Taxies leave the station $X$ for station $Y$ every $10 \mathrm{~min}$. Simultaneously, a taxi also leaves the station $Y$ for station $X$ every $10 \mathrm{~min}$. The taxies move at the same constant speed and go from $X$ and $Y$ or vice-versa in $2 \mathrm{~h}$. How many taxies…

Question

Taxies leave the station $X$ for station $Y$ every $10 \mathrm{~min}$. Simultaneously, a taxi also leaves the station $Y$ for station $X$ every $10 \mathrm{~min}$. The taxies move at the same constant speed and go from $X$ and $Y$ or vice-versa in $2 \mathrm{~h}$. How many taxies coming from the other side will meet each taxi enroute from $Y$ and $X ?$

Aicharya · Accepted Answer

We will first assume that 2 hours have passed since the first taxi started from $X$.
Let us consider one taxi starting from $\mathrm{Y}$ at this exact moment, at this moment of time there will be taxis at 10-minute intervals along the way.
Now all the taxis that are in the way and started from $X$ including the taxi that just reached $Y$ and all the taxis that will start from $\mathrm{X}$ till the time this taxi from $\mathrm{Y}$ reaches $\mathrm{X}$ will cross this taxi from $\mathrm{Y}$ at some point of time.
So, first we will calculate the number of taxis that are in the way.
As all the taxis are 10 minutes apart and it takes 2 hours for a one-way trip, there will be 11 taxis on the way.
The taxi that reached $\mathrm{Y}$ and the taxi that will just begin at $\mathrm{X}$ are not counted as they are not enroute. After this taxi starts and takes 2 hours to reach $\mathrm{X}, 12$ more taxis must have departed from $\mathrm{X}$ which will all cross this taxi from $Y$.
So, in total 23 taxis coming from $X$ will meet each taxi coming from $Y$ and vice-versa.

Alternate Solution:

We can also solve this problem using relative velocity.
The relative velocity of each taxi with reference to a taxi coming from the other side.
So, there are only 5-minute intervals in crossing each taxi. In the journey of 2 hours, there are 245 -minute intervals.
But in the last interval no taxi will cross, so we get the same answer i.e. 23.

Taxies leave the station \(X\) for station \(Y\) every \(10 \mathrm{~min}\). Simultaneously, a taxi also…

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