another math puzzle

[Albert]

$\dfrac{\square}{\square \square}+\dfrac{\square}{\square \square} +\dfrac{\square}{\square\square}=1------(*)$

fill in the above 9 blanks with 1~9 without repetition and satisfy (*)

(you should find it using mathematical analysis,and show your logic,don't use any

program)

in fact there is only one possibility

 

[MarkFL]

My solution:

Spoiler

I would begin with:

\(\displaystyle \frac{9}{12}\)

since this is the largest addend possible. This leaves \(\displaystyle \frac{1}{4}\).

Observing that:

\(\displaystyle \frac{1}{4}=\frac{17}{68}=\frac{2\cdot5+7}{68}= \frac{5}{34}+\frac{7}{68}\)

We now have:

\(\displaystyle \frac{9}{12}+\frac{5}{34}+\frac{7}{68}=1\)