Pairs of irreducible fractions that add up to a given irreducible fraction

Question

Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an expression (in terms of $a$ and $b$) for all the $\frac c d$ and $\frac e f$, with $c,d,e,f \in \mathbb N$, such that $\frac c d + \frac e f = \frac a b$.

Answer 1

The best you can do is simplify your expression:

$$\frac{c}{d} + \frac{e}{f} = \frac{cf + ed}{df} = \frac{a}{b}$$

The numerator and denominator will give you a system of two linear equations. With $a$ and $b$ fixed there are four unknowns $c,d,e,f$. As $4>2$ this means we have infinitely many solutions (as would be expected):

$$a = cf + ed$$ $$b = df$$

Now, with two parameters $T, S \in \mathbb R$ and substituting $c = T, d = S$ (for example):

$$a = Tf + Se$$ $$b = Sf$$

gives you

$$f = \frac{b}{S}$$ $$e = \frac{a - T\frac{b}{S}}{S}$$ $$d = S$$ $$c = T$$

for any real numbers $S, T$.