You have two identical ropes and three mice. Each mouse bites off the rope at a different biting speed. All three biting speeds are constant.
Mouse(A) bites off one rope in 60 minutes.
Mouse(B) bites off one rope in 90 minutes.
Mouse(C) bites off one rope in 180 minutes.
You do not have any time or distance measuring tools and the length of the rope is unknown. How can you measure 75 minutes?
Answer
Mark the two ropes as Rope(A) and Rope(B).
1. Let Mouse(A) and Mouse(C) start biting off Rope(A) at the same time, starting each from one end (moving in opposite directions). At the same time, let Mouse(B) start biting off Rope(B).
Finding relative speeds between Mouse(A) and Mouse(C):
Distance = Speed * Time
X = S(A) * 60, where S(A) is biting speed for Mouse(A) and X is Rope(A) length
X = S(C) * 180, where S(C) is biting speed for Mouse(C) and X is Rope(A) length
Solving the two above equations gives:
S(A)/S(C) = 3:1, which means that for each 3 units Mouse(A) will move, Mouse (C) will move 1 unit.
Moving in the opposite directions (assume Mouse(A) from left to right and Mouse(C) from right to left), the two mice will meet after Mouse(A) bites off three times what Mouse(C) bit off, which divides Rope(A) length into 4 units.
Mouse(A) bitten off portion = 3/4 X (where X is Rope(A) length)
Mouse(C) bitten off portion = 1/4 X
Finding the time spent before Rope(A) is completely bitten off by the two mice:
If Mouse(A) needs 60 minutes to bite off X, How much time is needed to bite off 3/4 X?
Minutes needed = 3/4 * 60 = 45 minutes is needed to bite off Rope(A) fully by Mouse(A) and Mouse(C). At that point in time, Mouse(B) would have bitten off exactly half the length of Rope(B) (needing 90 minutes to bite off the full rope), and needs an additional 45 minutes to bite off the remaining rope half.
2. Let Mouse(B) continue biting off Rope(B) and make Mouse(C) start biting off the other end of Rope(B).
Finding relative speeds between Mouse(B) and Mouse(C):
Distance = Speed * Time
X/2 = S(B) * 90, where S(B) is biting speed for Mouse(B) and X/2 is the remaining half of Rope(B) length
X/2 = S(C) * 180, where S(C) is biting speed for Mouse(C) and X/2 is the remaining half of Rope(B) length
Solving the two above equations gives:
S(B)/S(C) = 2:1, which means that for each 2 units Mouse(B) will move; Mouse(C) will move 1 unit.
Moving in the opposite directions (assume Mouse(B) from left to right and Mouse(C) from right to left), the two mice will meet after Mouse(B) has bitter off two times what Mouse(C) bit off, which divides the remaining half of Rope(B) length to 3 units.
Mouse(B) bitten off portion = 2/3 * X/2 =X/3
Mouse(C) bitten off portion = 1/3 * X/2 =X/6
Finding the time spent before the remaining half of Rope(B) is completely bitten off by the two mice:
If Mouse(B) needs 90 minutes to bite off X, how much time is needed to bite off X/3?
Minutes needed = 1/3 * 90 = 30 minutes are needed to fully bite off the remaining half of Rope(B) by Mouse(B) and Mouse(C).
At that point in time, 75 minutes would have passed (45 minutes consumed from biting off Rope(A) and another 30 minutes consumed from biting off Rope(B)).
