In the second hour, the number of towels given out is 50 x 1.20 = <<50*1.20=60>>60.
In the third hour, the number of towels given out is 60 x 1.25 = <<60*1.25=75>>75.
In the fourth hour, one-third of the guests of the third hour is 75 / 3 = <<75/3=25>>25.
In the fourth hour, the number of towels given out is 75 + 25 = <<75+25=100>>100.
In total, including the first hour guests, they need to wash this many towels: 50 + 60 + 75 + 100 = <<50+60+75+100=285>>285.
#### 285