Understanding Markup and Discount for No Profit No Loss

This problem involves calculating the maximum discount percentage that can be offered on an article after marking up its price, such that the final selling price is equal to the original cost price. This scenario is often referred to as a 'no profit, no loss' situation.

Defining Key Terms

• Cost Price (CP): The original price at which the article was bought or manufactured.
• Marked Price (MP): The price written on the article, usually higher than the cost price, after applying a markup.
• Selling Price (SP): The price at which the article is finally sold after applying a discount on the marked price.
• Markup: The increase in price from the Cost Price to the Marked Price, usually expressed as a percentage of the Cost Price.
• Discount: The reduction in price from the Marked Price to the Selling Price, usually expressed as a percentage of the Marked Price.

Setting up the Problem

We are given that the article is marked 33.33% above the Cost Price. The condition for 'neither profit nor loss' is that the Selling Price must be equal to the Cost Price (SP = CP).

Calculating the Marked Price

Let's assume a base Cost Price for easy calculation. A common approach is to assume CP = 100 units.

The markup is 33.33% of the Cost Price.

We know that \(33.33\%\) is approximately equivalent to the fraction \(1/3\).

Therefore, Markup Amount \( = 33.33\%\) of CP \( = \frac{33.33}{100} \times \text{CP} \approx \frac{1}{3} \times \text{CP}\).

If CP = 100, Markup Amount \( = \frac{1}{3} \times 100 = 33.33\).

The Marked Price is the Cost Price plus the Markup Amount.

MP \( = \text{CP} + \text{Markup Amount}\).

If CP = 100, MP \( = 100 + 33.33 = 133.33\).

Calculating the Required Selling Price

For 'neither profit nor loss', the Selling Price must be equal to the Cost Price.

SP \( = \text{CP}\).

If CP = 100, then the required SP \( = 100\).

Calculating the Discount Amount

The discount is the difference between the Marked Price and the Selling Price.

Discount Amount \( = \text{MP} - \text{SP}\).

Using our calculated values: Discount Amount \( = 133.33 - 100 = 33.33\).

Calculating the Maximum Discount Percentage

The discount percentage is always calculated on the Marked Price.

Discount Percentage \( = \left(\frac{\text{Discount Amount}}{\text{MP}}\right) \times 100\%\).

Using our values:

Discount Percentage \( = \left(\frac{33.33}{133.33}\right) \times 100\%\).

Alternatively, using the fraction \(1/3\) for 33.33%:

If CP = x, then Markup \( = \frac{1}{3}x\).

MP \( = x + \frac{1}{3}x = \frac{4}{3}x\).

For no profit/loss, SP \( = x\).

Discount Amount \( = \text{MP} - \text{SP} = \frac{4}{3}x - x = \frac{1}{3}x\).

Discount Percentage \( = \left(\frac{\text{Discount Amount}}{\text{MP}}\right) \times 100\% = \left(\frac{\frac{1}{3}x}{\frac{4}{3}x}\right) \times 100\%\).

The 'x' cancels out:

Discount Percentage \( = \left(\frac{1/3}{4/3}\right) \times 100\% = \left(\frac{1}{3} \times \frac{3}{4}\right) \times 100\% = \frac{1}{4} \times 100\% = 25\%\).

The maximum discount that can be allowed is 25%.

Summary of Calculations

The calculated discount percentage is 25%, which means offering a 25% discount on the marked price (which was 33.33% above the cost price) will result in the selling price being equal to the cost price, achieving the 'neither profit nor loss' condition.

Revision Table: Profit, Loss, Markup, and Discount Concepts

Additional Information on Profit and Loss Calculations

Understanding the base on which percentages are calculated is crucial in profit, loss, markup, and discount problems. Profit and loss percentages are typically calculated on the Cost Price, while markup percentage is calculated on the Cost Price, and discount percentage is always calculated on the Marked Price.

• When a price is marked up by a certain percentage, say M%, above CP, the MP = CP * (1 + M/100).
• When a discount of D% is given on MP, the SP = MP * (1 - D/100).
• For the condition of 'neither profit nor loss', SP = CP. This implies CP = MP * (1 - D/100).
• Substituting the expression for MP: CP = [CP * (1 + M/100)] * (1 - D/100).
• If CP is not zero, we can divide both sides by CP: 1 = (1 + M/100) * (1 - D/100).
• This equation 1 = \(\left(1 + \frac{M}{100}\right) \times \left(1 - \frac{D}{100}\right)\) relates the markup percentage (M) and the discount percentage (D) when there is no profit or loss.
• In our problem, the markup M = 33.33%. We need to find D.
• Using \(M = 33.33\% \approx 1/3\): \(1 = \left(1 + \frac{1}{3}\right) \times \left(1 - \frac{D}{100}\right)\).
• \(1 = \frac{4}{3} \times \left(1 - \frac{D}{100}\right)\).
• \(\frac{3}{4} = 1 - \frac{D}{100}\).
• \(\frac{D}{100} = 1 - \frac{3}{4} = \frac{1}{4}\).
• \(D = \frac{1}{4} \times 100 = 25\).
• So, the discount percentage D is 25%.

This algebraic approach confirms the result obtained using assumed values and provides a general formula connecting markup and discount in a no-profit-no-loss scenario.