Problem 52 - Intermediate Algebra

Question

The number of women in a Statistics class is 8 more than twice the number of men. Let \(m\) represent the number of men. Write an expression for the number of women.

Step-by-Step Solution

Answer

The expression for the number of women is \(2m + 8\).

Step 1 - Identify the Variables

Represent the number of men in the class with the variable \(m\). This indicates how many men there are in the Statistics class.

Step 2 - Understand the Relationship

According to the problem, the number of women in the class is 8 more than twice the number of men. This relationship helps to form the expression.

Step 3 - Construct the Expression

To express the number of women as an algebraic expression, take twice the number of men, \(2m\), and then add 8: \(2m + 8\).

Key Concepts

Variables

In algebra, variables are symbols used to represent unknown values. They help us describe and solve real-life problems through mathematical expressions. In this exercise, we use the variable \( m \) to represent the number of men in a Statistics class. It's important to choose a letter that makes sense and clearly reflects what it stands for. Here, \( m \) is chosen because it corresponds to 'men', making it easier to remember.
Variables are flexible and can be manipulated through various operations like addition, subtraction, multiplication, and division. Identifying the right variable is the first step towards solving any algebraic problem.

Linear Relationships

A linear relationship is a type of relationship that can be represented by a straight line on a graph. In this exercise, we see a linear relationship between the number of men and women in the class.
The problem states that the number of women is 8 more than twice the number of men. This relationship can be mathematically represented as an equation: \( 2m + 8 \).
The 'twice the number of men' part means we multiply the variable \( m \) by 2, resulting in \( 2m \). Then, we add 8 to account for the additional 8 women. Such relationships are straightforward to graph and solve, making them a fundamental concept in algebra.

Problem-Solving

Problem-solving in algebra involves breaking down a given problem into manageable steps. Here’s how we solved the exercise:

• Step 1: Identify the variable \QA_1m\ representing the number of men.
• Step 2: Understand the relationship given in the problem. The number of women is described as 8 more than twice the number of men.
• Step 3: Construct the expression \(2m+8\) to represent the number of women.

Let's revisit these steps to deepen our understanding. First, by identifying the variable, we clearly define what needs to be found. Then, understanding the relationship helps us form the correct mathematical model. Finally, constructing the expression allows us to solve for the desired value quickly.
Problem-solving is a critical skill in algebra, requiring clear thinking and structured approaches. Always follow these steps to simplify and solve complex problems.