Problem 64 - Elementary and Intermediate Algebra

Question

Answer with an algebraic expression. See Example 4. A caravan of \(b\) cars, each carrying 5 people, traveled to the state capital for a political rally. How many people were in the caravan?

Step-by-Step Solution

Answer

The expression is \(5b\).

Step 1: Understanding the Problem

First, we need to understand the problem. We know that there is a caravan consisting of multiple cars and each car carries a certain number of people. Our task is to determine the total number of people in the caravan.

Step 2: Identify Variables

Let's identify the variables given in the problem. We have: - The number of cars in the caravan as \(b\).- Each car carries 5 people.

Step 3: Calculate Total People

To find the total number of people in the caravan, multiply the number of cars \(b\) by the number of people each car carries. Therefore, the total number of people is calculated as: \[ ext{Total People} = 5 \times b \]

Step 4: Writing the Expression

The algebraic expression that represents the total number of people in the caravan is \(5b\). This expression tells us that if we know the number of cars \(b\), we can find the total number of people by multiplying \(b\) by 5.

Key Concepts

Variables in Algebra

In algebra, variables act as placeholders for numbers, helping us represent general relationships and rules in math. They are usually denoted by letters like \(b\). In the context of the exercise, the variable \(b\) stands for the number of cars in the caravan. Variables offer a flexible way to solve problems because they can adapt to different scenarios and values.

With variables, we move away from specific numbers and create general formulas that work in a variety of situations. This makes algebra incredibly powerful and versatile in different fields such as science, engineering, and everyday problem-solving.

• Variables can be any letter, but common ones include \(x\), \(y\), and \(b\).
• They allow us to express general solutions that are applicable in many situations.

Understanding variables is foundational to learning algebra and helps in forming equations that can model real-world situations.

Multiplicative Relationships

Multiplicative relationships are all about scaling quantities using multiplication. In our exercise, we see this with each car carrying 5 people. The relationship between the cars and the total number of people is shown through multiplication: if you know how many cars there are, you multiply by how many people each car carries to find the total.

This type of relationship is key to solving many algebra problems because it shows how one quantity affects another. It's essential for understanding how quantities grow and change.

• By multiplying, we scale one quantity by another.
• Multiplicative relationships often appear in real-world contexts like speed, density, and pricing.

Using multiplication helps simplify complex real-world situations by breaking them down into manageable mathematical expressions.

Problem-Solving with Algebra

Algebra is a tool that allows us to systematically solve problems by turning words into equations. In our problem, we're asked to find the total number of people in a caravan. We used algebra to express this situation as an equation \(5b\), representing the total number of people.

Solving problems with algebra often involves:

• Understanding and identifying the relationship between different parts of the problem.
• Deciding on the variables to represent the unknowns.
• Constructing equations and expressions based on these relationships.

Algebra provides a structured way to tackle problems by setting up models that represent reality. This approach enables anyone to methodically solve for unknowns, making it a fundamental skill in mathematics and beyond.