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24 Mar 2020, 06:21
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A certain line segment rotates consistently in one direction in 45° increments around one of its endpoints E. The segment loses half of its length every 45° that it rotates, and it stops rotating when it reaches less than 5 percent of its starting length. Did the segment stop rotating?
(1) The segment rotated at least 180°.
(2) The segment made at most 5 movements.
(1) The segment rotated at least 180°.
(2) The segment made at most 5 movements.
26 Mar 2020, 00:41
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From the first two statements of the question, we can make out that the length of the line segment becomes 1/4th of itself, every time the line segment completes rotating 90 degrees.
If the original length of the line segment is 100 units, it will reduce to 25 units by the time the line segment has completed 90 degrees of rotation. Similarly, the length will reduce to 6.25 units by the time it completes 180 degrees of rotation.
Therefore, the line hasn’t stopped rotating when it has rotated by 180 degrees; in other words, it has not stopped rotating when it has made 4 movements. Keeping this in mind, let us analyse the statements.
From statement I alone, we know that the segment rotated by at least 180 degrees i.e. ≥180 degrees. If the segment rotated exactly 180 degrees, it hasn’t reached less than 5 percent of its starting length and hence hasn’t stopped rotating. If the segment rotated 225 degrees, it has reached less than 5 percent of its starting length and has stopped rotating.
Statement I alone is insufficient to obtain a definite YES or a definite NO. Answer options A and D can be eliminated. The possible answer options are B, C or E.
From statement II alone, the segment made at most 5 movements i.e. ≤ 5 movements.
If it made 4 movements, it hasn’t stopped rotating. If it made 5 movements, it has stopped rotating. Statement II alone is insufficient to obtain a definite YES or a definite NO. Answer option B can be eliminated. The possible answer options are C or E.
Combining statements I and II together, we have the following:
From statement I, we know that the angle rotated could be 180 degrees or more. From statement II, the segment makes a maximum of 5 movements.
180 degrees corresponds to 4 movements and the segments hasn’t stopped rotating.
225 degrees corresponds to 5 movements and the segments has stopped rotating.
The combination of statements is insufficient to obtain a definite YES or NO. Answer option C can be eliminated.
The correct answer option, IMO, is E.
Hope that helps!
If the original length of the line segment is 100 units, it will reduce to 25 units by the time the line segment has completed 90 degrees of rotation. Similarly, the length will reduce to 6.25 units by the time it completes 180 degrees of rotation.
Therefore, the line hasn’t stopped rotating when it has rotated by 180 degrees; in other words, it has not stopped rotating when it has made 4 movements. Keeping this in mind, let us analyse the statements.
From statement I alone, we know that the segment rotated by at least 180 degrees i.e. ≥180 degrees. If the segment rotated exactly 180 degrees, it hasn’t reached less than 5 percent of its starting length and hence hasn’t stopped rotating. If the segment rotated 225 degrees, it has reached less than 5 percent of its starting length and has stopped rotating.
Statement I alone is insufficient to obtain a definite YES or a definite NO. Answer options A and D can be eliminated. The possible answer options are B, C or E.
From statement II alone, the segment made at most 5 movements i.e. ≤ 5 movements.
If it made 4 movements, it hasn’t stopped rotating. If it made 5 movements, it has stopped rotating. Statement II alone is insufficient to obtain a definite YES or a definite NO. Answer option B can be eliminated. The possible answer options are C or E.
Combining statements I and II together, we have the following:
From statement I, we know that the angle rotated could be 180 degrees or more. From statement II, the segment makes a maximum of 5 movements.
180 degrees corresponds to 4 movements and the segments hasn’t stopped rotating.
225 degrees corresponds to 5 movements and the segments has stopped rotating.
The combination of statements is insufficient to obtain a definite YES or NO. Answer option C can be eliminated.
The correct answer option, IMO, is E.
Hope that helps!
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