The bigger the decision you need to make, the more likely it is that you should take a sanity check. A sanity check is a simple attempt to verify that something makes sense without going through a detailed process that offers certainty. Be patient with the mathematical examples below. There are other applications.
One place where this strategy comes in handy is a multiple choice question on a timed test. Without doing the math, which of the answers to the following problem can you dismiss?
Answer d is too large because it is more than 2856*1000 =2856000 which can be done in your head. If you estimate taking away 10% of that as 26..... it is clear that answer a is too small. Finally, looking at the units digit 6*8 comes out 8 meaning answer b is wrong. Of the choices offered only c could be correct.
There is a mathematical magic trick where you ask a person to calculate the cube of any two digit number and tell you only the result. In less time than it rakes him to multiply the numbers together, you announce the original number. Cube root calculation is not involved.
One of my favorite places is Perkins Observatory, the heart of which has a large sheet metal dome. As we planned an exhibit on large numbers someone asked if a billion grains of rice could fit in the dome or not. Once we realized that billion items make up a cube 1000 by 1000 by 1000 and that 1 cm * 1000= 10 meters, a cube 10 meters on a side would hold a billion items 1 cm on a side. Our very large dome could do that, so a billion grains of rice would fit with room to spare.
This really can work on much more difficult problems. After NASA launched the Hubble Space Telescope they discovered a serious flaw in the shape of the mirror. Since the mirror was designed to work in 0g space, a precise optical test would have cost a fortune. Since the smoothness of the surface was extremely good they concluded the expensive test for shape wasn't needed.
Fortunately the project team was able to sendvup an adaptive camera, basically a corrective lens analagous to a pair of glasses, to save a significant portion of the project. It wasn't cheap but it avoided a disaster.
The footnote here, something the project managers could not imagine, is that the mirror was so far off that a graduate student with simple tools would easily have seen it. Sometimes a sanity check us far more useful than a magic trick.
Sanity checks are not limited to math and physics problems. If you can make reasonable assumptions about your expectations, those assumptions provide the basis for the bounds you define. How does this apply to issues in your life?
Practice estimating upper and lower limits for problems that face you. Sometimes an approximate answer will save a lot.
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