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Mathematical Journeys: Inverse Operations, or "The Answer is Always 3"

 
Four years ago, I came up with this math trick. Take a look at it, and at the end I'll show you why it works!

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Let's play a game. I’m going to let you make up a math problem, and I will be able to tell you the answer from here. I can’t see what you’re doing, I’m not even in the same room as you, but I will still be able to tell you the correct answer.

Trust me. I’m a professional. Ready?

Okay. First, pick a number. It can be any number you wish, large or small. Now add 5 to that number. Got it? Okay, now double your new number (multiply by 2). Alright, now subtract 4 from the double.

Next, divide your new number by 2. Now, finally, subtract your original number from this new quotient. Got it? Okay. Here comes the cool part. Ready?

The answer is 3. Nifty, huh? What’s that? How’d I do it? Oh, magic.

Okay, okay, it’s not magic. The answer will always be 3, no matter what number you pick. Let’s illustrate this by writing it out as an algebraic expression.

Pick a number, any number. Since your number could be anything and is therefore a variable, we’ll call it b.

Add 5.

b + 5

Double that.

2(b + 5)

Subtract 4.

2(b + 5) – 4

Divide by 2.

[2(b + 5) – 4] / 2

Now subtract your original number.

([2(b + 5) – 4] / 2) — b

Okay, so let’s simplify this expression and see what we get.

([2(b + 5) – 4] / 2) — b

Let’s get that fraction out of there. Divide each term in the numerator by 2.

(b + 5) – 2 – b

That’s better. Now simplify that.

b – b + 5 – 2

5 – 2

3

See? It doesn’t matter what number you pick, because the variable cancels itself out at the end. The answer is always 3. Now, go forth and amaze your friends!

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This game is a perfect example of the concept of inverse operations. Inverse operations are operations that cancel each other out; what I sometimes refer to as “undoing” each other. Addition undoes subtraction and vice versa, multiplication undoes division. Early in the problem you double your mystery number, and then later on you divide it by two. Those two actions cancel each other out – one makes the number larger and the other shrinks it back down.

In an algebraic equation, you can effectively move a term from one side of the equals sign to the other by performing the inverse operation to both sides. Y = x + 5 becomes y – 5 = x, which can tell you the value of x instead of y. Algebra, at its heart, is the process of using these inverse operations to rewrite an equation so that it tells you the piece of information you want to know.
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