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 \( (3t + 2) / 4 \), meaning the correct answer is option d.

Step-by-step solution

Set up the equation

Let the longer piece be \( l \) and the shorter piece be \( s \). From the problem, we know that the total length of the twine is \( t \). So, we set up the equation as \( l + s = t \). Moreover, we know that the length of the longer piece is 2 yards greater than 3 times the length of the shorter piece, so \( l = 3s + 2 \).

Substitute \( l \) into the equation

Substitute the second equation (\( l = 3s + 2 \)) into the first equation (\( l + s = t \)) to express everything in terms of \( s \). Hence, it gives \( 3s + 2 + s = t \). Simplifying the equation gives \( 4s + 2 = t \).

Solve for \( l \)

Substitute \( s = (t - 2) / 4 \) into the equation \( l = 3s + 2 \). Simplifying the expression gives \( l = (3t + 2) / 4 \).

Key concepts

Algebraic Equations

Algebraic equations are a way to express relationships between different variables. In this problem, we've used algebraic equations to relate the lengths of two pieces of twine.
The pieces have lengths denoted by \( l \) and \( s \), with a total length of \( t \). These equations help us find the length of the longer piece by understanding how one piece is related to the other.
An algebraic equation can be simple, like \( a + b = c \), or complex with multiple variables and operations. Here, using equations simplifies working with various conditions given in the problem, and it's an invaluable tool for solving mathematical queries like these.

Equation Solving Steps

Following methodical steps to solve equations ensures we don't miss any minute detail in calculations, which is crucial for problems like these.
Initially, identifying what our variables stand for is essential. In this case, \( l \) and \( s \) are the longer and shorter piece of twine, respectively.

• We start by writing equations from the given information, i.e., \( l + s = t \) and \( l = 3s + 2 \).
• Next, we substitute the second equation into the first to consolidate into a single variable, allowing us to express one unknown in terms of the other.
• After substituting, we end up with \( 4s + 2 = t \). Further manipulation and substitution help us find \( l = (3t + 2) / 4 \).

These steps help break down the problem into manageable parts until we reach our desired result.

Mathematical Reasoning

Mathematical reasoning plays a crucial role in solving algebraic problems. It's about logical thinking and understanding relationships between different entities rather than just applying formulas.
In this scenario, reasoning begins by interpreting the word problem and organizing the given information into mathematical terms. We know that the longer piece is described as being 2 yards more than three times the shorter. This requires understanding and transforming this statement into algebraic language.
Practicing reasoning helps in making connections between concepts which results in an enhanced ability to formulate solutions. It prepares you to tackle problems not just systematically but creatively.