Practical significance of $\frac{17}{21}$=$\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{7}$

Question

In an article of ancient maths I found that Babylonians also had a great idea of maths. For practicle purposes, if I want to divide 17 grain bushels among 21 workers, the equation would be $\frac{17}{21}$≈0.8. It means I have to give 0.8 grain bushels to each worker. Since $\frac{17}{21}$ can be written as $\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{7}$, what does this equation implies to the real world? There is also written that it has practical meaning then I am wondered what's that practical meaning of this equation.

Answer 1

The point is just that everyone can get the right amount of grain if I give everyone half a bushel, a sixth of a bushel, and a seventh of a bushel. The article you originally linked goes on to note that the obvious greedy algorithm outputs 1/2 + 1/4 + 1/17 + 1/1428, which is an extremely unhelpful division (who has the ability to divide a bushel into 1428ths?).

Answer 2

I will give you some simple real world applications which might help you.

1. Thin lens Equation: If the lenses of focal lengths $f_1, f_2$ are “thin”, the combined focal length $f$ of the lenses is given by $$\dfrac1f=\dfrac1{f_1}+\dfrac1{f_2}$$ while if the lenses are separated in air by some distance $d$ then the combined focal length is given by $$\dfrac{d}{f_1f_2}=\dfrac1{f_1}+\dfrac1{f_2}-\dfrac1f.$$

2. Electric Circuits: In a parallel circuit the reciprocals in individual resistances/inductances are added to find the total resistance/inductance. A similar equation used to compute the capacitance of capacitors in series.

3. Harmonic Mean: This is another central tendency measure like arithmetic mean and geometric mean etc. Specially when rates and ratios are involved the harmonic mean gives the most correct value of the mean. This is useful in statistics and financial mathematics.