Yes, you need to know the densities, but the models outputted the correct densities even as part of the wrong answer.

Not sure what you mean by volumes "not adding". One way to calculate it is like:

    density_alcohol = 0.789g/ml
    density_water = 1g/ml
    weight_total = 16g
    (density = weight / volume)
    
    volume_alcohol = 0.4 * volume_total
    volume_water = 0.6 * volume_total
    
    weight_total = weight_alcohol + weight_water
    
    weight_alcohol =
      density_alcohol * volume_alcohol =
      0.789 * volume_alcohol =
      0.789 * (0.4 * volume_total) =
      0.3156 * volume_total
    
    weight_water =
      density_water * volume_water =
      volume_water =
      0.6 * volume_total
    
    weight_total =
      0.3156 * volume_total + 0.6 * volume_total =
      0.9156 * volume_total
    
    volume_total =
      weight_total / 0.9156 = 
      16 / 0.9156 =
      17.4749 (rounded)
    
    volume_alcohol =
      0.4 * volume_total =
      6.99 ml (rounded)

Not sure which school grade problems like this are taught in, but I remember similar ones from waaay back. This particular one is a real world problem caused by having a kitchen scale but not a measuring cup, and me wrongly assuming that just asking ChatGPT would help. (ChatGPT 4 from about 2 months ago)