A taxi leaves the station X for station Y every 10 minutes. Simultaneously, a taxi leaves station Y also for station X every 10 minutes. The taxis move at the same constant speed and go from X to Y or vice-versa in 2 hours. How many taxis coming from the other side will each taxi meet enroute from Y to X A. 10 B. 11 C. 12 D. 23
Hint: We will make necessary assumptions and then divide the problem in two parts, the number of taxis already on the way when this taxi departs and the number of taxis that will depart in the time this taxi reaches it destination and then we will add them all.
Complete step-by-step answer: We will first assume that 2 hours have passed since the first taxi started from X. Let us consider one taxi starting from 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 X till the time this taxi from Y reaches X will cross this taxi from 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 Y and the taxi that will just begin at X are not counted as they are not enroute. After this taxi starts and takes 2 hours to reach X, 12 more taxis must have departed from 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.
So, the correct answer is “Option D”.
Note: 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 24 5-minute intervals. But in the last interval no taxi will cross, so we get the same answer i.e. 23.
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