Does mass really follow the additive property? I've heard people talking about how photons, individually, are massless but when they're together as a group, they exhibit mass. This clearly shows that mass generally doesn't follow the additive property. Is this fact true?
If yes, why do we use the additive property of masses while solving problems?
 A: Mass as a definition exists in two different physics frameworks. The classical framework where there is Euclidean geometry, where it is defined by $F=ma$, the second law of motion . The kinematics are defined by three vectors( x,y,z) and time independently  by clocks.  In this framework masses are additive .
Then there is the framework of special relativity, which works  with four vectors, where space and time define one four vector. In this framework the "length" of the  (energy, momentum) four  vector  defines the invariant mass, which does not change under Lorentz transformations . In this framework, adding algebraically the four vectors gives a new four vector with a new invariant mass that is not the addition of the individual masses, but larger. This is the framework of nuclear reactions and high energies and velocities. Photons as moving with the velocity of light are described by four vectors each with zero invariant mass. The addition of two photon four vectors has an invariant mass. The meson pi0 decays into two photons, the four vectors of those two photons added have the invariant mass of the pi0, even though the photons have zero invariant mass.
For low energies and velocities, the two definitions coincide  mathematically within errors. 
A: The people you have heard talking about this are wrong. All photons are massless, but they do possess momentum, and that momentum gives rise to reaction forces whereby light exerts a push on things it strikes. 
For ordinary, everyday circumstances we experience on earth, mass is additive. 
