Final answer:
A 'clever integer' is defined as an even integer between 20 and 120 with digits summing to 9. The clever integers that meet these criteria are 24, 42, 60, 78, and 96, but only 42 and 78 are divisible by 27, giving us a fraction of 2/5.
Step-by-step explanation:
The question asks us to determine the fraction of clever integers that are divisible by 27. We define a clever integer as an even integer greater than 20, less than 120, and with digits summing to 9. First, we identify all such clever integers, and then determine which of these are divisible by 27.
Even integers that have a sum of digits equal to 9 and fall within the specified range are 24, 42, 60, 78, and 96. Out of these, only 42 and 78 can be divided by 27 with no remainder. Therefore, there are 2 clever integers divisible by 27 out of a total of 5 clever integers. This gives us a fraction of 2/5.
The fraction of all clever integers that is divisible by 27 is 2/5.