If n is the smallest number, the nine consecutive numbers can be expressed as:
n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8
We are told that they all add up to 2007
so n + n+1 + n+2 + n+3 + n+4 + n+5 + n+6 + n+7 + n+8 = 2007
so 9n + 36 = 2007
9n = 1971
n = 219
If the smallest number is 219, the largest will be 219 + 8 = 227
Alternatively, the average of the 9 numbers is 223, so they are 219, 220,..., 226, 227.
So the last number is 227.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.