Question

Critical Thinking You are the manager of a clothing store and have just purchased 100 dress shirts for $20.00 each. After 1 month of selling the shirts at the regular price, you plan to have a sale giving 40% off the original selling price. However, you still want to make a profit of 4 on each shirt at the sale price. What should you price the shirts at initially to ensure this? If, instead of 40% off at the sale, you give 50% off, by how much is your profit reduced?

Short answer

The initial price should be \(40 to ensure a \)4 profit with a 40% discount. With a 50% discount, the profit is reduced by $4, resulting in no profit.

Step-by-step solution

Determine the Sale Price with 40% Discount

First, calculate the sale price after a 40% discount on the initial price (P). The sale price would be 60% of the initial price since 100% - 40% = 60%. The equation for the sale price is: \( \text{{Sale Price}} = 0.6P \)}, {

Determine the Target Sale Price for Profit

Now, we calculate the target sale price which ensures a profit of \(4 per shirt at the sale price. Since each shirt was originally purchased for \)20, the sale price must be the cost plus the profit. The target sale price for profit is: \( \text{{Target Sale Price}} = \text{{Cost Price}} + \text{{Profit}} = 20 + 4 = 24 \)}, {

Determine the Initial Price (P)

Since the sale price is 60% of the initial price (P), we set up the equation: \[ 0.6P = 24 \] To find the initial price, solve for P: \[ P = \frac{24}{0.6} = 40 \]}, {

Calculate the Profit with 50% Discount

To find the profit if a 50% discount is given, first calculate the sale price with the 50% discount: \( \text{{Sale Price with 50% Off}} = 0.5 \times 40 = 20 \). Since each shirt is purchased for \(20, the profit is: \( \text{{Profit with 50% Off}} = 20 - 20 = 0 \). Compare this to the profit from the 40% discount: \( 4 - 0 = 4 \). The profit is reduced by \)4 per shirt.

Key concepts

price discount calculation

Understanding how to calculate price discounts is essential in retail. A price discount reduces the original selling price of an item to make it more attractive to customers.
To calculate the sale price after a given discount percentage, you need to subtract the discount percentage from 100%, then multiply this result by the original price. Here, the equation for calculating the sale price after a 40% discount looks like this: \( \text{Sale Price} = 0.6 \times \text{Original Price} \).
For instance, if a shirt originally costs \( P \), the sale price after a 40% discount is \( 0.6P \). This process helps businesses plan sales and make products more affordable to customers.

profit margin analysis

Profit margin is a crucial metric for understanding how much money is made on a product after accounting for its cost. To ensure a profit, you need to set a target sale price that includes both the cost of the product and the desired profit.
For example, if the cost price of a shirt is \( 20 \) and you desire a profit of \( 4 \) per shirt, the target sale price needed is \( \text{Cost} + \text{Profit} = 20 + 4 = 24 \).
However, this target sale price needs to align with the discount percentage applied during sales. In this example, with a 40% discount, the original price (\( P \)) must be high enough to still achieve the target sale price of \( 24 \) after the discount.

percentage discount problems

Percentage discounts can affect profitability in various ways. When the discount percentage changes, the profit also changes. Let's compare the impact of a 40% and a 50% discount on the original price.
For a 40% discount, the sale price is \( 0.6P \) and to ensure a \( 4 \) profit, setting \( 0.6P = 24 \) leads to an initial price of \( 40 \). For a 50% discount, the sale price is \( 0.5 \times 40 = 20 \). Given the cost price is \( 20 \), the profit is zero, resulting in a reduction of \( 4 \) in profit per shirt compared to a 40% discount.
This demonstrates how important it is to carefully consider the discount percentage to maintain profitability.