## Ask Professor Puzzler

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Erin shared the following image, and wants to know if this works. (Click on the image to see a larger version).

Well, Erin, there are really two questions here, and I'm going to try to answer both of them. The first question is the one you asked - *Does it work? *The second question is one you didn't ask - *Is it practical?*

## Does it Work?

When you ask if it works, I assume you really mean, "Does it work all the time? Or just for this specific example?"

The answer to that is, yes, it always works. I can show you some simple algebra to help you see that this always works.

Suppose you have the equation A - B = C (In this case, A = 5000, B = 2384, and C = 2616)

If we subtract 1 from the number we start with, and the number we're subtracting from, we get:

(A - 1) - (B - 1). If we distribute that negative in front of (B - 1), we get an equivalent expression: A - 1 - B + 1. But the -1 and the +1 are like terms which can be combined, to give A - B, which is still C. So yes, this always works.

What's more, an altered version of this process works with addition:

Suppose you wanted to add 9999 + 4352. Add one to 9999 to get 10000 and subtract one from 4352 to get 4351. Now do 10000 + 4351, and you have 14351.

## Is It Practical?

If you are trying to do the problem mentally, then yes, I would say this is practical. However, if you're trying to do it on paper, you should consider that it actually takes *more *writing to write the entire problem over again, instead of making the few marks required to indicate where you've borrowed. Of course, if you don't understand regrouping/borrowing, then the method shown here will be very appealing to you, even if it takes more writing.

I was trying to think of a real world application where this would be useful, and it didn't take much thought. Suppose I'm at the store, and I have a ten dollar bill. I want to buy an item that costs $3.54. What will my change be?

Instead of mentally trying to do $10.00 - $3.54, I can mentally do $9.99 - $3.53 = $6.46.

Not bad.