From the three equations we see that $(\mathrm{abc}{)}^2=2\times 24\times 3=144$ and so, since $\mathrm{abc}$ is positive, $\mathrm{abc}=12$. Then the third equation tells us that $a=1/2$, the second that $b=4$ and the first that $c=6$. Therefore $a+b+c=10\frac{1}{2}$.