If you have 137.8 mL of a 38.4% aqueous solution of ethanol, what volumes of ethanol and water are in the solution?

Really, we don't know unless we know 38.4% w/v, v/v, w/w.

If it is v/v, then
we have 38.4 mL ethanol/(100 mL solution). That means 100-38.4 = 61.6 mL water.
So ethanol = 38.4 x (137.8/100) = ??
and water = 61.6 x (137.8/100) = ??