Twelve soldiers had to get to a place twenty miles distant with the quickest possible dispatch, and all had to arrive at exactly the same time. They requisitioned the services of a man with a small car. "I can do twenty miles an hour," he said, "but I cannot carry more than four men at a time. At what rate can you walk?" "All of us can do a steady four miles an hour," they replied. "Very well," exclaimed the driver, "then I will go ahead with four men, drop them somewhere on the road to walk, then return and pick up four more (who will be somewhere on the road), drop them also, and return for the last four. So all you have to do is to keep walking while you are on your feet, and I will do the res t." They started at noon. What was the exact time that they all arrived together?
Answers
Given : Twelve soldiers had to get to a place twenty miles distant with the quickest possible dispatch, and all had to arrive at exactly the same time.
To find : exact time that they all arrived together
Solution:
Let say People walked for 5k hrs
then distance covered by them by walk = 4 * 5k = 20k miles
Distance covered by them in car = 20 - 20k miles
time traveled by each people in car = (20 - 20k)/20 = 1 - k hr
Total time = 5k + 1 - k = 1 + 4k hrs
first group of 4 people will be dropped after 1 - k hrs
Distance covered by car = 20(1 - k) miles
Distance covered by 8 people during that time = 4(1 - k) miles
Distance between 8 people & car =20(1 - k) - 4(1 - k) = 16 ( 1- k ) miles
pick up point for 4 people will be after t hrs of that time
( both 8 people and car will be approaching each other)
=> 4t + 20t = 16(1 - k) => t = (2/3)(1 - k ) hr
Distance covered in (2/3)(1 - k ) hr by 8 people = (8/3)(1 - k) miles
so their total Distance covered = 4(1 - k) + (8/3)(1 - k) = (20/3)(1 - k) Miles
and now 4 people out of these will picked up from (20/3)(1 - k) miles
and dropped after ( 1- k ) hrs
and Distance covered in those (1 - k) hr by remainig 4 people = 4(1 - k) miles
and then distance covered by these people when car is coming back for pickup = (8/3)(1 - k) miles
So these 4 people pickup point would be
= (20/3)(1 - k) + 4(1-k) + (8/3)(1 - k)
= (40/3)(1 - k) miles
now these 4 people has to travel in car for (1 - k) hrs to reach at same time
Distance covered in car by these 4 people = 20(1- k) miles
=> (40/3)(1 - k) + 20(1 - k) = 20 ( total Distance)
=> 100(1 - k) = 60
=> 1 - k = 0.6
=> k = 0.4
total time 1 + 4k = 1 + 4 * 0.4 hrs
= 2.6 hrs
= 2 hrs 36 mins
=> they will reach at 2 : 36 PM
First pickup of 4 people at origin
First Drop point at 12 miles
time taken by car = 12/20 = 0.6 hr
time taken by people for remaining 8 miles = 8/ 4 = 2 hrs
then next pickup of 4 people would be at 4 miles
Next drop point would be 16 miles
last pickup would be at 8 miles
last drop point 20 miles
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