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?
Multiply by $\,2\,.$
Add $\,5\,.$
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:

Already had birthday:
Haven't yet had birthday:

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! VOILA!!