Useful mental models
Yesterday evening, we had a family reunion after a long time of not being all together. As it happens, our conversation gradually became more and more philosophical and at one point, we gave my brother a fun physics riddle with the following setting:
A car travels from point A to point B with speed 50 km/h. How fast does it have to travel back to achieve an average speed 100 km/h for the whole trip.
He was thinking a long time about it and tried to solve it on paper. He was resilient but after a while, he asked for help. He almost gotting and with just a little hint, he solved it. Then dad showed us a really nice visual explanation of why the result is so with two riders. It goes as follows:
One goes with speed 100 km/h and one with 50 km/h. You can imagine, that you want to arrive at the end at the same time as the second rider. But where does the second rider end up, where we are at point B? He is already back to point A, so we would have to teleport in order to get to the same average speed as him.
This nice explanation lead to the main topic of the conversation: Can one learn how to come up with such ideas or is it just intelligence?
My brother asked the question above. He told us, that he is now thinking if there are some methods of thinking that can help one live better. He called these mental models. The conversation that followed was quite interesting and I tried to take note of the models we thought about. Here is what I noted:
1. Thinking about the limit cases
This is closely related to the question above and the root of the conversation. One can use this methods when solving some problem that involves some quantity that has to be calculated or estimated.
For example consider you are making plans for a vacation and what to figure out how much money you will spend. Many times, one tries to get a very exact solution before one thinks about a very rough estimate. Sometimes it is good to just take one limiting case as taking some low amount of days for the duration of the vacation and low amount of money spent per day. And then do the oposit for an enormously high amount.
First consider the low budget case: Lets say we think our vacation will last roughly 10 days. In the limiting case, we can only consider 9 days multiplied by a low amount of money spent per day. Lets say 500 CZK. Then we can really fast come up with the number 4,500 that gives us a rough low estimate of what we should expect at least to pay.
Now the high budget: 11 days where each day we spend 2,000 crowns. We will get 22,000.
So by these two limits above, we bounded the result and can get a rough feel of how much we will have to spend. This is useful since we don’t have to think about complicated details.