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Chapter 13: Problem 5

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

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

Step by step solution

01

Define the Variables

Let's denote the length of the shorter piece as \( s \) and the length of the longer piece as \( l \).
02

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.
03

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 \( l = 3 * \frac{t - 2}{4} + 2 = \frac{3 t - 6 + 8}{4} = \frac{3t + 2}{4} \).
04

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.

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