Brain Teasers
Buses
Every day, Tom walks from home to school, and buses that travel in opposite directions commute on the way at constant intervals. Tom sets off from his home while one bus is just going past him and another bus is coming toward him. Tom arrives at school in 48 minutes while one bus is just going past him and another bus is coming toward him. It's known that buses are 6 times as fast as Tom, and Tom has seen a total of 26 buses on his way.
How long is the time interval between successive buses? Put another way, if you just miss your bus when you arrive at the bus stop, how long would you have to wait until your next bus comes along?
How long is the time interval between successive buses? Put another way, if you just miss your bus when you arrive at the bus stop, how long would you have to wait until your next bus comes along?
Answer
Let the buses' time interval be t and Tom's walking speed be v.A bus goes past him every 6vt/(6v-v)=6t/5, and a bus comes toward him every 6vt/(6v+v)=6t/7.
Therefore, there are a total of 48/(6t/5)+1=40/t+1 buses that went past him, and 48/(6t/7)+1=56/t+1 buses that came toward him.
40/t+1+56/t+1 implies t=4.
On each bus route, there is a 4 minute interval between buses.
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Comments
It's easier. Ignore first two buses as they represent zero mark. The encounters with the other buses mark the ends of intervals. The average of those intervals is two minutes (48 minutes / 24 buses). Because that involves both directions, a bus going our way leaves every four minutes.
I supposed that the number of buses Tom seen going past him and coming toward him is the same for his whole trip. So 26 / 2 = 13. And as stil said, ignore the first two buses as he sets off from his home (13 - 1 = 12, 12 each going past him and coming toward him). So 48 / 12 = 4 minutes!
pating-
You forgot that some of the buses he will meet are already on the road between home and school. The buses which will pass Tom have yet to even reach his home.
You forgot that some of the buses he will meet are already on the road between home and school. The buses which will pass Tom have yet to even reach his home.
I did it by figuring that the distance he traveled D = 48v. The speed of the busses towards Tom is 7v and he sees 26 so the distance each bus travels in that time is 26*7v = 192v. So 192v/48v = 192/48 = 4.
Not quite as elegant as Stil's approach, but simpler than the given solution.
Not quite as elegant as Stil's approach, but simpler than the given solution.
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