Your birth year + your age = 111. How?

There's a meme flying round the internet at the moment that tells you to...

... add the last two digits of your birth year to the age you will turn on your birthday this year (2011, for any time travellers). It tells you the answer will be 111.

For me, born in 1982 and turning 29 in October, this would be 82 + 29 = 111. Try it with your own details...

It works!

But how?
To complete this task, you need to know two pieces of information: the last two digits of your birth year, and the age you will turn this year.

Your birth year is easy- just take the two digits off the end: for me, it's 82.

You probably know your own age, but if we're going to figure out how this works, we need to think about it in a different way. To find out your age knowing only the year you were born in and what year it is now, you could subtract your birth year from this year. Using my info, that'd be...

 2011 - 1982 = 29

... which I can confirm is correct!

The thing is, we're only supposed to be using the last two digits of the year, so lets try...

 11 - 82

That gives us a strange value: -71. But consider that the leading two digits of the years we're using are different: 20 and 19 respectively, and remember that this is because we're in different centuries: a century is 100 years, so add this on and what do we get? 29!

So to find out the age this year of anyone who was born during the previous century using only the last two digits of the year, we could do the following:

 Age = 11 - the last two digits of their birth year + 100

This looks a bit clumsy, so I'm going to use the letter 'x' to represent 'the last two digits of their birth year':

 Age = 11 - x + 100

Lets clean it up further: We start off with 11, take something from it, and then add on 100. Why not deal with those two numbers at the same time? If I start with 11 and then later add 100, I may as well start off with 111 in the first place:

 Age = 111 - x

Right, that's the slightly complicated bit sorted. Lets put it all together:

The info we need:

• Last two digits of birth year: we're calling this x.
• Age this year: we're saying this is 111 - x

What we have to do:

• birth year + age;
• Using the notation we've defined above, that's: x + 111 - x

But wait... we're starting off with whatever x is, then adding on 111, then taking x off again. Whatever x turns out to be, it doesn't really matter because we're just getting rid of it, leaving just...

111

... all by itself!

An important point...
... just pointed out to me by @justfin is that, due to the fact that two-digit years go in 100-year cycles, anyone who's over 100 years old this year will find that they get an answer of 211 instead of 111!