Total length of the twine = t

longer piece : l

shorter piece= t-l

Given , l=2+3(t-l)

by solving , t= (3t+2)/4

Ans E

I Algebra

\(t=\) length of twine
\(S=\) length of shorter piece
\(L=3S + 2 =\) length of longer piece
\(L - 2 = 3S\)
\(\frac{L-2}{3}=S\)

\(L+S=t\)
\(L + \frac{(L-2)}{3}=t\)
\(3L +(L-2)=3t\)
\(4L=3t+2\)
\(L=\frac{(3t+2)}{4}\)

Answer E

II. Assign values
Let smaller length, S = 1 yard
Longer length, L = (3S + 2) = (3 + 2) = 5 yds
Twine length, t = (S + L) = (1 + 5) = 6 yds

Using t=6, find the answer that yields L = 5

(A) \(\frac{(t+3)}{3}=\frac{(5+3)}{3}=\frac{8}{3}\) NO

(B) \(\frac{(3t+2)}{3}=\frac{(3*6)+2}{3}=\frac{20}{3}=6.xx\) NO

(C) \(\frac{(t-2)}{4}=\frac{(6-2)}{4}=1\) NO

(D) \(\frac{(3t+4)}{4}=\frac{(3*6)+4}{4}=\frac{22}{4}=5.5\) NO

(E) \(\frac{(3t+2)}{4}=\frac{(3*6)+2}{4}=\frac{20}{4}=5\) YES

Answer E
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First algebraically, SP= shorter piece, LP=longer piece.
3(SP)=LP-2yards
LP+SP=t (the variable t is given by the author).
We need to show LP in terms of t variable.
If LP+SP=t, therefore SP=t-LP. So now we can plug this into our above equation.
3(t-LP)=LP-2yards --> 3t-3LP=LP-2 --> 3t=4LP-2 --> 4LP=3t+2 --> LP= 3t+2/4
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A piece of twine of length t is cut into two pieces.
Let x = the length of the LONGER piece in yards
So, t - x = the length of the SHORTER piece in yards

The length of the longer piece is 2 yards greater than 3 times the length of the shorter piece.
In other words: (longer piece) = 3(shorter piece) + 2
In other words: x = 3(t - x) + 2

Which of the following is the length, in yards, of the longer piece?
So, we must solve for x
Take: x = 3(t - x) + 2
Expand right side: x = 3t - 3x + 2
Add 3x to both sides: 4x = 3t + 2
Divide both sides by 4 to get: \(x = \frac{3t + 2}{4}\)

Answer: E
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We can create the equation:

L = 2 + 3S

and

L + S = t

S = t - L

Substituting, we have:

L = 2 + 3(t - L)

L = 2 + 3t - 3L

4L = 2 + 3t

L = (2 + 3t)/4

Answer: E
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Whenever you see variables in options, try to plug in.

Say length of shorter piece is 1. Then length of longer piece is 3*1 + 2 = 5
And total length t = 1 + 5 = 6

Now put t = 6 in the options and see which one gives you 5.

Only (E) does.
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