HOW DOES “CHOCOLATE MATH&lrdquo; WORK?

So, how does that Chocolate Math thing work?

How is it that we do these seemingly random arithmetic operations, and always end up with such a predictable answer:
a three-digit number, where the first digit is the number we started with, and the last two digits are our age?

It's a beautiful example of:   NUMBERS HAVE LOTS OF DIFFERENT NAMES!

Here are the details, for those of you with inquiring minds:

Let $\,x\,$ be the number of times that you like to eat chocolate each week $\,(1 \le x \le 9)\,$.

Write your age as $\,10T + W\,$.
For example, if you're $\,46\,$, then $\,46 = 10\cdot 4 + 6\,$, so $\,T = 4\,$ and $\,W = 6\,$.

Okay, now let's go through each of the steps of “Chocolate Math” with an algebraic expression depending on $\,x\,$, $\,T\,$, and $\,W\,$:

 How many times a week do you like to eat chocolate? $x$ Multiply by $\,2\,$. $2x$ Add $\,5\,$. $2x+5$ Multiply by $\,50\,$. $50(2x + 5)$ Subtract the year you were born. Already had birthday:   Haven't yet had birthday:

Now, let's look at each of the resulting expressions:

In both cases, we get the expression $\,100x + 10T + W\,$,
which is the digit $\,x\,$ in the hundreds place,
the digit $\,T\,$ in the tens place,
and the digit $\,W\,$ in the ones place!