Solar spectra of ultraviolet bursts and flare ribbons from the Interface Region Imaging Spectrograph (IRIS) have suggested high electron densities of $>10^{12}$ cm$^{-3}$ at transition region temperatures of 0.1 MK, based on large intensity ratios of Si IV $lambda$1402.77 to O IV $lambda$1401.16. In this work a rare observation of the weak O IV $lambda$1343.51 line is reported from an X-class flare that peaked at 21:41 UT on 2014 October 24. This line is used to develop a theoretical prediction of the Si IV $lambda$1402.77 to O IV $lambda$1401.16 ratio as a function of density that is recommended to be used in the high density regime. The method makes use of new pressure-dependent ionization fractions that take account of the suppression of dielectronic recombination at high densities. It is applied to two sequences of flare kernel observations from the October 24 flare. The first shows densities that vary between $3times 10^{12}$ to $3 times 10^{13}$ cm$^{-3}$ over a seven minute period, while the second location shows stable density values of around $2times 10^{12}$ cm$^{-3}$ over a three minute period.
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