Problem 35 - Intermediate Algebra

Question

Use linear functions. A retailer has a number of items that she wants to sell and make a profit of \(40 \%\) of the cost of each item. The function \(s(c)=c+0.4 c=1.4 c\), where \(c\) represents the cost of an item, can be used to determine the selling price. Find the selling price of items that cost \(\$ 1.50\), \(\$ 3.25, \$ 14.80, \$ 21\), and \(\$ 24.20\).

Step-by-Step Solution

Answer

Selling prices: $2.10, $4.55, $20.72, $29.40, and $33.88.

Step 1: Identify the Selling Price Function

The function given is \( s(c) = 1.4c \). This function represents the selling price \( s \) of an item as a function of its cost \( c \).

Step 2: Calculate Selling Price for $1.50 Item

Substitute \( c = 1.50 \) into the function: \( s(1.50) = 1.4 \times 1.50 \). This equals \( 2.10 \).

Step 3: Calculate Selling Price for $3.25 Item

Substitute \( c = 3.25 \) into the function: \( s(3.25) = 1.4 \times 3.25 \). This equals \( 4.55 \).

Step 4: Calculate Selling Price for $14.80 Item

Substitute \( c = 14.80 \) into the function: \( s(14.80) = 1.4 \times 14.80 \). This equals \( 20.72 \).

Step 5: Calculate Selling Price for $21 Item

Substitute \( c = 21 \) into the function: \( s(21) = 1.4 \times 21 \). This equals \( 29.40 \).

Step 6: Calculate Selling Price for $24.20 Item

Substitute \( c = 24.20 \) into the function: \( s(24.20) = 1.4 \times 24.20 \). This equals \( 33.88 \).

Key Concepts

Profit Calculation

In business, profit calculation is crucial as it helps determine whether a business is making money or not. Profit is essentially the difference between sales revenue and the costs associated with generating that revenue. To make a profit, a business must sell its products for more than they cost to produce or purchase.

• **Profit** = Selling Price - Cost
• In the context of the example, the desired profit is 40% of the cost of an item.

Using a linear function, the retailer calculated a selling price that included this profit margin. For instance, if an item's cost is \( c \), then the selling price must be \( c + 0.4c = 1.4c \). The "1.4" here means the item is being sold for 140% of its cost.
This ensures that 40% over the cost is earned as profit. Understanding how to set a profit margin using percentages and functions can guide pricing strategies effectively.

Function Application

Function application in mathematics enables us to simplify and automate calculations by using predefined formulas or relationships. In this scenario, the function \( s(c) = 1.4c \) directly gives us the selling price for any cost \( c \).

• It's efficient - we can calculate numerous selling prices quickly.
• It's consistent - each calculation follows the same rule, so results are uniform.

The function is versatile and can be applied to any item's cost to determine its selling price with a 40% profit margin.
All you need to do is substitute the item's cost into the function. For example, if an item costs \\(3.25, you find the selling price by simply calculating \(1.4 \times 3.25 \), which equals \\)4.55.
This process showcases how linear functions can simplify complex profit calculations and ensure accuracy and consistency in results.

Cost and Selling Price

Understanding the relationship between cost and selling price helps businesses make informed pricing decisions.

• The **cost** is the price paid to produce or purchase an item.
• The **selling price** is the amount a retailer charges customers for an item.

The difference between the two is crucial in understanding profitability. Setting the right selling price ensures that the cost is covered, and a sufficient profit margin is achieved over that cost.
In our scenario, by setting the linear function \( s(c) = 1.4c \), the retailer guarantees a selling price that covers the cost plus a 40% profit.
For instance, if an item costs \\(21, the selling price is \(1.4 \times 21 = \\)29.40\). Setting prices using such a function is strategic as it automatically integrates the profit margin directly into pricing.
This strategy is simple yet effective in maintaining desired profit levels across various items.