Question

A piece of twine with length of \(t\) is cut into two pieces. The length of the longer piece is 2 yards greater than 3 times the length of the shorter piece. Which of the following is the length, in yards, of the longer piece? a. $$\frac{t+3}{3}$$ b. $$\frac{3 t+2}{3}$$ c. $$\frac{t-2}{4}$$ d. $$\frac{3 t+4}{4}$$ e. $$\frac{3 t+2}{4}$$

Short answer

The length of the longer piece is \( \frac{3t + 2}{4} \).

Step-by-step solution

Define the Variables

Let's denote the length of the shorter piece as \( s \) and the length of the longer piece as \( l \).

Formulate the Equations

According to the problem, two equations can be created. First, \( s + l = t \), because the sum of the lengths of both the short piece and the long piece equals total length of the twine, which is \( t \). The second equation is \( l = 3s + 2 \), which is from the problem's relation that the length of the longer piece is 2 yards greater than 3 times the length of the shorter piece.

Solve for \( l \)

First, replace \( l \) in the first equation with the expression found for it in the second equation. So we substitute \( 3s + 2 \) for \( l \) in \( s + l = t \) and get \( s + 3s + 2 = t \). This simplifies to \( 4s + 2 = t \), or \( 4s = t - 2 \), and then to \( s = \frac{t - 2}{4} \). To find \( l \), we substitute \( s \) into the second equation, getting \( l = 3 * \frac{t - 2}{4} + 2 = \frac{3 t - 6 + 8}{4} = \frac{3t + 2}{4} \).

Match the Result with the Choices

Looking at the multiple choice options, it can be seen that answer option e. \( \frac{3t + 2}{4} \) is the formula calculated for the length of the longer piece.

Key concepts

algebraic equations

Algebraic equations are the building blocks of math problems, where relationships between variables are expressed in the form of equations. In the twine cutting math problem, we use algebraic equations to describe how the total length of twine is divided into two parts. The key is understanding and setting up these equations correctly.
Let's break it down further:

• An equation is a mathematical statement showing that two expressions are equal, an important tool for problem-solving.
• In this problem, we first identify the unknowns, here represented as variables—specifically, the lengths of the twine pieces.
• The equation \( s + l = t \) represents the sum of the two pieces equalling the total length \( t \).
• Another equation is formulated to show the relationship between these pieces: \( l = 3s + 2 \), indicating the longer piece is derived from manipulating the shorter piece's length.

Thus, setting up these equations correctly is pivotal for solving such math problems as it reflects the narrative of the problem in mathematical terms.

problem solving

Problem-solving in math involves carefully analyzing the situation, understanding the relationships between elements, and using systematic methods to reach a solution. Let's see how it applies to this twine problem:

• Break down the problem: Identify what you know and what you're trying to find. Here, we know the total length \( t \) and must find the longer piece.
• Formulate relationships: Use equations to describe the links between variables—in this case, how the shorter and longer twine pieces relate to each other and the overall length.
• Solve systematically: Substitute known data into your equations to simplify and gradually solve for unknowns.
• Verify your solution: Check that the solution aligns with all given conditions of the problem.

Each of these steps ensures a structured approach to tackle complex problems, leading to a successful and logical solution. Practice these steps and they'll become second nature, making even challenging math problems manageable.

twine cutting math problem

The twine cutting problem is about dividing a total length into parts with specific conditions linking them. Here, the focus is on understanding the constraints and relationships described in the problem.
Consider the problem setup:

• The total length of the twine, \( t \), needs to be divided such that one piece is longer, specifically, it is 2 yards more than three times the other piece.
• This condition is mathematically expressed as \( l = 3s + 2 \), where \( l \) is the longer piece.
• By formulating this condition as an equation, you demonstrate how algebra simplifies real-world situations.
• The key is substituting and rearranging these equations systematically to isolate the desired variable, providing the final answer.

With practice, problems like this become less about the specific numbers involved and more about understanding fundamental relationships, encouraging a deeper comprehension of algebra as a tool to solve real-world problems.