Question

The rectangular rug shown in the figure above has an accent border. What is the area of the portion of the rug that excludes the border?
(1) The perimeter of the rug is 44 feet.
(2) The width of the border on all sides is 1 foot.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (E)

Statements (1) and (2) together are not sufficient.

• Considering statement 1, it is given that the perimeter of the rug is $44$ ft, then the rectangular rug could have area $121$ sq ft (length $=11$ ft and width $=11$ ft) and the rectangular rug could have area $96$ sq ft (length $=16$ ft and width $=6$ ft). Moreover, no information is given about the width of the accent border, so even if the area of the rectangular rug could be determined, it would still not be possible to determine the area of the portion of the rug that excludes the border. Thus, statement 1 is insufficient.
• Considering statement 2, it is given that the width of the border is $1$ ft, the area of the portion of the rug that excludes the border is $(L–2)(W–2)$ sq ft, where $L$ ft and $W$ ft are the length and width, resp, of the rug. If $L=W=11$, then $(L–2)(W–2)=81$. But, if $L=16$ and $W=6$, then $(L–2)(W–2)=56$. Thus, statement 2 is insufficient.
• Considering statement 1 and 2, , the area could be $81$ sq ft or $56$ sq ft as shown by the examples above. Thus, both the statements together are insufficient.