Discounts and Profit Analysis

This problem involves understanding how multiple sequential discounts affect the final selling price and how profit is calculated based on the cost price. We need to find the profit percentage if the item was sold after only the first discount, given the final selling price after three discounts and the overall profit.

Key terms:

• Marked Price (MP): The initial price tag on the item.
• Discount: A reduction applied to the price. Consecutive discounts mean each discount is applied to the reduced price after the previous discount.
• Selling Price (SP): The final price the customer pays after all discounts.
• Cost Price (CP): The price the shopkeeper originally paid for the item.
• Profit: The financial gain, calculated as SP minus CP when SP is higher.
• Profit Percentage: Profit expressed as a percentage of the Cost Price. The formula is: \(\text{Profit \%} = \frac{\text{Profit}}{\text{CP}} \times 100\).

Step-by-Step Calculation: Discounts and Profit

Let's break down the calculation step-by-step to determine the profit:

1. Final Selling Price Calculation

Assume the Marked Price (MP) of the article is '\(M\)'. The shopkeeper applies three consecutive discounts: 10%, 20%, and 25%.

First, calculate the price after the 10% discount:

\(SP_1 = M \times \left(1 - \frac{10}{100}\right) = M \times (1 - 0.10) = 0.90M\)

Next, apply the 20% discount to the resulting price (\(SP_1\)):

\(SP_2 = SP_1 \times \left(1 - \frac{20}{100}\right) = 0.90M \times (1 - 0.20) = 0.90M \times 0.80 = 0.72M\)

Finally, apply the 25% discount to the price \(SP_2\) to get the final Selling Price (SP):

\(SP = SP_2 \times \left(1 - \frac{25}{100}\right) = 0.72M \times (1 - 0.25) = 0.72M \times 0.75 = 0.54M\)

This means the final selling price is 54% of the marked price.

2. Cost Price Determination

We are told that this final selling price (\(SP = 0.54M\)) results in a profit of 8% on the Cost Price (CP). The relationship between SP, CP, and profit percentage is:

\(SP = CP \times \left(1 + \frac{\text{Profit \%}}{100}\right)\)

Substitute the known values:

\(0.54M = CP \times \left(1 + \frac{8}{100}\right)\)

\(0.54M = CP \times (1 + 0.08)\)

\(0.54M = 1.08 \times CP\)

Now, solve for the Cost Price (CP) in terms of the Marked Price (M):

\(CP = \frac{0.54M}{1.08}\)

\(CP = \frac{54}{108} M = \frac{1}{2} M = 0.50M\)

Therefore, the cost price is exactly half of the marked price.

3. Selling Price After First Discount

The question asks for the profit if the article had been sold after only the *first* discount of 10%. From Step 1, we found this selling price (\(SP_1\)) to be:

\(SP_1 = 0.90M\)

4. Profit Calculation After First Discount

To find the profit in this scenario, we compare the selling price \(SP_1\) with the cost price \(CP\). We have \(SP_1 = 0.90M\) and \(CP = 0.50M\).

The profit would be:

\(\text{Profit} = SP_1 - CP\)

\(\text{Profit} = 0.90M - 0.50M = 0.40M\)

Now, calculate the profit percentage using the profit and the cost price:

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

\(\text{Profit \%} = \frac{0.40M}{0.50M} \times 100\)

The variable '\(M\)' cancels out:

\(\text{Profit \%} = \frac{0.40}{0.50} \times 100 = \frac{4}{5} \times 100\)

\(\text{Profit \%} = 0.8 \times 100 = 80\%\)

Result Comparison with Options

The calculated profit percentage, if the article were sold after only the first discount, is 80%. We compare this result with the given options:

• Option 1: 20%
• Option 2: 40%
• Option 3: 50%
• Option 4: None of the above

Since 80% is not listed among the first three options, the correct choice is "None of the above".