Before lunch, the cook had used 2/5*300 = <<2/5*300=120>>120 carrots.
The number of remaining carrots after the cook used 2/5 of them is 300-120 = <<300-120=180>>180.
The cook had used 3/5*180 = <<3/5*180=108>>108 of the remaining carrots by the end of the day.
The number of carrots that the cook did not use that day is 180-108 = <<180-108=72>>72.
#### 72