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A question in my text book:-

A circular field has a circumference of $360 \;km$. Three cyclists start together and can cycle $48$, $60$ and $72$ km a day, round the field. When will they meet again?

Solution: We first find out the time taken by each cyclist in covering the distance.

Number of days $1^{st}$ cyclist took to cover $360 \;km = 360/48 = 7.5$ days.

Number of days taken by $2^{nd}$ cyclist to cover same distance $= 360/60 = 6$ days.

Number of days taken by $3^{rd}$ cyclist to cover this distance =$ 360/72 = 5$ days.

Now, $LCM$ of $7.5, 6$ and $5 = 30$ days

My question is: Why is the distance covered taken as $360 \;km$? Does this involve the pre-assumption that they are going to meet at the starting point only.
Another question, can we solve this question by taking LCM in terms of distances? Thanks.

amWhy
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Shekhar Malhotra
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1 Answers1

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You are right. It is not necessarily the case that the first time they meet happens at the starting location. The solution method you quote seems to make that assumption so will not always give the correct answer.

Imagine you are the slowest rider. The other two riders ride $12$ and $24$ km more than you each day. They will meet you again if they travel a whole number of laps more than you. The second rider laps you every $360/12=30$ days, and the third rider laps you every $360/24=15$ days. They first time they both lap you at the same time occurs after $\text{LCM}(30,15)=30$ days.

It just so happens that in those $30$ days you have travelled a whole number of laps ($4$ laps actually) so that it is the starting point where you all meet, but that would not have been the case with different speeds, e.g. $40, 52, 64$.

Note that every time the second rider laps you, so does the third, so the LCM does not do much in the calculation above, but if the speeds were different, it could have taken a lot longer before their laps coincided.

Jaap Scherphuis
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