Simple word problem puzzle

Recently I read this question online:

The sum of two numbers is 91. And the difference is 31. What are the two numbers?

The first  answer to it simply  gave the answer as,  “The two numbers are 66 and 25.”

I think the whole thing is a waste of time for all concerned, so far. The asker has learned nothing, and the answerer has taught nothing.

Just giving an answer is “show–and–tell” teaching. It serves no purpose except to show off that you have an answer. It doesn’t teach anything. It’s the old phenomenon of,”Give a man a fish and you make him dependent on you.”

Let’s see if we can make this meaningful with math, by figuring out how we could come up with this answer.

You could try this by guess-and-check, But you wouldn’t really learn a system that would help you solve these kind of things in the future. So let’s use what we know to discover what we don’t.

We know that the two numbers have a difference of 31. So both numbers are unknown at the moment, but we do know that there is a relationship between the two.

So we can call one number x, and describe the other one as its relationship to x. In this case, we know that if one number is x, the other one is x+31. Alternately we could call them x and x -31.

I will randomly choose the first pair:  x and x + 31.

So now we can write it as an equation, using only one variable to express what we don’t know.

x (one number) + x + 31 (plus some other number that is 31 more than the first number) = 91.

That’s

• x + x + 31 = 91.

That’s the same as 2x + 31 = 91.

• Subtract 31 from both sides (because according to the Addition Principle, if two sides of an equation are equal, you can add or subtract equal amounts to each side, and the solution set will remain the same) and you’ll get 2x = 60.

Can you take it from there? I’m sure you can, but for the sake of completion, let’s do it together.

• Divide both sides by 2 (because according to the Multiplication Property of Equality, if two sides of an equation are equal, you can multiply or divide equal both sides by the same non-zero number, and the solution set will remain the same) and you will get x = 30.

• You know the answer to one of the numbers  (x = 30). You also know its relationship to the other number, which is x + 31.  Add 31 to 30, and you will get 61.

Therefore,  the numbers are 30 and 61. Check it.  Do 30 and 61 add up to 91? Yes. Is the difference between 30 and 61 31? It is.

Do we want to leave it at that? I don’t think so. It would be good to check if other numbers could fit in the equation, too, so we know if our answer is unique.

If you make either number higher, you would have to make the other one lower in order for them to still equal 91. But if you make one lower and the other higher, you will increase the difference between them, so the difference could not remain at 31.

In other words there is no way that you could change either one of the addends and still get the sum of 91.

So we have gotten the answer, and proven that it can be the only answer.

Why do we care if it is the only answer? Ah, here is where we try to make this meaningful in real life…

There are many questions, decisions, etc. that we are faced with in life that can have more then one answer. Simple people  (by this I mean “simpletons,” not “regular people”) like simple answers to complex questions. Say, for example, there are many ways to solve an argument. One is by violence. It might work. But logic might work, flattery might work, mutual benefit might work, as well as many other things. But a simpleton would grasp for the first thing he or she could think of, and tell themselves, “I feel threatened by this situation – If I club the other person, it will end the threat.”

Yes, it might. And it might not. And other things might work better. If the simpleton stopped with the first answer that might work, they wouldn’t realize that there are other solutions that are better for everyone, the simpleton included.

If you filter out all the answers that don’t work, you are left with a much clearer choice. Math is a great way to learn effective decision-making strategy.

Thinking through things, and working them out together, and making sure we understand each step of the way together is a lot more productive than having someone hand you the answer, isn’t it? If we teach this way to our children, it encourages them, enables them to explore their own minds, and demonstrates how to use critical thinking to evaluate solutions to problems. It also encourages them to check their answers, and not just blindly accept “whatever works.”

This is what Math Mojo is about – making math meaningful.

Even if you knew all of this before you read this, I hope you got something out of it, even if it was just to reaffirm your healthy appreciation of critical thinking.

All the best to you in your learning/teaching endeavors,

Brian (a.k.a. Professor Homunculus)

Afterthoughts: Can any reader here see how understanding why we don’t divide by 0 in arithmetic also illustrates the lesson from this post? Leave a comment if you do.