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07 Jul 2025, 09:50
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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A ribbon 40-inches long is to be cut into three pieces, each of whose lengths is a different integer number of inches.
Quantity A
The least possible length, in inches, of the longest piece
Quantity B
15
Quantity A
The least possible length, in inches, of the longest piece
Quantity B
15
19 Jul 2025, 09:05
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OE
To minimize the length of the longest piece, maximize the lengths of the remaining pieces, subject to the constraints that they be shorter than the longest piece and different from each other. Suppose that the longest piece were 14 inches long (the largest possible length that is still less than the 15 in Quantity B). That would leave 40 − 14 = 26 inches to be accounted for by the other two pieces.
Because each piece must be a different number of inches long, those pieces cannot each be 13 inches long. This, in turn, implies that one of the two remaining pieces would have to be more than 13 inches long—but then, that piece would be 14 inches long, again violating the constraint that each piece be of a different length. Thus, it’s not possible for the longest piece to be 14 inches or shorter. The longest piece must be at least 15 inches long. In the case that the longest piece is exactly 15 inches long, the shorter pieces could then be 12 and 13 inches long, for a total of 40 inches.
Therefore, the correct answer is (C): The two quantities are always equal.
To minimize the length of the longest piece, maximize the lengths of the remaining pieces, subject to the constraints that they be shorter than the longest piece and different from each other. Suppose that the longest piece were 14 inches long (the largest possible length that is still less than the 15 in Quantity B). That would leave 40 − 14 = 26 inches to be accounted for by the other two pieces.
Because each piece must be a different number of inches long, those pieces cannot each be 13 inches long. This, in turn, implies that one of the two remaining pieces would have to be more than 13 inches long—but then, that piece would be 14 inches long, again violating the constraint that each piece be of a different length. Thus, it’s not possible for the longest piece to be 14 inches or shorter. The longest piece must be at least 15 inches long. In the case that the longest piece is exactly 15 inches long, the shorter pieces could then be 12 and 13 inches long, for a total of 40 inches.
Therefore, the correct answer is (C): The two quantities are always equal.
