Problem 19 - Algebra 2 and Trigonometry

Question

In \(18-23,\) write and solve an equation or an inequality to solve the problem. In an algebra class, 3 students are working on a special project and the remaining students are working in groups of five. If there are 18 students in class, how many groups of five are there?

Step-by-Step Solution

Answer

There are 3 groups of five students.

Step 1: Define Variables

Let's define the variables that will be used in the equation. Let \( x \) represent the number of groups of five students.

Step 2: Create an Equation

We know there are 18 students in total, and 3 students are working on a special project. The remaining students, \(18 - 3\), should work in groups of five. This gives us the equation: \(5x = 18 - 3\).

Step 3: Simplify the Equation

First, simplify the right side of the equation: \(18 - 3\) equals 15. So, the equation becomes \(5x = 15\).

Step 4: Solve for x

To find \( x \), divide both sides of the equation by 5: \(x = \frac{15}{5}\).

Step 5: Calculate the Solution

Perform the division: \(x = 3\). This is the number of groups of five students.

Key Concepts

Equations

In algebra, an equation is a mathematical statement that asserts the equality of two expressions. An equation consists of two sides separated by an equal sign. The challenge is to find the values of unknowns, or variables, that make the equation true. For example, in this exercise, we created the equation \(5x = 18 - 3\).
This equation states that the expression on the left, which involves the variable \(x\), is equal to the simplified result of the expression on the right.
Solving an equation involves finding the value of the variable(s) that satisfy the equation, and it typically involves a series of logical steps such as adding, subtracting, or dividing both sides by the same number.
When you have an equation, you're essentially balancing both sides to understand how components relate to one another.

Variables

Variables are symbols used in mathematics to represent unknown or changeable values. They are like placeholders for numbers that can vary. In algebra, we often use letters like \(x\), \(y\), or \(z\).
In our problem, we defined \(x\) to represent the number of groups of five students. By introducing a variable, we simplify complex problems into manageable parts, making equations easier to solve.
Variables allow us to set up equations and inequalities to model real-world problems. They help in translating everyday situations into mathematical language, which can then be solved systematically.
In this context, determining the value of \(x\) gives us insight into how students are grouped in the class.

Grouping in Mathematics

Grouping in mathematics involves organizing elements into sets based on specific criteria. In this problem, grouping meant organizing students into smaller groups for a project or activity.
The equation we derived, \(5x = 15\), indicated that students were organized into groups of five after accounting for the three students working on a different project.
By grouping, you simplify complex information, allowing for easier analysis and problem-solving. It's an efficient way to manage and break down larger sets of data.
This technique is important in various mathematical disciplines, including probability, statistics, and algebra, as it provides insights into patterns and relationships within a dataset.