Problem 35 - Prealgebra

Question

The booster club at Jefferson High School sells football programs for \(\$ 1\) each. The costs to make the programs are \(\$ 60\) for page layout plus \(\$ 0.20\) for printing each program. If they print 400 programs, how many programs must the Club sell to make at least \(\$ 200\) profit?

Step-by-Step Solution

Answer

Sell at least 340 programs.

Step 1: Determine Total Cost

Calculate the total cost of producing 400 programs. This includes the layout cost and the printing cost per program.\[\text{Total Cost} = 60 + (0.20 \times 400) = 60 + 80 = 140\]

Step 2: Set Up Profit Equation

Write an equation representing the profit. Profit is defined as total revenue minus total cost. Revenue is generated from selling programs at $1 each.\[\text{Profit} = \text{Revenue} - \text{Cost} = 1x - 140\] where \(x\) is the number of programs sold.

Step 3: Solve for Required Sales

We need a profit of at least \(\$200\). Therefore, solve the inequality:\[1x - 140 \geq 200\]Add 140 to both sides:\[x \geq 340\]

Step 4: Conclusion

The Club must sell at least 340 programs to achieve a minimum profit of \(\$200\).

Key Concepts

Cost Analysis in Profit Calculation

Whenever a business or an organization wants to calculate profits, a thorough cost analysis is essential. This involves breaking down all the expenses associated with producing an item or service. In our example, the Jefferson High School's booster club needs to understand its costs to determine how many programs they need to sell to make a desired profit.
The costs can be categorized as fixed costs and variable costs. Fixed costs, such as the \\(60\\) for page layout, do not change with the number of programs produced. On the other hand, variable costs, which are \\(0.20\\) per program for printing, rise with the production quantity.
To calculate the total cost, add the fixed and variable costs together. For 400 programs, the total cost is calculated as follows:\[\text{Total Cost} = \text{Fixed Cost} + \text{Variable Cost} \times \text{Number of Programs} = 60 + (0.20 \times 400) = 140\]By understanding these costs, the club can make better financial decisions and set realistic sales targets.

Using Inequalities in Business Calculations

Inequalities are a crucial mathematical tool used in business to set and understand constraints and goals. In the context of the exercise, the inequality helps Jefferson High School's booster club to determine the minimum number of programs they need to sell to achieve at least a certain profit.
The inequality represents the relationship between revenue, cost, and profit. The club wants a profit of at least \\(200.\\) By substituting the profit equation into an inequality, you get: \[1x - 140 \geq 200\] Here, \\(1x\\) represents the revenue from selling \(x\) programs, and \\(140\\) is the total cost.
Solving inequalities involves finding the range of values that satisfy the condition. In this case, the solution shows that the club must sell at least 340 programs: \[x \geq 340\] This means they need to clear this threshold of program sales to meet or exceed their profit goal.

Effective Problem-Solving Steps

Solving a problem efficiently requires following a structured approach, which often involves several steps. In the Jefferson High School booster club's scenario, the steps help break down the profit calculation problem into manageable parts.
- **Step 1: Calculate Costs** - Start by determining the total cost involved, by adding fixed and variable costs. - **Step 2: Formulate Profit Equation** - Next, express the financial goal in terms of an equation. Here, the profit is revenue minus cost. - **Step 3: Set Up and Solve Inequality** - Use inequalities to encapsulate the objective, for example, earning at least the desired profit. - **Step 4: Interpretation** - Finally, interpret the solution to understand what actions are necessary. Following these steps ensures that complex problems become more straightforward, allowing one to develop a methodical approach to reaching desired outcomes. By using these problem-solving steps, you can address various issues in both personal and professional contexts.