  The thermodynamic quantities of the ideal gas of hadrons and the
$(2+1)$--flavor lattice QCD scaled by the effective degeneracy factors of the
corresponding models are compared. We have found that in terms of the scaled
variables the quark-hadron duality of the lattice QCD and the hadron resonance
gas (HRG) model disappears. However, we have unexpectedly revealed that the
scaled variables lead to the quark-hadron duality of the lattice QCD and the
quantum ideal gas of kaons and antikaons, namely, the ideal gas of those
hadrons that contain all the three quarks $u,d,s$ and their antiquarks.
Satisfactory agreement between the scaled results of the kaon ideal gas and the
lattice QCD data is achieved at large values of the volume in the entire
temperature range. In the ideal gas of kaons there is no any phase transition.
Nevertheless, in our calculations the scaled thermodynamic quantities of the
ideal gas and the lattice QCD follow the same qualitative behavior and are
consistent with each other especially at high temperatures in the perturbative
region.
