Problem 8 - Elementary Algebra

Question

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Step-by-Step Solution

Answer

There are 3 adult leaders in Pack 645.

Step 1: Understand the problem

Identify the key information and what needs to be found. We have 18 Cub Scouts and need to find the number of adult leaders. The number of scouts is three more than five times the number of adult leaders.

Step 2: Define a variable

Let the number of adult leaders be represented by the variable, say, Let \( x \) be the number of adult leaders.

Step 3: Set up an equation

Translate the word problem into an equation. Given that the number of Cub Scouts is three more than five times the number of adult leaders, we write:\( 18 = 5x + 3 \)

Step 4: Solve the equation

Isolate the variable \( x \) by performing algebraic operations:\( 18 - 3 = 5x \)\( 15 = 5x \)\( x = 3 \)

Step 5: Write the answer in a complete sentence

There are 3 adult leaders in Pack 645.

Key Concepts

linear equations

Linear equations are at the heart of many word problems in algebra, and they're essential for solving this particular problem. A linear equation is a type of equation where the highest power of the variable is one. This means the graph of the equation is always a straight line. In our problem, we translated the relationship between the number of Cub Scouts and adult leaders into the linear equation: \( 18 = 5x + 3 \) Here, \( x \) represents the number of adult leaders. It's a simple linear equation because the variable \( x \) is raised to the power of 1. By solving this equation, we can find the exact number of adult leaders needed to match the Cub Scouts.

variable representation

Representing quantities with variables makes it easier to handle complex relationships in word problems. In our problem, we used the variable \( x \) to stand for the number of adult leaders. This step is crucial as it turns a wordy problem into a mathematical expression that is much easier to manipulate. By letting \( x \) be the number of adult leaders, we could clearly state the relationship between the number of Cub Scouts and adult leaders as \( 18 = 5x + 3 \). This equation translates the words into a precise mathematical form, paving the way for straightforward problem-solving.

problem-solving strategy

Using a structured problem-solving strategy simplifies the process of tackling algebra word problems. Here are the steps we followed in this problem:

• Understand the problem: Identify what is known and what needs to be found.
• Define a variable: Assign a variable to the unknown quantity.
• Set up an equation: Translate the word problem into a mathematical equation.
• Solve the equation: Use algebraic operations to find the value of the variable.
• Write the answer: State the solution in a complete sentence.

This clear, step-by-step approach avoids confusion and ensures a logical progression towards the solution.

algebraic operations

Algebraic operations are the tools we use to manipulate equations and solve for variables. In the context of our problem, we used simple arithmetic operations to isolate the variable and solve the equation:

• First, we subtracted 3 from both sides of the equation \( 18 = 5x + 3 \) to get \( 15 = 5x \).
• Then, we divided both sides by 5 to solve for \( x \), yielding \( x = 3 \).

These steps highlight basic algebraic principles such as maintaining equality and isolating the variable. Understanding and applying these operations is crucial for solving any algebra problem efficiently.