The ATIC balloon-borne experiment measures the energy spectra of elements from H to Fe in primary cosmic rays from about 100 GeV to 100 TeV. ATIC is comprised of a fully active bismuth germanate calorimeter, a carbon target with embedded scintillator hodoscopes, and a silicon matrix that is used as the main charge detector. The silicon matrix produces good charge resolution for protons and helium but only partial resolution for heavier nuclei. In the present paper, the charge resolution of ATIC was improved and backgrounds were reduced in the region from Be to Si by using the upper layer of the scintillator hodoscope as an additional charge detector. The flux ratios of nuclei B/C, C/O, N/O in the energy region from about 10 GeV/nucleon to 300 GeV/nucleon obtained from this high-resolution, high-quality charge spectra are presented, and compared with existing theoretical predictions.