Abstemiously. It’s 12 letters long and the letter y is considered a vowel in certain cases, and is included. Facetiously is another word that has all the vowels in order, but it’s only 11 letters long.
The problem works out to a set of three equations: b + c + d = 22 a + c + e = 22 a + b + c + d + e = 30
Solving for c = 14, leaving d = 8 – b and e = 8 – a. In other words, c must be 14, but the other two numbers just have to add up to 8. The requirement that they be unique rules out 4 + 4, so you’re left to choose from the following combations for b + d and a + e: 0 + 8 1 + 7 2 + 6 3 + 5
I have seven letters and am something you eat. My only anagram can help your pain. If you remove my first 2 letters I wear things down. Removing my first 3 letters is an adjective and removing my first 4 letters leaves a measure of time.
Sausage. You eat sausage, assuage is the only single-word anagram and provides relief from pain. Usage wears things down, sage is an adjective and age is a measure of time.