In some situations, you can use last and past interchangeably with no difference in meaning:
The economy has improved in the past month.
The economy has improved in the last month.
When used with time, the word past always requires the or this - but the word last does not.
Last weekend I went to the beach.
This past weekend I went to the beach.
Past weekend I went to the beach.
I started a new job last year.
I started a new job past year.
I've been very busy the last few days. or I've been very busy the past few days.
The company has hired several new people in the past month. or The company has hired several new people in the last month.
There is also a difference between "last year" and "in the last/past year."
If it is September of 2014.
"last year" means the year 2013 (January - December) o "in the last/past year" means "in the most recent 12 months up to now: (September 2013 - September 2014).
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