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 \( l = (3t + 2) / 4 \).