Let the 3 integers be n-1, n and n+1.
Actually the smallest integer is not referred to in the problem at all, so the
problem can be solved for any 2 consecutive integers too. But that does not 
matter.
Break down the statement given step by step and derive the algebraic expression
for each.
1) Largest of the number = n+1
2) Half the largest = 
This is the first part.
3) Thrice the second integer = 3*n
4) 24 less than thrice the second integer = 
5) 1/4th of, 24 less than thrice the second integer = 
This is the second part.
Equating the 2, we get

Cross multiplying and simplifying



So the 3 numbers are 25, 26, 27
:)