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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?

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30 days

15 days

10 days

25 days

answer is A.

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Detailed Solution

Given,
Circumference

= 360 km

Distance covered by first-person

= 48 \frac{km}{h}

Distance covered by the second person

= 60 \frac{km}{h}

Distance covered by third-person

= 72 \frac{km}{h}

The time taken

1^{st}

person in one round in hours

= \frac{Circumference}{Speed} \times 24

\Rightarrow \frac{360}{48} \times 24

hours
⇒

\frac{360}{2}

hours
⇒

180

hours
Therefore, the time taken by the 1st person in one round of hours is 180 hours.
The time taken

2^{st}

person in one round in hours

= \frac{Circumference}{Speed} \times 24

\Rightarrow \frac{360}{60} \times 24

hours

\Rightarrow 6 \times 24

hours

\Rightarrow 144

hours
Therefore, the time taken by the 2nd person in one round of hours is 144 hours.
The time taken

3^{st}

person in one round in hours

= \frac{Circumference}{Speed} \times 24

\Rightarrow \frac{360}{72} \times 24

hours

\Rightarrow 5 \times 24

hours

\Rightarrow

120 hours
Therefore, the time taken by 3rd person in one round of hours is 120 hours.
LCM of 180, 144 and 120 are as follows.

2 180, 144, 120 2 90, 72, 60 2 45, 36, 30 2 45, 18, 15 3 45, 9, 15 3 15, 3, 5 5 5, 1, 5 1,1,1

LCM = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5

\Rightarrow LCM = 720

Minimum time taken by three persons

= \frac{LCM (180, 144, 120)}{24}

\Rightarrow \frac{720}{24}

days

\Rightarrow 30

days
Therefore, the minimum time taken by three persons to meet at the same point is 30 days.
Hence, option 1 is correct.

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