Problem 33 - Elementary Algebra

Question

Set up an equation and solve each problem. (Objectives 2 and 3) The owner of a pizza parlor wants to make a profit of \(55 \%\) of the cost for each pizza sold. If it costs \(\$ 8\) to make a pizza, at what price should it be sold?

Step-by-Step Solution

Answer

The selling price should be $12.4.

Step 1: Identify the Known Values

The objective is to determine the selling price of the pizza. We are given the cost of making a pizza, which is $8, and the desired profit percentage, which is 55% of the cost.

Step 2: Set Up the Equation

To find the selling price, we need an equation that includes both the cost and the profit. Let \(x\) be the selling price of the pizza. The profit can be expressed as \(0.55 \times 8\). Therefore, the equation for the selling price becomes: \[x = 8 + 0.55 \times 8\]

Step 3: Calculate the Profit

Calculate the profit by multiplying the cost of making the pizza by the profit percentage: \[0.55 \times 8 = 4.4\] This means the owner wants to make a profit of $4.4 per pizza.

Step 4: Substitute and Solve for Selling Price

Substitute the calculated profit back into the equation and solve for \(x\): \[x = 8 + 4.4 = 12.4\] Thus, the selling price of the pizza should be $12.4.

Key Concepts

Profit and Loss

Profit and loss are fundamental concepts in business. They help determine whether a business venture is successful. **Profit** refers to the financial gain obtained when the revenue from sales exceeds the costs associated with production and operations. It is calculated as:

• Profit = Selling Price - Cost

In the given pizza parlor problem, the owner wants to make a 55% profit on each pizza. Understanding profit includes knowing both the percentage of profit desired and how it adds up to the overall price. **Loss**, on the other hand, occurs when costs outweigh revenue. Fortunately, our example focuses on profit. By calculating the profit, the business owner ensures that the price covers costs while also achieving financial goals. This forms the basis for creating a sustainable pricing strategy and keeps the business thriving.

Percentage Calculations

Percentage calculations are everywhere in real life. They allow us to understand relationships and changes in quantitative terms. A percentage is essentially a fraction of 100. When you see a 55% profit, it means that the profit is 55 parts out of every 100 of the cost. To express this calculation, you multiply the initial amount by the percentage.

• Profit = 55% of Cost
• Percentage Calculation = 0.55 \( \times \) Cost

In the pizza parlor example, this calculation was performed as:

• 0.55 \( \times \) 8 = 4.4

This simple multiplication helps identify how much extra needs to be charged on top of the cost to achieve the desired profit margin. Mastering percentage calculations is crucial, as it can be applied to discounts, interest rates, and many other financial contexts.

Basic Algebra

Basic algebra involves using symbols and letters to represent numbers and quantities within formulas and equations. It's a powerful tool, allowing us to solve for unknown values by forming relationships between known quantities. In our exercise, the **unknown variable** is the selling price of the pizza, represented by \( x \). We derived the equation:

• \( x = \text{Cost} + \text{Profit} \)

Substituting known values into this equation:

• \( x = 8 + 4.4 \)
• \( x = 12.4 \)

This shows how algebra can solve real-world problems. Basic algebra is essential for setting up equations that model and solve complex situations easily. It goes beyond numbers, helping interpret relationships in various fields—science, engineering, and even economics.