1/3 is a calculation the same way 42 is a calculation (4*10^1 + 2*10^0). Nothing is real except sets containing sets! /j
Computations are exactly the place where fractions shine over (repeating) decimals.
The disadvantage of fractions is that there is an infinite number of ways to represent each rational number - 1/3 is the same number as 2/6 or as 818171/2454513. Also, comparisons are harder as well. It's easy to tell that 1/3 is bigger than 1/7, but is 2/3 greater or smaller than 5/7? Here you have to do a computation to really tell.
Irrational numbers have a similar problem, btw. The square root of two is the same number as the fourth root of 8, but you can't tell this without performing some computations.
Well I don't suggest that adding 1/3, 1/5 and 1/7 isn't more precisely done by keeping them as fractions and multiply them out to get (35+21+15)/105=71/105. In this case it's relatively easy to get an idea of the resulting value but the next computation with other fractions could give me 83742/36476 as you say which is harder to judge without doing the division as well, so the price for doing fractions as non-decimal fractions is that in order to get the sum of three numbers I'll generally have to perform 2 multiplications (ab)c to find a common denominator, than do 2*3=6 different multiplications with three different numerators to get the normalized numerators, then do 2 additions to get the resulting numerator, and finally 1 division to get a normalized answer. That's a whopping 8 multiplications, two additions and 1 division for the sum of three numbers.
If I am given the fractions as above I could also do three divisions to get their decimal expansions, followed by 2 additions to get the decimal expansion of the normalized result; this result will be imprecise in the general case but it can be as precise as I want to.
If I am given the decimal expansions right away then I can do 3+2+1=6 immediately to get 0.6 as an approximate answer which is not bad given that the correct answer is more like 0.6761904... and all I did was looking at the figures. The slightly harder 33+20+14 is already much closer with it's result, 0.67. There's no denying the fact that many mathematical problems are better done with fractions than with decimals but when doing things like physical measurements, decimal expansions are IMO more practical.
