Balancing chemical equations: Organic alcohol redox reactions

Directly using principle of atom conservation (POAC) on carbon and chromium, the following can be written:

$a = d$

$2b = e$

Using POAC on hydrogen:

$8a + c = 6d + 2f \implies 8a + c = 6a + 2f \implies f = a + \dfrac{c}{2}$

Using POAC on oxygen:

$a + 7b = 2d + f \implies a + 7b = 2a +f \implies 7b = 2a + \dfrac{c}{2}$

Using conservation of charge:

$-2b + c = 3e = 6b \implies c = 8b$

The last three equations consist of only $4$ variables, which is sufficient information as we are only interested in the ratios. Solving the above three equations results in:

$a:b:c:f = 3:2:16:11$

$\therefore f = \boxed{11}$