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Question

Dear Mitch,

There is a kind of question on a lot of the big tests including the SAT called "alpha-numeric" or something like that. And the part I understand is that they use letters instead of numbers, AND they use numbers too, and somehow you have to figure out what numbers the letters stand for. BUT, somehow it's different from algebra, even though it looks just like it with the letters. And none of the rules I know from algebra work. What am I doing wrong? And PLEASE can you tell me what makes these letter and number problems different from regular algebra?

Thank You,

Christopher J.

Green Bay, WI

Answer

Dear Christopher,

This is one of those questions I was hoping someone would send in and, I find it surprising that we haven't come across one just like it until now.

Excellent question! Yes, there is a class of questions on a lot of standardized tests called 'alphanumeric', and, just as you say, they almost always have both letters and numbers in them, just like the algebraic questions. (It should be noted, though, that sometimes instead of letters these alphanumeric questions have little symbols to represent numbers, such as a tiny equilateral triangle or a square. It should also be noted that one sometimes comes across an alphanumeric question that has no numbers in it, just letters or symbols, but even those usually have at least one or two numbers in the answer choices -- if it is presented in the multiple choice format).

The difference between algebraic expressions and equations and ALPHANUMERIC ones is THIS: In algebra, letters represent numbers, but in alphanumeric expressions/equations the letters represent DIGITS – one letter per digit. And that difference causes a SECOND DIFFERENCE: You need to do a different procedure once you've replaced the letter or letters with numbers (or digits).

Now we back up to the beginning to untangle what that means.

First, there is something that you must understand.

QUESTION: What is the difference between a 'number' and a 'digit'?

ANSWER: there are an infinite number of numbers, but thee are only ten digits in our math world. The digits are these: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. That's it, you now have the complete collection of digits. So, for example, the number one-hundred is written like this: 100. By convention, we say that it has three digits (even though two of them happen to be the same digit, a zero, but it is only ONE NUMBER.) One more example: 27. That is one number, called twenty-seven, but there are two digits – a 2 and a 7.

O.K. Now, the second difference, which I referred to as 'caused' by the first difference is this: in algebra, since a letter gets replaced by a complete number, the rule is to multiply it by whatever number it is then next to. EXAMPLE: If you see 3Y, and you learn that Y equals 2, you then have 3 x 2, which equals 6. In alphanumeric, that is NOT THE CASE. If you see a 3Y, and you learn that Y = 2, you replace the Y with a 2 and you get 32. That's it.

Now we come to the interesting part, which is how on earth do you figure out what digit a letter is representing in an alphanumeric question? It is true that you typically cannot go through a standard routine of steps to figure them out the way you usually would in algebra. No, for alphanumeric questions you have to problem-solve a bit more and often try different digits before settling on the correct one. BUT there are lots of little 'tricks' or clues along the way. AND NOTE: ALPHANUMERIC QUESTIONS COME IN ALL LEVELS, SO THEY CAN BE AMONG THE EASIEST QUESTIONS ON THE TEST OR THEY CAN BE AMONG THE MOST DIFFICULT. Personally, though, in both cases, they're always fun to solve. And I'm not kidding.

Here is an example:

In the following correctly solved problem, each letter represents a different digit.

  AB
+ CB

DEA

QUESTION: What is the value of D?

What? How are you supposed to know that? Even when they don't bother to tell you anything about any of the other numbers?

How? By thinking.

You have two 2-digit numbers, and a plus sign, and their sum (DEA). That sum is a three-digit number. I promise you that if you think about it long and hard enough, you will realize that there is a rule...

RULE: Any time that two 2-digit numbers are added together and produce a 3-digit number, the first digit of that three-digit number is a 1.

Why?

Well, the largest 2-digit number there is happens to be 99. And 99 + 99 = 198. After that, try a few others and see for yourself!

NOTE: Alphanumeric problems are usually accompanied by the phrase, "Each letter represents a digit" or something to that effect. However, there is an elaboration which should be included (but rarely is), and that is this: It is true that once you figure out the value of a letter, say a B for example, then you can replace all the B's with that value, BECAUSE EVERY DIGIT REPRESENTED BY THE SAME LETTER IS THE SAME DIGIT, SO IF B = 4, THEN B8B = 484. HOWEVER, THERE IS A CATCH: Two different letters can end up equaling the same digit unless the instructions for the problem inform you otherwise. So if B = 4, and there is a C and a D, C might also equal 4, and C might equal 3 or 9 or 7 (or whatever).

Well, that's just one tip out of the fifty or so I could probably outline here, but then, it won't be a 'tip' and no one will read it. In fact, this is already feeling long for a 'tip'.

So, hope it helps, good luck!

Mitch