Is the perimeter of a given rectangle greater than 8 inches? (1) The two…

Concept

In a Data Sufficiency question, a statement is sufficient only if it lets you answer the Yes/No question with the same certain answer for every case consistent with that statement. For a rectangle, perimeter = 2 × (length + width). A Yes/No threshold question can sometimes be settled even by a partial or relational constraint, as long as it forces the sum (length + width) to always stay on one side of the threshold.

Application

• Statement (1) alone:

  1. The two shorter sides are 2 inches long, so width = 2 inches.

  2. Because these are explicitly the SHORTER sides, the length must be strictly greater than the width: length > 2 inches.

  3. Perimeter = 2 × (length + width) = 2 × (length + 2). Since length > 2, perimeter > 2 × (2 + 2) = 8 inches for every valid rectangle. The answer is always “Yes” → sufficient.

• Statement (2) alone:

  1. Let width = w. Then length = w + 2.

  2. Perimeter = 2 × (length + width) = 2 × (w + 2 + w) = 4w + 4.

  3. w is not fixed by this statement: if w = 0.5, perimeter = 6 inches (“No”); if w = 10, perimeter = 44 inches (“Yes”). Different valid values give different Yes/No answers, so this statement alone is NOT sufficient.

Cross-check

Plug back into Statement (1): width = 2, length = 3 (valid since 3 > 2) gives perimeter = 2 × 5 = 10 > 8 (“Yes”). width = 2, length = 2.001 gives perimeter = 2 × 4.001 = 8.002 > 8 (“Yes”). Every valid case agrees.

For Statement (2), the two test values (w = 0.5 giving “No” and w = 10 giving “Yes”) confirm it gives conflicting answers and is therefore insufficient.

Result

Statement (1) ALONE is sufficient, but Statement (2) is not sufficient.