Is there a way to quickly estimate the reciprocal of a number?

Solution 1:

$ 1/0.388 = 1/(388/1000)=1000/388=250/97$. Now do it by hand, as precise as you need.

Solution 2:

It depends what facts you have in your head. Probably you know that 1/2=0.5, 1/3=0.3333, 1/4=0.25, 1/5=0.2. It is less common to know 1/6=0.16666, 1/7=0.142857, 1/8=0.125, 1/9=0.11111. Leading zeros just make a factor of 10. You can just "look down the list" to see 1/.388 is between 2 and 3. Then 0.388 is about 15% bigger than 0.333, so 1/0.388 is about 15% less than 3, or about 2.55. Here we are using that $(1+x)^{-1}\approx 1-x$ for $x$ small compared with 1. This is only 1% off

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