Question

Two articles are sold at the same price. One at a profit of \(75 \%\) and another one at a loss of \(30 \%\). What is the overall profit or loss? (a) \(22.5 \%\) profit (b) \(57.5\) profit (c) \(13 \frac{2}{7} \% \operatorname{los}\) (d) none of these

Short answer

Answer: The overall profit or loss is a loss of \(13 \frac{2}{7} \%\).

Step-by-step solution

Find the cost price of each article

Let the selling price of both articles be S. For the first article, the profit is \(75 \%\). For the second article, the loss is \(30 \%\). For the first article, Selling price = Cost price × (1 + Profit percentage) S = CP1 * (1 + 75/100) S = CP1 * (1 + 0.75) S = CP1 * 1.75 CP1 = S / 1.75 For the second article, Selling price = Cost price × (1 - Loss percentage) S = CP2 * (1 - 30/100) S = CP2 * (1 - 0.3) S = CP2 * 0.7 CP2 = S / 0.7

Calculate the overall cost price of both articles

Now, we will calculate the overall cost price of both articles: Total_Cost_Price = CP1 + CP2

Find the overall profit or loss

Since both articles are sold at the same price, the total selling price of both articles is 2S. To find the overall profit or loss, we will compare the total selling price with the overall cost price: Total_Selling_Price = 2S Profit_or_Loss = Total_Selling_Price - Total_Cost_Price = 2S - (CP1 + CP2) = 2S - (S/1.75 + S/0.7) = 2S - S(1/1.75+1/0.7) Now, we need to find the profit or loss percentage: Profit_or_Loss_Percentage = (Profit_or_Loss / Total_Cost_Price) * 100 = ((2S - S(1/1.75+1/0.7))/(S/1.75 + S/0.7)) * 100 = 100 * (1 - (1/1.75+1/0.7)) After calculating the expression, we get: = \(-13 \frac{2}{7} \ \%\) Since the result is negative, it means there is a loss. The overall profit or loss is (c) \(13 \frac{2}{7} \% \operatorname{loss}\).

Key concepts

Cost Price Calculation

Understanding the cost price is crucial when dealing with profit and loss problems. Cost price (CP) is the amount paid to acquire a product or service, which becomes the basis for determining profits or losses.

To calculate the CP when the selling price (SP) and the profit percentage are known, you can use the formula: \
\[\text{CP} = \frac{\text{SP}}{1 + (\text{Profit Percentage}/100)}\]

In the same vein, if a loss occurs, modify the denominator to account for it like so: \
\[\text{CP} = \frac{\text{SP}}{1 - (\text{Loss Percentage}/100)}\]

These formulas simply adjust the selling price based on the percentage of profit or loss to find the amount initially invested, or the cost price. In our exercise, for the first article sold at a 75% profit, the CP is effectively 1.75 times smaller than the SP, whereas for the second article with a 30% loss, the CP is greater than the SP.

Selling Price Determination

Determining the selling price is an equally important concept in business math problems. The selling price (SP) is the amount for which the product is sold to the customer. The selling price can be determined based on the desired profit or to cover a loss.

To arrive at the SP when the CP and profit are known, use: \
\[\text{SP} = \text{CP} \times (1 + (\text{Profit Percentage}/100))\]

If aiming to recover a loss, the formula inverts the addition to subtraction: \
\[\text{SP} = \text{CP} \times (1 - (\text{Loss Percentage}/100))\]

These equations highlight the direct relationship between SP and profit or loss percentages. In our example, both articles are sold at the same SP, implying that despite different CPs, they ended up with the same final sales price.

Profit or Loss Percentage

The concept of profit or loss percentage is integral to understanding business transactions. It's a measure of how much money has been earned (profit) or lost (loss) as a percentage of the cost price.

The general formula for calculating profit or loss percentage is: \
\[\text{Profit or Loss Percentage} = \left( \frac{\text{Profit or Loss}}{\text{Total Cost Price}} \right) \times 100\%\]

When the percentage is positive, it signifies a profit; when negative, it represents a loss. In cases where multiple items are involved, like in our exercise, you'd calculate the overall profit or loss percentage by first finding the total CP and SP, then applying the formula above. The difficulty often comes from combining the effects of different transactions. As we've seen, even when articles are sold at the same SP, one can be a source of profit, the other a source of loss—their interplay determining the overall outcome.