Answer
The ratio of the distances that each block travels is 9. The smaller block travels 9 times farther than the larger block.
Work Step by Step
Let the masses of the two blocks be $m$ and $3m$.
We can use conservation of momentum to find the ratio of the two speeds after the explosion:
$m~v_1 = 3m~v_2$
$v_1 = 3v_2$
Since the surface is rough, a sliding object will decelerate because of the force of friction. Let $M$ be the mass of this object.
$Ma = F_f$
$Ma = Mg~\mu_k$
$a = g~\mu_k$
Note that the acceleration $a$ is the same for both blocks of wood since the acceleration does not depend on the mass.
We can find an expression for the distance traveled.
In general: $d = \frac{v^2}{2a}$
$d_1 = \frac{v_1^2}{2a}$
$d_2 = \frac{v_2^2}{2a}$
We can then find the ratio of the distances traveled by the two blocks of wood.
$\frac{d_1}{d_2} = \frac{(v_1^2)/(2a)}{(v_2^2)/(2a)} = \frac{v_1^2}{v_2^2} = \frac{(3v_2)^2}{v_2^2} = 9$
The ratio of the distances that each block travels is 9. The smaller block travels 9 times farther than the larger block.