ₹5,000 and SP = ₹6,250, Profit = ₹6,250 - ₹5,000 = ₹1,250.
• If CP = ₹5,000 and SP = ₹3,750, Loss = ₹5,000 - ₹3,750 = ₹1,250.

Since the profit (₹1,250) equals the loss (₹1,250), our calculated cost price is correct.