Let the two even numbers be n and n+2.  Then translating the
words,



Multiply both sides by 4 to clear of fractions:















So the first even number n could be is 20

So the two consecutive even numbers are 20 and 22. 

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Checking:

We need to find half of the sum of the two 
consecutive even numbers.

Their sum is 20+22 or 42

Half their sum is 21

That must be at least 15 more than a quarter of the second number

A quarter of the second number is  or 

Is 21 at least 15 more than ?  Yes it is, for 
is  so 21 is  more than .

So it checks.  Are 20 and 22 the least?

Let's try 18 and 20.

Their sum is 18+20 or 38

Half their sum is 19

That must be at least 15 more than a quarter of the second number

A quarter of the second number is  or 

Is 19 at least 15 more than ?  No it is't, for 
is  so 19 is only  more than .

So it doesnt check.  So 20 and 22 are the least.

Answer 20 and 22.

Edwin