...The only way of avoiding this that I can see is to know the multiples of 143.8 up to 9*143.8, but I'd be curious to here other ideas. Can it be made purely mechanical. ...
Well, it *is* purely mechanical: you just gave the procedure yourself.
Right. It's just that I've never seen it explicitly suggested. My son's courses presented it as partly trial and error. I don't remember what I did as a kid, but it could be that the problems weren't hard enough that I couldn't get each digit of the quotient right first time.
This does relate to various interesting issues (that might make good discussion points with interested children) though. For example, we can get an approximation of the answer by using approximations of the arguments, and we can bound the error, but nevertheless, knowing when we have been precise enough to be sure about the first digit can be tricky. If you don't have access to full information about the arguments, it can be more than tricky, and one can go off into study of the on-line computable functions (which do not include division, IIRR)...
Ha! I have been led to exactly these mental meanderings inspired by my son's 4th/5th grade maths (except I don't know the definition of on-line computable functions, but I can guess based on the context).
I've been frustrated by the presentation of long division. Since there is a mechanical method, they should present it, even if laborious. You can still use faster methods (and/or mental arithmetic) where appropriate. Most questions in mathematics can't be solved by mechanical methods, but when they can, it's worth knowing about.
