Three cyclists Raman, Mohan and Nitin ride around a circular course

Question

Three cyclists Raman, Mohan and Nitin ride around a circular course 85 km around at the rate of 8, 12 and 20 km an hour. Raman and Mohan ride in the same direction and Nitin in the opposite direction. In how many hours will they meet again?

Answer 1

Mohan will meet Nitin every $\frac{85}{20+12} = \frac{85}{32}$ hours

Raman will meet Nitin every $\frac{85}{20+8} = \frac{85}{28}$ hours

Now when will both meet Nitin ?

Just take the LCM of $\frac{85}{32}$ and $\frac{85}{28} = 21.25$ hours

PS

I hope that you remember that in taking the LCM of a fraction, you take the LCM of the numerator but the HCF of the denominator, thus $\frac{85}{4}$

Btw, you can check that Raman will have completed exactly 2 laps, Mohan 3 laps, and Nitin 5 laps (of course, in reverse direction)

Answer 2

Let's consider Nitin ($N$) is the reference cyclist.

We have then (in $km.h^{-1}$) :
$v_N=20$, $v_R=8$, $v_M=12$ where $R$ is Raman and $M$ is Mohan.
The circuit length is $D=85km$

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The event "Raman and Nitin meet" is equivalent to "the distance cycled by Raman + the distance cycled by Nitin since the last time they meet is 85km".
How much time passes between two successive meetings ?

Using $t=\frac dv$, we deduce that : $$t_{RN}=\frac{D}{v_R+v_N} = \frac{85}{28}$$

Now, what point of the circuit do they meet at ? (We use $d=v\times t$ here)

After $t_{RN}$, Raman has cycled $$d_R=v_r\times t_{RN}=\frac{170}7km$$ that is $\frac27$ of the circuit.
This result comforts the logical reasoning that Nitin is $2,5$ times faster as Raman ($=v_N/v_R=20/8$), so when Raman rides $2$ units, Nitin rides $5$ units.

With the same reasoning and the same notations,

After $t_{MN}$, Mohan has cycled $\frac38$ of the circuit

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The only thing left to figure out is two integers $k_1$ and $k_2$ so that $$\frac57 k_1 = \frac58 k_2$$ Which are obviously $k_1=7$ and $k_2=8$. This means that the Moment we are looking for is the seventh time Raman and Nitin meet, as well as the eighth time Mohan and Nitin meet.

From that we deduce that Raman has cycled $d_{R_tot}=\frac27\times7=2$ laps, Mohan has cycled $d_{M_tot}=\frac38\times8=3$ laps, Nitin has cycled $d_{N_tot}=\frac57\times7=\frac58\times8=5$ laps.