Problem 47 - Elementary Algebra Within Reach

Question

Two types of tickets are to be sold for a concert. One type costs \(\$ 15\) per ticket and the other type costs \(\$ 25\) per ticket. The promoter of the concert must sell at least 15,000 tickets, including at least 8000 of the \(\$ 15\) tickets and at least 4000 of the \(\$ 25\) tickets. Moreover, the gross receipts must total at least \(\$ 275,000\) in order for the concert to be held. Write a system of linear inequalities that describes the different numbers of tickets that can be sold.

Step-by-Step Solution

Answer

The system of inequalities representing the number of tickets that can be sold is \(x \geq 8000\), \(y \geq 4000\), \(x + y \geq 15000\), and \(15x + 25y \geq 275000\), where \(x\) represents the number of $15 tickets and \(y\) represents the number of $25 tickets.

Step 1: Define the Variables

Let's denote by \(x\) the number of tickets sold for $15 and by \(y\) the number of tickets sold for $25.

Step 2: Formulate the inequalities from the conditions

Given that at least 15,000 tickets must be sold and this includes at least 8000 of the $15 tickets and at least 4000 of the $25 tickets, we get the inequalities: \(x \geq 8000\), \(y \geq 4000\), and \(x + y \geq 15000\). The total receipts from selling these tickets should be at least $275,000, giving the inequality: \(15x + 25y \geq 275000\).

Step 3: Summarize the system of inequalities

The system of inequalities that describes the different numbers of tickets that can be sold is: \(x \geq 8000\), \(y \geq 4000\), \(x + y \geq 15000\), and \(15x + 25y \geq 275000\).

Key Concepts

Algebraic Formulation

The challenge of translating a real-world scenario into mathematical terms is at the heart of algebraic formulation. In our exercise, this involves defining variables to represent different types of concert tickets and setting up inequalities to reflect the constraints on ticket sales.

For instance, by representing the number of \(15 tickets as variable x and the number of \)25 tickets as variable y, we establish a basis for our mathematical model. The inequalities—x ≥ 8000, y ≥ 4000, and x + y ≥ 15000—translate the conditions into a solvable system, expressing the minimum ticket numbers required. Lastly, the inequality 15x + 25y ≥ 275000 encapsulates the financial goal for the concert.

Understanding the importance of choosing appropriate variables and accurately representing contractual stipulations is an essential learning outcome from this algebraic formulation.

Inequality Problem

Inequality problems deal with finding solutions that satisfy certain conditions, rather than specific values. In our ticket sales example, the inequalities define a range of acceptable solutions, each of which represents a feasible ticket sales strategy.

The inequalities x ≥ 8000 and y ≥ 4000 ensure that both ticket types meet a minimum sales quota, while x + y ≥ 15000 guarantees the overall ticket sales threshold. 15x + 25y ≥ 275000 ensures the monetary target is achieved. Students must understand that the solution to an inequality problem is a set of values, represented graphically as a region, that satisfy all conditions, illustrating the practical nature of inequality problems in various disciplines, including economics, business analytics, and event planning.

Ticket Sales Analysis

Ticket sales analysis involves the use of algebraic inequalities to forecast sales and revenue to ensure the feasibility of an event such as our concert. By constructing a set of inequalities based on minimum ticket sales and target revenue, analysts can determine which combinations of ticket sales will meet the concert's financial goals.

With a system of linear inequalities, we can also determine a safe margin for sales, ensuring profitability and avoiding potential losses. The role of algebra comes into sharp focus when adjusting different variables to simulate various outcomes. Through analyzing solutions that fulfill these inequalities, decision-makers can plan more effectively and minimize financial risk.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operations. In the context of our concert problem, expressions like 15x and 25y symbolize the total revenue from each ticket type, while x + y represents the total number of tickets sold.

These expressions become part of larger equations or inequalities that define the relationships between variables in accordance with the given conditions. Students should recognize how to manipulate these expressions, deal with like terms, and understand their role within an equation or inequality. Mastering algebraic expressions is crucial for solving more complex problems in algebra and beyond.