Problem 4 - College Algebra

Question

If a man sawed a \(10-f\) board into two sections and one section was \(n \mathrm{ft}\) long, how long would the other section be in terms of \(n\) ?

Step-by-Step Solution

Answer

10 - n

Step 1: Understanding the Problem

We are given a board that is 10 feet long, which is split into two parts. One part is "n" feet long. We need to find the length of the other section in terms of n.

Step 2: Setting Up the Equation

Since the total length of the board is 10 feet and it is divided into two sections, the equation relating the lengths of the two pieces is: length of the first section (n) + length of the second section (unknown) = 10.

Step 3: Solving the Equation

Let's denote the length of the second section as "x" feet. So, the equation becomes: \[ n + x = 10 \]To find x, subtract n from both sides of the equation:\[ x = 10 - n \]

Step 4: Conclusion

The length of the other section of the board, in terms of n, is expressed as 10 - n.

Key Concepts

Equations

Equations serve as foundational tools in algebra to express relationships between different quantities. In the context of slicing a board, the equation helps us determine the length of a segment when we know the length of another. Imagine we have a board that is 10 feet long in total. We measure one part of it as \( n \) feet. To find out how long the other part is, we establish an equation like this: \( n + x = 10 \).
This equation indicates that the sum of the two sections (one known and one unknown) equals the entire length of the board, which is 10 feet.
Using equations allows us to remove the unknown and solve for it logically. By rearranging the equation, we isolate the unknown variable \( x \) to find its value. Knowing how to set up this kind of equation can simplify even more complex problems later on.

Problem Solving

Problem solving in algebra often involves a series of logical steps that lead to the solution. Here, we are faced with a straightforward problem: dividing a board into two sections. We know the length of one section, \( n \) feet, and we want to find the other section's length.
The approach to solving this problem involves clear steps:

• Understand the problem: Identify what is given and what needs to be found.
• Set up an equation to represent the relationship between known and unknown quantities.
• Solve the equation using algebraic techniques.
• Verify the solution to ensure it's logically sound and fits all given details.

This structured approach not only helps in algebra but is a useful strategy for tackling a variety of problems in mathematics and beyond. Each step builds confidence and enhances critical thinking skills.

Variables

Variables are symbols, often letters, used in math to represent unknown or changing quantities. In the board-cutting example, the variable \( n \) represents the known length of one section, while \( x \) is the unknown length of the other section.
Using variables simplifies complex expressions and calculations, allowing us to write general equations like \( n + x = 10 \). This is more efficient than trying to express every possibility in words.
Variables are essential in any equation, as they allow for flexibility and adaptability in approaching problems. Understanding variables helps you translate real-world situations into mathematical models, vastly expanding your ability to solve diverse problems.