What would happen to the density of a substance whose mass was cut in half

  I agree it should be the same, provided the material was homogeneous (uniform) prior to cutting. In other words, if the mass was evenly distributed, and you cut the object in half, you would have half the mass in half the volume. But if it were say, a chocolate chip cookie, then you could imagine cutting it in such a way that more chips were in one half and the density of that half could be of significantly higher density.
hope this helps

You don't cut a substance in half. 
You cut a piece of it, or a sample of it, or a blob of it, or a drop of it, in half.

The definition of density is         

   Density = (mass of a piece or a sample) / (volume of the same piece or sample) .

There's no mention of what size the piece or the sample has to be.
The density is the density of the substance, and it's the same density
no matter how much of the substance you have.  It could be a microscopic
speck of it or a dump truck full of it.  The density is the same.

If you measured the density of a supertanker full of it, and then split up
the supertanker cargo into pieces and mailed them to a million scientists
all over the world, every scientist would measure the same density of his
tiny share of it.

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