Problem 28 - Algebra for College Students

Question

Zack wants to sell five items that cost him \(\$ 1.20, \$ 2.30\), \(\$ 6.50, \$ 12\), and \(\$ 15.60\). He wants to make a profit of \(60 \%\) of the cost. Create a function that you can use to determine the selling price of each item, and then use the function to calculate each selling price. See below

Step-by-Step Solution

Answer

The selling prices are \$1.92, \$3.68, \$10.40, \$19.20, and \$24.96.

Step 1: Determine the Profit Function

To determine the selling price of an item with a 60% profit, we first need to find the profit. The profit is calculated as 60% of the cost price. To express this mathematically, the profit is given by the formula:\[\text{Profit} = 0.60 \times \text{Cost Price}\]

Step 2: Create the Selling Price Function

The selling price of an item is the sum of its cost price and the profit. Therefore, we can write the selling price function as:\[\text{Selling Price} = \text{Cost Price} + \text{Profit}\]Substituting the expression for profit, the function becomes:\[\text{Selling Price} = \text{Cost Price} + (0.60 \times \text{Cost Price})\]Simplifying, we have:\[\text{Selling Price} = 1.60 \times \text{Cost Price}\]

Step 3: Apply the Function to Each Cost Price

Now, apply the selling price function to each cost price:1. For \(\\(1.20\):\[\text{Selling Price} = 1.60 \times 1.20 = \\)1.92\]2. For \(\\(2.30\):\[\text{Selling Price} = 1.60 \times 2.30 = \\)3.68\]3. For \(\\(6.50\):\[\text{Selling Price} = 1.60 \times 6.50 = \\)10.40\]4. For \(\\(12\):\[\text{Selling Price} = 1.60 \times 12 = \\)19.20\]5. For \(\\(15.60\):\[\text{Selling Price} = 1.60 \times 15.60 = \\)24.96\]

Key Concepts

Percentage Profit

Understanding percentage profit is crucial when calculating how much you want to earn from selling an item. Percentage profit represents the ratio of the profit to the original cost price, expressed as a percentage. To put it simply, percentage profit tells you how much extra money you're making on top of the item’s cost price.
If an item cost you \(\\(1\) and you sell it for \(\\)1.60\), your percentage profit would be 60%. This means that for every dollar you spent, you made an extra \(60\) cents. Calculating this is easy!

• First, determine the profit by subtracting the original cost price from the selling price.
• Next, divide this profit by the cost price.
• To find the percentage, multiply the result by \(100\).

Mathematically, if the cost price is \(C\) and the selling price is \(S\), the formula for percentage profit is: \[\text{Percentage Profit} = \left( \frac{S-C}{C} \right) \times 100\] This concept is essential for anyone wanting to set a target profit when pricing items for sale.

Selling Price Function

The selling price function is a mathematical tool that helps you determine what price to set for your goods to achieve a desired profit margin. Essentially, it combines your cost price with the percentage profit to give you a straightforward formula for pricing.
To determine the selling price of an item, you need to add the desired profit to the cost price. From the example given, the desired profit was 60% of the cost price.
Let's break it down:

• Start with the cost price of the item.
• Calculate the profit by multiplying the cost price by the desired profit percentage, which is expressed as a decimal (e.g., 60% as 0.60).
• Add this profit to the original cost price to determine the selling price.

The resulting selling price function is given by: \[\text{Selling Price} = \text{Cost Price} + (0.60 \times \text{Cost Price})\]
Which simplifies to:\[\text{Selling Price} = 1.60 \times \text{Cost Price}\] This formula allows sellers to effortlessly calculate how much they should charge to secure the desired profit margin.

Cost Price Analysis

To effectively utilize the selling price function and understand your profit margins, analyzing the cost price is fundamental. Cost price is simply what you originally pay to acquire a product, without any additional markup or profit added.
Analyzing the cost price involves more than just knowing the figure; it helps you to:

• Assess the minimum price at which you should sell your item to avoid a loss.
• Determine if cost-cutting measures are needed to increase your profit margin.
• Compare with historical cost data to strategize pricing better.

Knowing the cost price also allows you to incorporate fixed and variable expenses into your selling price, aiming for maximum profitability. For instance, if you're selling an item with a cost price of \(\$1.20\), keeping the analysis in mind, you might decide to include additional expenses like packaging or distribution. This comprehensive understanding ensures that every cost is considered, protecting your profit margin and leading to better financial decisions.