Fractions with multiple representations

If a fraction is simultaneously in the forms $7k - 5\over5k - 3$ and $6l - 1\over4l - 3$ we must have ${7k - 5\over5k - 3}={6l - 1\over4l - 3}$ .

This simplifies to This simplifies to $kl + 8k + l - 6 = 0$ . We can write this in the form $(k + 1)(l + 8) = 14$ .

Now $14$ can be factored in $8$ ways: $1 × 14,\;\; 2 × 7, \;\;7 × 2, \;\;14 × 1, \;\;(-1) × (-14), \;\;(-2) × (-7),\;\;(-7) × (-2)$ and $(-14) × (-1)$ . Thus we get 8 pairs:

$(k, l) = (13, -7),(6, -6),(1, -1),(0, 6),(-15, -9),(-8, -10),(-3, -15),(-2, -22).$

This gives the required fractions to be:

$\frac{43}{31}$ , $\frac{31}{27}$ , $\frac{1}{1}$ , $\frac{55}{39}$ , $\frac{5}{3}$ , $\frac{61}{43}$ , $\frac{19}{13}$ , $\frac{13}{9}$

Adding gives $\frac{ 5117122}{467883}\approx10.93675555$