I31-10

Question

Two autonomous cleaning robots move along a long circular corridor. They start at the same time from points diametrically opposite each other on the corridor, one moving clockwise and the other moving counterclockwise. They continue without stopping at their own constant speeds. After how many seconds after they start do they meet for the second time?

(1) They meet for the first time 26 seconds after they start.

(2) One robot moves at a speed that is \(\frac{3}{4}\) the speed of the other.

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Official Answer

A

(1) They meet for the first time 26 seconds after they start.
 
(2) One robot moves at a speed that is  \frac{3}{4}  the speed of the other.

Bunuel · Answer

Official Solution:

Two autonomous cleaning robots move along a long circular corridor. They start at the same time from points diametrically opposite each other on the corridor, one moving clockwise and the other moving counterclockwise. They continue without stopping at their own constant speeds. After how many seconds after they start do they meet for the second time? 
 
The robots start from points directly opposite each other on a circular corridor and move in opposite directions at their own constant speeds. For the first time, they will meet each other after together covering half the corridor’s length. Following this first meeting, they continue to move and will meet each other again after together covering an additional full corridor length. Consequently, by the time of their second meeting, they would together have covered 0.5 + 1 = 1.5 corridor lengths in total.
 
(1) They meet for the first time 26 seconds after they start.
 
This says that their initial meeting takes place 26 seconds after they start. This implies that together it takes them 26 seconds to cover half the corridor length. Hence, to cover 1.5 corridor lengths, they would need 26 * 3 = 78 seconds. Sufficient.
 
(2) One robot moves at a speed that is  \frac{3}{4}  the speed of the other.
 
This is irrelevant because it fails to provide specific details about their individual speeds or the actual time each of them spent moving. Not sufficient.
 
Answer: A

BongBideshini · Answer

I like the solution - it’s helpful.

Anshulyadavvv · Answer

I don’t quite agree with the solution. It says they move at their own constant speeds. Meaning the speeds are constant but not the same as each other. How can you infer they meet at their own constant exact middle from their starting point (1/4 of the diameter).

Bunuel · Answer

You are missing the point. The solution does not assume they meet at the midpoint of either robot’s path. It only uses the fact that, since they start diametrically opposite and move in opposite directions, by the first meeting  their combined distance  must equal half the corridor. How that half is split between them depends on their speeds.

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