Which number produces a irrational number when added 0.25

I think the easiest way to think about this is to put 0.25 into fractional form. 0.25 = 1/4 because 4 divided into 1 = 0.25

So 1/4 added to 2/9 will give us a fraction (rational number). Of course to determine what the fraction is, we have to get a common denominator.
(In case you wanted to know what that rational number is: The least common denominator would be 36. So changing 1/4 by multiplying top and bottom by 9 gives 9/36 and changing 2/9 by multiplying top and bottom by 4/4 gives 8/36 and adding 9/36 to 8/36 gives 17/36 - a rational number.)