Two consecutive whole numbers cannot have an even sum, so the solution cannot be whole numbers.
This means we're dealing with mixed numbers, that is, they must contain a fraction. But then the word "consecutive" has an unclear meaning: how can two such numbers be consecutive?
We need to define consecutive: Two consecutive numbers may have to be represented by n and n+1, which have the sum 2n+1=24. Therefore 2n=23 and n=23/2 or 11½. The two numbers are 11½ and 12½.
Two consective even numbers or two consecutive odd numbers will have an even sum. Let's go with that:
n+n+2=2n+2=24, n+1=12, n=11; so 11+13=24. So 11 and 13 are two consecutive odd numbers with a sum of 24.