\(a + \frac 1b = 5\)
⇒ \(a = 5 - \frac 1b\)
\(= \frac{5b -1}b\) .....(1)
\(b + \frac 1c = 12\) .....(2)
\(c + \frac 1a = 13\) .....(3)
From (1) & (3),
\(c + \frac b{5b -1} = 13\)
⇒ \(c = 13 - \frac b{5b - 1} \)
\(= \frac{65b - 13 -b}{5b - 1}\)
\(\therefore \frac 1c = \frac{5b -1}{64b - 13}\)
From (2),
\(b + \frac{5b -1}{64 b - 13} = 12\)
⇒ \(64b^2 - 8b - 1 = 768b -156\)
⇒ \(64b^2 -776b + 155 = 0\)
\(\therefore b =\frac{776 \pm \sqrt{776^2 - 4\times 64 \times 155}}{2\times64}\)
\(= \frac{776 \pm 750}{128}\)
\(= \frac{26}{128} \;or\; \frac{763}{64}\)
\(= \frac{13}{128} \;or\; \frac{763}{64}\)
\(\therefore a = 5 - \frac 1b \)
\(= 5 - \frac{128}{13} \)
\(= \frac{-63}{13}\)
or \(= 5 - \frac {64}{763} \)
\(= \frac{3751}{763}\)
\(\therefore c = 13 - \frac 1a \)
\(= 13 + \frac {13}{63} \)
\(= \frac{832} {63}\)
or \(= 13 - \frac {763}{3751} \)
\(= \frac{48000}{3751}\)
\(\therefore abc = \frac{-63}{13} \times \frac {832}{63} \times \frac {13}{128} = -6.5\)
or \(abc = \frac{3751}{763} \times \frac {763}{64} \times \frac{48000}{3751} = 750\)
\(\therefore abc + \frac 1{abc} = -6.5 - \frac1{6.5} = -6.6538\)
or \(abc + \frac 1{abc} =750 + \frac 1{750} = 750.0013\)
