First overtone of a closed pipe is the 3rd harmonic ($f_c = 3v/4L_c$). First overtone of an open pipe is the 2nd harmonic ($f_o = 2v/2L_o = v/L_o$). Setting $f_c=f_o$ gives $3v/4L_c = v/L_o$, so $L_c/L_o = 3/4$.