On the first day, Alina sent 120 - 20 = <<120-20=100>>100 messages.
The total number for the first day is 100+120 = <<100+120=220>>220 messages.
On the second day, Lucia sent 1/3 *120 = <<1/3*120=40>>40 messages.
Alina sent 100*2 = <<100*2=200>>200 messages on the second day.
The total for the second day is 40+200 = <<40+200=240>>240 messages.
For the three days, they sent 220+220+240 = <<220+220+240=680>>680 messages.
#### 680