Problem 57 - Elementary Algebra Within Reach

Question

In Exercises \(55-58\), write an algebraic equation. Do not solve the equation. A high school earned \(\$ 986\) in revenue for a play. Tickets for the play cost \(\$ 10\) for adults and \(\$ 6\) for students. The number of students attending the play was \(\frac{3}{4}\) the number of adults attending the play. How many adults and student attended the play?

Step-by-Step Solution

Answer

The algebraic equation that represents the problem is: \(10x + 6(\frac{3}{4}x) = 986\). Here, \(x\) represents the number of adults who attended the play.

Step 1: Defining Variables

Firstly, define variable \(x\) to represent the number of adults who attended the play, and \(\frac{3}{4}x\) to represent the number of students. Now, \(x\) and \(\frac{3}{4}x\) represents number of adults and students respectively.

Step 2: Creating the Equation

Then, an equation can be created from the given information: revenue equals to number of adults times price per adult ticket plus number of students times price per student ticket. Therefore, equation becomes: \(10x + 6(\frac{3}{4}x) = 986\).

Key Concepts

Variable Representation

Understanding variable representation is essential when translating real-life situations into algebraic language. In algebra, variables are symbols that stand in for unknown values. They can represent numbers, quantities, or even whole expressions. Let’s look at the example of the high school play revenue, where tickets have different prices based on whether the attendees are adults or students.

First, we decide on a variable to represent the number of adult attendees. Here, we chose variable \(x\). This choice is arbitrary, but once selected, it should be consistently used throughout the problem. Then, to represent the students, whose number is tied to that of the adults, we used the expression \(\frac{3}{4}x\). This fraction directly applies the given relationship between the number of students and adults.

In summary, good variable representation follows these guidelines:

• Choose a symbol that does not conflict with other standard notations or given values.
• Be consistent with the chosen symbols throughout the problem.
• Clearly define what each variable represents before creating your equations.

Creating Equations from Word Problems

Creating equations from word problems is a skill that combines reading comprehension with algebraic reasoning. To start, carefully read and identify the key components of the problem: quantities, relationships, and the unknowns that need solving.

In the case of the high school play, the key pieces of information are the revenue collected, the price of tickets, and the relationship between the number of student and adult attendees. With this information, an equation can be constructed that models the situation. Here's the thought process we follow:

• Identify the total revenue given (\$986), which is the target value our equation will equal to.
• Note the prices of tickets for adults (\$10) and students (\$6).
• Understand the relationship between the number of students and adults, which allows us to express the number of students as \(\frac{3}{4}x\), where \(x\) is the number of adults.

The final equation is derived by multiplying the number of each type of ticket by its respective price and summing them up to match the total revenue: \(10x + 6(\frac{3}{4}x) = 986\). This equation incorporates all the given information in an algebraic form.

Revenue Calculation in Algebra

Revenue calculation in algebra is an application of linear equations in real-world scenarios. These problems often involve finding out how much money is made by selling goods or services at given prices.

In our high school play example, calculating revenue involves setting up an equation to match the total income from ticket sales. The total revenue equation is formed by adding the product of the number of adult tickets sold and the price per adult ticket to the product of the number of student tickets sold and the price per student ticket.

Here is a breakdown of the revenue calculation process:

• Multiply the number of adult tickets (represented by \(x\)) by the price per adult ticket (\$10).
• Multiply the number of student tickets (represented by \(\frac{3}{4}x\)) by the price per student ticket (\$6).
• Add these two products together to set up the equation for total revenue.

This yields \(10x + 6(\frac{3}{4}x) = 986\), which is the algebraic representation of the revenue calculation for this scenario. This equation encapsulates all the details needed to understand how the total revenue is affected by the number of tickets sold to adults and students.