You need to set the bitrate. Open the movie in QuickTime Player then do File->Export, select MPEG-4 as the file format and click the Options button. There you see the bitrate per second. Let's say your movie is 180 seconds and the bitrate is 256kB/s then the video will take 45 MB. You want it to be 20 MB or 20480 kB and that divided by 180 seconds, so you need a bitrate of 113 kB/s. You have to consider that audio also takes some space, so make it 100 kB/s and it will match more or less (of course you can also calculate and set the bitrate for the audio so you can get it to match better).