Data from the Multiple EUV Grating Spectrograph component of the Extreme-ultraviolet Variability Experiment (EVE) on board the Solar Dynamics Observatory (SDO) were used to quantify the contribution of continuum emission to each of the extreme ultraviolet (EUV) channels of the Atmospheric Imaging Assembly (AIA), also on SDO, during an X-class solar flare that occurred on 2011 February 15. Both the pre-flare-subtracted EVE spectra and fits to the associated free-free continuum were convolved with the AIA response functions of the seven EUV passbands at 10 s cadence throughout the course of the flare. It was found that 10%-25% of the total emission in the 94 Å, 131 Å, 193 Å, and 335 Å passbands throughout the main phase of the flare was due to free-free emission. Reliable measurements could not be made for the 171 Å channel, while the continuum contribution to the 304 Å channel was negligible due to the presence of the strong He II emission line. Up to 50% of the emission in the 211 Å channel was found to be due to free-free emission around the peak of the flare, while an additional 20% was due to the recombination continuum of He II. The analysis was extended to a number of M- and X-class flares and it was found that the level of free-free emission contributing to both the 171 Å and 211 Å passbands increased with increasing GOES class. These results suggest that the amount of continuum emission that contributes to AIA observations during flares is more significant than stated in previous studies which used synthetic, rather than observed, spectra. These findings highlight the importance of spectroscopic observations carried out in conjunction with those from imaging instruments so that the data are interpreted correctly.