Problem 27 - Algebra for College Students

Question

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

The selling prices are \$2.10, \$4.55, \$20.72, \$29.40, and \$33.88.

Step 1: Understand the Function

The function given is \( s(c) = 1.4c \). This function calculates the selling price \( s \) based on the cost \( c \) of an item, including a profit margin of 40\%.

Step 2: Calculate Selling Price for \( c = \$1.50 \)

Substitute \( c = 1.50 \) into the function: \( s(1.50) = 1.4 \times 1.50 = 2.10 \). The selling price for an item costing \( \\(1.50 \) is \( \\)2.10 \).

Step 3: Calculate Selling Price for \( c = \$3.25 \)

Substitute \( c = 3.25 \) into the function: \( s(3.25) = 1.4 \times 3.25 = 4.55 \). The selling price for an item costing \( \\(3.25 \) is \( \\)4.55 \).

Step 4: Calculate Selling Price for \( c = \$14.80 \)

Substitute \( c = 14.80 \) into the function: \( s(14.80) = 1.4 \times 14.80 = 20.72 \). The selling price for an item costing \( \\(14.80 \) is \( \\)20.72 \).

Step 5: Calculate Selling Price for \( c = \$21.00 \)

Substitute \( c = 21.00 \) into the function: \( s(21.00) = 1.4 \times 21.00 = 29.40 \). The selling price for an item costing \( \\(21.00 \) is \( \\)29.40 \).

Step 6: Calculate Selling Price for \( c = \$24.20 \)

Substitute \( c = 24.20 \) into the function: \( s(24.20) = 1.4 \times 24.20 = 33.88 \). The selling price for an item costing \( \\(24.20 \) is \( \\)33.88 \).

Key Concepts

Linear Functions

Linear functions are mathematical expressions where each term is either a constant or the product of a constant and a single variable. The general form is often written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In simple terms, a linear function creates a straight line when graphed.
In this scenario, we have a function \(s(c) = 1.4c\). Here, \(c\) is the variable representing the cost, and \(1.4\) is the constant multiplier indicating the total percentage increase applied to the cost. There’s no additional constant added or subtracted, meaning there's no y-intercept other than the origin, making it a proportional increase.

• Using this function, whenever we change the value of \(c\), the resulting value of \(s(c)\) is directly proportional to \(c\).
• This function helps in determining a new value based on a fixed percentage markup or adjustment on the input cost.

Linear functions are handy because they are easy to work with and can be applied to a wide range of real-world problems, like financial calculations and statistics.

Percentage Increase

A percentage increase is a way of expressing a change in a value where the new value is higher than the original value.
This concept is critical in business for calculating profits, price adjustments, and more. It’s calculated by comparing the difference between the new and original value, divided by the original value, then multiplied by 100 to get a percentage.
In this problem, the retailer wants a profit of \(40\%\). This percentage becomes part of the function \(s(c) = 1.4c\). The point four, \(0.4\), specifically represents the \(40\%\) increase added to the original cost \(c\). The function \(1.4c\) hence represents the total activity: original price plus the added \(40\%\) profit.
For example, if you have an item that costs \(\\(1.50\), adding \(40\%\) would involve multiplying \(\\)1.50\) by \(1.4\), resulting in \(\$2.10\), which reflects the cumulative amount.

Cost and Selling Price Calculation

Cost and selling price are essential components in profit calculations. The cost price refers to the amount spent to purchase or produce goods, while the selling price is what customers pay to purchase them. The difference between these two figures indicates the profit.
In this exercise, you need to calculate the selling price from the cost price, including a \(40\%\) profit. Using the linear function \(s(c) = 1.4c\), you'll multiply the cost by \(1.4\) to find this new price, ensuring that the profit margin is automatically included.
Let’s see some examples:

• The calculation for an item costing \(\\(3.25\) results in \(\\)4.55\) as the selling price.
• If an item is \(\\(14.80\), the selling price becomes \(\\)20.72\).
• For an item with a \(\\(21\) cost, the resulting selling price will be \(\\)29.40\).
• Lastly, an item that cost \(\\(24.20\) will sell for \(\\)33.88\).

The key takeaway is the ease with which one can adjust costs to prices, ensuring all necessary increases are accounted for without multiple, complicated steps.