  Context. Class II methanol masers at 6.7 and 12.2 GHz occur close to
high-mass young stellar objects (HMYSOs). When they are observed
simultaneously, such studies may contribute to refining the characterisation of
local physical conditions.
  Aims. We aim to search for the 12.2 GHz methanol emission in 6.7 GHz methanol
masers that might have gone undetected in previous surveys of northern sky
HMYSOs, mainly due to their variability. Contemporaneous observations of both
transitions are used to refine the flux density ratio and examine the physical
parameters.
  Methods. We observed a sample of 153 sites of 6.7 GHz methanol maser emission
in the 12.2 GHz methanol line with the Torun 32 m radio telescope, using the
newly built X-band receiver.
  Results. The 12.2 GHz methanol maser emission was detected in 36 HMYSOs, with
4 of them detected for the first time. The 6.7 GHz to 12.2 GHz flux density
ratio for spectral features of the contemporaneously observed sources has a
median value of 5.1, which is in agreement with earlier reports. The ratio
differs significantly among the sources and for the periodic source
G107.298+5.639 specifically, the ratio is weakly recurrent from cycle to cycle,
but it generally reaches a minimum around the flare peak. This is consistent
with the stochastic maser process, where small variations in the physical
parameters along the maser path can significantly affect the ratio. A
comparison of our data with historical results (from about ten years ago)
implies significant (> 50%) variability for about 47% and 14% at 12.2 GHz and
6.7 GHz, respectively. This difference can be explained via the standard model
of methanol masers.
