Problem 20 - Elementary Algebra Within Reach

Question

Ticket Sales Ticket sales for a play totaled \(\$ 1700\). The number of tickets sold to adults was three times the number sold to children. The prices of the tickets for adults and children were \(\$ 5\) and \(\$ 2\), respectively. How many of each type were sold?

Step-by-Step Solution

Answer

The number of adult tickets sold was 300 and child tickets sold was 100.

Step 1: Setup the equation

According to the given exercise, the total sales amount to \$1700 with adult tickets costing \$5 each and child tickets at \$2 each. This gives the first equation as: 5a + 2c = 1700\n\nAdditionally, it's mentioned that the number of adult tickets sold was three times the child tickets, resulting in the second equation: a = 3c

Step 2: Substitute Equation

Substitute the second equation into the first to eliminate one variable. Substituting a = 3c into the first equation results in: 5(3c) + 2c = 1700 which simplifies to: 15c + 2c = 1700

Step 3: Solve for c

The previous step results in a simple linear equation that can be solved by adding the c terms together then dividing both sides by that sum: 17c = 1700\nAfter dividing both sides by 17, we get c = 100

Step 4: Find the value of a

Using the solution for c, substitute c = 100 into equation 2 to find the value for a: a = 3(100) which simplifies to a = 300

Key Concepts

Understanding Word Problems

Word problems are story-like problems that describe real-life situations using equations. They challenge us to identify the key information given and translate it into mathematical language. In this exercise, the context is a ticket sale event. The total revenue from selling adult and child tickets helps in forming equations based on given conditions.

For instance, knowing that adult tickets cost \( \\(5 \) each and child tickets cost \( \\)2 \) each is essential. Moreover, realizing that the number of adult tickets sold is three times the number of child tickets provides the relationships needed to form equations.
- Identify the quantities involved (e.g., number of adult and child tickets).- Determine the relationships between these quantities (e.g., thrice as many adult tickets as child tickets).- Formulate equations that express these relationships mathematically.By breaking word problems into these parts, you simplify complex scenarios into manageable mathematical tasks.

Linear Equations and Their Role

Linear equations are equations where the highest power of the variable is one. They are straightforward to solve and provide the backbone for solving more complex problems. In this problem, linear equations are used to represent the relationships between variables.

The first equation, \( 5a + 2c = 1700 \), uses the ticket prices as coefficients, while the second equation, \( a = 3c \), conveys the relationship between the number of tickets sold.

• Linear equations have a clear structure: each variable is raised to the first power.
• Variables can be represented by any alphabetic letter, symbolizing unknown quantities to be found.
• Linear equations can represent many relationships, such as price, quantity, or rate problems.

Linear equations are particularly useful in representing real-world word problems, as demonstrated in this situation.

Substitution Method Simplified

The substitution method is a powerful technique used to solve systems of equations. By solving one equation for one variable, you can substitute that expression into another equation, reducing the number of variables. This method is particularly useful when dealing with two variables.

Here, we start with the relationship \( a = 3c \) and substitute this expression into the price equation \( 5a + 2c = 1700 \). This allows us to express everything in terms of \( c \) and solve:- Substitute \( a = 3c \) into \( 5a + 2c = 1700 \).- This simplifies to \( 15c + 2c = 1700 \), which further reduces to \( 17c = 1700 \).- Solving for \( c \), we find \( c = 100 \).Then, we use \( c = 100 \) to find \( a \) by substituting back, leading us to \( a = 300 \).
The substitution method systematically reduces variables and offers a clear path to find the solution of two interrelated equations.