Problem 36 - Elementary Algebra

Question

Set up an equation and solve each problem. (Objectives 2 and 3) If a box of candy costs a retailer \(\$ 2.50\) and he wants to make a profit of \(50 \%\) based on the selling price, what price should he charge for the candy?

Step-by-Step Solution

Answer

The candy should be sold for $5.

Step 1: Understand the Problem

The problem requires us to determine the selling price of a box of candy that results in a 50% profit based on the selling price. We know the cost of one box is \( \$2.50 \).

Step 2: Define Variables

Let \( x \) be the selling price of the candy box. The profit is then \( x - 2.50 \).

Step 3: Set Up the Equation

Since the profit is 50% of the selling price \( x \), we can set up the equation for profit: \( x - 2.50 = 0.50x \).

Step 4: Simplify the Equation

To find \( x \), simplify the equation: \( x - 2.50 = 0.50x \). Subtract \( 0.50x \) from both sides to get: \( 0.50x = 2.50 \).

Step 5: Solve for the Selling Price

Divide both sides by 0.50: \( x = \frac{2.50}{0.50} \). This results in \( x = 5 \).

Step 6: Verify the Solution

Check that a selling price of \( \\(5 \) indeed results in a 50% profit based on the selling price. The profit would be \( \\)5 - \\(2.50 = \\)2.50 \). Since \( 0.5 \times 5 = 2.50 \), the calculation confirms a 50% profit.

Key Concepts

Profit Calculation

When calculating profit, it's essential to determine how much money is made over and above the cost of goods. In this exercise, the retailer pays $2.50 for a box of candy and wants a profit of 50% based on the selling price.
The basic formula for profit is:

• Profit = Selling Price - Cost Price

If the selling price provides a 50% profit, then the profit is said to be half of the selling price. This means for every $1.00 of selling price, 50 cents is profit.
By knowing this relationship, you can figure out the necessary selling price that makes a specific profit margin. This principle is crucial for businesses to ensure they earn enough on their sales to cover costs and make a profit.

Algebraic Equations

Algebraic equations are powerful tools for solving problems related to unknown quantities. In this problem, we use an equation to find the selling price of a candy box. The cost of the box is known, but the selling price isn't.
We start by assigning a variable, let's say \( x \), to the unknown selling price. The equation set up to find \( x \) was:

• \( x - 2.50 = 0.50x \)

This equation essentially says that the profit (\( x - 2.50 \)) is equal to 50% of the selling price. Solving such equations involves isolating \( x \) by performing arithmetic operations that help unveil the unknown. The final step involves simple division to reach the solution for \( x \).
Using algebra makes it easy to solve for variables systematically, allowing more complex problem-solving down the road.

Problem Solving

Solving math problems often involves a series of strategic steps. Here’s a general approach used in this exercise:

• **Understand the Problem:** Grasp what is being asked. In this exercise, it's about finding the right selling price for profit.
• **Define Variables:** Identify what is known and what needs to be discovered. Use variables for unknown elements, like the selling price.
• **Set Up Equations:** Use the relationships and information given to build equations. The equation expresses the problem mathematically.
• **Simplify and Solve the Equation:** Work through the equation to find the unknown. Use arithmetic operations to isolate the variable.
• **Verify the Solution:** Double-check to ensure the solution is correct, as was done by confirming the 50% profit in this exercise.

This method systematically approaches a problem, making it less overwhelming. By breaking it down into manageable parts, solutions become more attainable.