The only answer that makes sense is E.

You can do like this

A > B
D > B
B > C

where the alphabet represents the character.

One year after

A2 = A + 2
B2 = B + 4
D2 = D + 4
C2 = C + 3

Now, let check the answers

(A) Alan was shorter than Boris. => A2 < B2
or A + 2 < B + 4, this is possible if A is 10, B is 8

(B) Alan was shorter than Charles. => A2 < C2
or A + 2 < C + 3, this is possible too

(C) Boris was shorter than Dave. => B2 < C2
or B + 4 < D + 4, absolutely right

(D) Dave was shorter than Alan. => D2 < A2
or D + 4 < A + 2, this is possible too

(E) Dave was shorter than Charles. => D2 < C2
or D + 4 < C + 3, we have D > C at the beginning so this answer is wrong

The answer is E

You can do it in your mind very quickly. Just my explanation

sanju09 wrote:At the beginning of 2006, both Alan and Dave were taller than Boris, and Boris was taller than Charles. During the year, Alan grew 2 inches, Boris and Dave each grew 4 inches, and Charles grew 3 inches. Of the following, which could NOT have been true at the beginning of 2007?

(A) Alan was shorter than Boris.
(B) Alan was shorter than Charles.
(C) Boris was shorter than Dave.
(D) Dave was shorter than Alan.
(E) Dave was shorter than Charles.

Dave starts off taller than Boris. Boris is taller than Charles

If Dave and Boris grow 4 inches and Charles only grows 3 inches
Dave is still taller than Boris and Charles

E.

Please start posting the OA.

Consider the choices in turn. At the beginning of 2006, Alan was taller than Boris; during the year, Alan grew 2 inches and Boris grew 4 inches. Since Alan grew less than Boris, it is possible that Alan was shorter than Boris at the beginning of 2007. So choice (A) is not the correct answer.

At the beginning of 2006, Alan was taller than Boris, who was taller than Charles; thus, Alan was taller than Charles. During the year, Alan grew 2 inches and Charles grew 3 inches. Since Alan grew less than Charles, it is possible that Alan was shorter than Charles at the beginning of 2007. So choice (B) is not the correct answer.

At the beginning of 2006, Dave was taller than Boris; during the year, Boris and Dave each grew 4 inches. Thus Boris was still shorter than Dave at the beginning of 2007. So choice (C) is not the correct answer.

At the beginning of 2006, Alan and Dave were each taller than Boris, but we cannot determine whether Alan was shorter than Dave or Dave was shorter than Alan, nor what the difference in their heights was. So even though Dave grew 4 inches during the year while Alan grew only 2 inches, it is possible that Dave was shorter than Alan at the beginning of 2007. So choice (D) is not the correct answer.

At the beginning of 2006, Dave was taller than Boris, who was taller than Charles; thus, Dave was taller than Charles. During the year, Dave grew 4 inches and Charles grew only 3 inches, so Dave remained taller than Charles. Thus it could not be true that Dave was shorter than Charles at the beginning of 2007. Therefore, choice [spoiler](E)[/spoiler] is the correct answer.