Give SetPrecision a try

In[4]:= SetPrecision[2.40000032783267236526523651, 2]
Out[4]= 2.4

In[5]:= SetPrecision[2.40000032783267236526523651, 2] + 2.
Out[5]= 4.4

In particular, this seems crazy to me:

In[9]:= SubsetQ[Map[Round[#, .1] &, {2.400001}], {2.4}]

Out[9]= False

In[10]:= Round[2.400002, .1] == 2.4

Out[10]= True

Yes, I know I can write my own form of SubsetQ, but I would think there were be some way to get a real 2.4 out of 2.400001 or whatever.

In[3]:= NumberForm[2.40, {2, 1}]

Out[3]//NumberForm= \!( TagBox[ InterpretationBox["\<\"2.4\"\>", 2.4, AutoDelete->True], NumberForm[#, {2, 1}]& ])

Mathematica uses binary floating point representation for approximate numbers (i.e. numbers with a decimal point). That means that you can never exactly represent a rational number whose denominator is not a power of 2 with an approximate number.