Math competitions are all about logic. Practicing logic and intuition will prove to be most effective at preparing for math contests. So, let’s work on an intuition problem.
Starting at the same time on opposite shores of a lake, two boats cross back and forth for 35 minutes without stopping. One boat needs 5 minutes to cross the lake. The other boat needs 7 minutes to cross the lake. What is the number of times the faster boat passes the slower boat? (Source: MOEMS)
First, we need to realize that the number of times the faster boat passes the slower boat is the number of times the boats are at the same location. You can’t pass someone without being at the same spot as them!
To solve this, we can either draw a diagram and map every minute or use our intuition. Since this lesson is about our intuition, let’s go with the latter.
Let’s try to map out this situation. Set point A and point B as the starting and ending points on the lake. Say that the faster boat is at point A. Is it possible for the faster boat to never cross paths with the slower boat during its path to B?
To do this, we can look at the possible area where the slower boat can be, depending on the faster boat’s location. If the area is 0, then it is not possible.
At the beginning, the faster boat is at A. The slower boat can be anywhere between A and B.
Now, the faster boat is halfway between A and B. Let’s call this point H, for half. If the slower boat is in the area from A to H, that means the boats have had to pass each other at least once. Try to see why this is true.
This means that if the boat is at any point P, the slower boat has to be between points P and B so that they have not crossed yet.
Now, let’s say the faster boat is at point B. Where can the slower boat be? If it is anywhere behind B, it will have crossed paths. If it is at point B, it has just crossed paths. This means that the boats always cross paths at least once on a lap.
Now, let’s see if the boats can cross paths more than once on a lap. Let’s say that the boats have just crossed at point P. Now, the faster boat is in front of the slower boat.
The faster boat will reach the other side before the other boat, no matter what direction it is travelling. Try to see why this is the case.
Once the faster boat reaches the other side of the lake, the lap is over. This means that the faster boat crosses the slower boat 1 time every time it takes a lap. Using this information, we can now solve the problem.
If the faster boat takes 5 minutes per lap and there are 35 minutes, this means it can do 7 laps in the allotted time. This makes 7 crossings. Since they do not start at the same spot, we do not have to add an extra initial crossing. So, the answer is 7.