Problem 18 - Elementary Algebra

Question

A lumber company has contracted to cut boards into two pieces so that one piece is three times the length of the other piece. If a board is 12 feet long, what is the length of each piece after cutting?

Step-by-Step Solution

Answer

Answer: The lengths of the two pieces after cutting the 12 feet long board are 3 feet and 9 feet.

Step 1: Set up the equation

Let the length of the shorter piece be x, and the length of the longer piece be 3x. The equation representing the situation given is: x + 3x = 12

Step 2: Simplify and solve the equation

Combine like terms: 4x = 12 Now, divide by 4: x = 3

Step 3: Find the length of the longer piece

We know that the length of the longer piece is 3 times the shorter piece: 3x = 3 * 3 = 9

Step 4: State the lengths of the two pieces

The length of the shorter piece is x = 3 feet. The length of the longer piece is 3x = 9 feet. So, after cutting the.board it will have a 3 feet long piece and a 9 feet long piece.

Key Concepts

Linear Equations

Linear equations are mathematical statements that show the relationship between different quantities. These equations have variables, such as \( x \), and constants, and they form a straight line when graphed. In our example, we use a linear equation to find the lengths of the boards. The equation starts with \( x + 3x = 12 \).

Here, \( x \) represents the length of the shorter board piece, while \( 3x \) represents the longer piece, as it is three times the length of the shorter one. To determine their actual lengths, we solve this linear equation, simplifying it to \( 4x = 12 \) by combining \( x \) and \( 3x \). Further simplification by dividing both sides by 4 gives us \( x = 3 \). This reveals that the shorter piece is 3 feet long.

Linear equations are powerful tools because they provide a clear step-by-step approach to finding unknown values. By understanding and following these steps, we can solve a wide variety of real-world problems.

Problem-Solving

Problem-solving involves identifying a problem, breaking it down into manageable parts, and systematically finding a solution. The key steps often include understanding the problem, forming a plan, carrying it out, and evaluating the result. In our exercise, we begin by understanding the problem: splitting a 12-foot board into two pieces where one is three times the length of the other.

Next, forming a plan translates into setting up a mathematical equation—using variables to represent the unknown quantities. This becomes the linear equation \( x + 3x = 12 \).

Certainly, different methods exist to strategize solutions. However, breaking down the equation, simplifying, and solving systematically makes the plan solid. Finally, evaluating results helps confirm that the solution is correct. By checking that \( x = 3 \), and recognizing \( 3x = 9 \), our initial understanding of the problem holds true.

Problem-solving skills are crucial not only in mathematics but also in everyday situations. They help improve analytical thinking and the ability to tackle diverse challenges.

Proportion

Proportion involves the relationship between quantities where one quantity is a multiple of the other. In our board-cutting scenario, the concept of proportion tells us that the length of the longer piece should be three times that of the shorter piece.

By setting \( x \) as the length of the shorter piece and \( 3x \) for the longer piece, we create a proportionate relationship described by \( x:3x \). This proportion means that as long as the relationship \( 1:3 \) remains constant, the specific values of \( x \) can be determined with the total length equation.

Knowing proportions helps us understand how quantities relate to each other and maintain balance. This knowledge is widely applicable, for example, in cooking recipes, scale modeling, and mixing substances in chemistry. Proportional reasoning makes it easier to predict and adjust the outcomes in various fields and practical scenarios.