In this type I prefer to use number; however there are other ways to solve it -

Let
B = smaller side = 3

A = Larger side = 1 + (2*3) = 7

So the total length = 10.

Put the value of L in the option and we need 7 as an answer. Only option E yields the correct answer.

It is also advisable to start substituting the number L from option C as we will come to know whether we need to proceed towards option A or to option E.
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I was able to solve this using numbers, but it didn't work out using algebraic expressions. Can someone find out why? [quantity][/quantity]

Let's split up a line and translate the sentence given into an equation. Then the line can be broken up to a piece of length 2x+1 and a piece of length x. Then 3x+1 = L. Solving for L we get

x = (L-1)/3

Therefore, the length of the longer piece by substitution is 2x+1 = 2(l-1)/3

But this does not fit any of the answer choices. Please help. Thank you in advance.

We can let b= the length of piece B and (2b - 1) = the length of piece A and create the equation:

b + 2b - 1 = L

3b - 1 = L

3b = L + 1

b = (L + 1)/3

So the length of piece A is:

2(L + 1)/3 - 1

(2L + 2)/3 - 1

(2L + 2)/3 - 3/3 = (2L - 1)/3

Answer: E
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My approach for this exercise would be to draw it out.

L/3.........L/3................L/3
..........I...............I....................I

now I imagine we divide it into 3 pieces, since we are given the information that the length of one piece is twice the other.

Following this it gets really easy.-->we know that the longer one is twice the other, thereby 2*(L/3)-->2L/3 and since we know that we have +1-->(2L+1)/3

This problem can be solved algebraically.

The length of the board = L

It is cut into two pieces, one smaller and another larger.

Lets call the smaller piece S.

Now we know that the length of the longer piece is 2.S + 1.

But the choices are all representing the length of the longer piece in terms of L, the total length of the board.

So we need an equation relating S to L so that 2.S + 1 can be represented in terms of L, the total length, instead of the smaller piece S.

Since Length of the board = Length of the Smaller Piece + Length of the Larger Piece

L = S + 2.S + 1

L = 3.S + 1

3.S + 1 = L

3.S = L - 1

S = (L - 1)/3

Now we substitute the above equation into 2.S + 1, which is the length of the longer piece in terms of the shorter piece.

=> 2.[(L - 1)/3] + 1

= 2.[(L-1) + 3]/3

= (2.L - 2 + 3)/3

= (2.L + 1)/3

Answer is E.
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Plugging in numbers can lead us astray here.

Let's assume that the longer and shorter sides are A and B, respectively.

Let B = 1.
Let A = 2B + 1 = 2(1) + 1 = 2 + 1 = 3.

So, the total length is 3 + 1 = 4.

The correct choice should evaluate to produce the length of A.

(A) (4+2)/2 = 6/2 = 3 This equals the length of A.

(B) ((2*4) + 1)/2 = (8+1)/2 = 9/2 = 4.5

(C) (4 - 1)/3 = 3/3 = 1

(D) (2*4+3)/3 = (8+3)/3 = 11/3

(E) (2*4+1)/3 = (8+1)/3 = 9/3 = 3 This equals the length of A.

Both A and E equal 3, the length of the longer side, when we evaluate by picking numbers.

That being said, E is the original answer, so we have to find a way to get to that and only that reliably.

For that reason, I think the algebraic solution offered by HarishKumar was very good.

Cheers!