First represent your 3 consecutive integers as x, x + 1, and x + 2.
Since their sum is 27, our equation reads x + (x + 1) + (x + 2) = 27.
Now simplifying on the left side we get 3x + 3 = 27.
You should be able to easily solve for x and find that x = 8.
Make sure you list all your answers.
If x is 8, then x + 1 is 9 and x + 2 is 10.