Saturday, March 19, 2016

Another order of operations adventure

Here's a gem I've seen floating about in Facebookland lately. A lot of people are answering with either 58 or 10. What do you think?

Find this puzzle and more awesome stuff at Curiosity
What do you think? Did you get 58? Did you maybe get 10? 

If your answer is 10, you likely made a mistake in using the order of operations. Let's talk about what that mistake was.

The Order of Operations

As you may know from my previous posts (like Where the Math is Lacking), I love a good math problem. Problems like the one above are a great way to see whether people remember the Order of Operations for solving mathematical equations. 

In case you have forgotten the order of operations, then you might recall the mnemonic acronym "Please Excuse My Dear Aunt Sally", which stands for PEMDAS.

PEMDAS is an acronym for the order of operations and it stands for Parentheses-Exponents-Multiplication/Division-Addition/Subtraction:

This order of operations allows us to make sure that we're all following the same rules for solving or simplifying problems, or, at least, that's what it's supposed to do. Unfortunately, a lot of people have forgotten how to use the order of operations. Let's talk about that problem above again and how the order of operations applies to it.

We have the following problem:

62 ÷ 2(3) + 4 = ?

Let's step through each part of PEMDAS and see how the order of operations applies to this problem.

We have to start with the Parentheses part of the order. This is actually the step where most people get the wrong answer to this problem. The Parentheses step means that we have to solve terms or sets of terms within parentheses and brackets before solving the rest of the problem. 

For instance, if I had 2+(3+4), then I would solve the stuff inside of the parentheses first to get 2+(7). At that point, the parentheses are no longer necessary, since 2+(7)=2+7=9.

In our problem above, what's inside of the parentheses (the number 3) doesn't need to be operated on since it's already simplified. So, following the parentheses step, we still have the problem:

62 ÷ 2(3) + 4 = ?

Now moving on to the Exponents part of PEMDAS, we see that we have to solve the six squared term. That's pretty simple:

62 ÷ 2(3) + 4 = 36 ÷ 2(3) + 4 = ?

Then, the Multiplication and Division step tells us to solve all multiplications and divisions in the problem, but there's also a caveat there as well. Since many western languages read from left-to-right, we've also developed our modern mathematics to read left-to-right. This means that we have to sequentially solve multiplications and divisions reading from left to right in a problem. So for out current version of the problem,

36 ÷ 2(3) + 4 = ?

We have to solve the division of 36 by 2 first and then we can apply the multiplication by 3. Yielding:

36 ÷ 2(3) + 4 = 18(3) + 4 = ?

and then

18(3) + 4 = 54 + 4 = ?

And now we can take this thing home by solving the final step, Addition and Subtraction.

54 + 4 = 58

Bump bump baaaaaaah! The answer to this problem is 58

So how did some people get 10? 

Like I said, most people who had trouble with this problem made their mistake in the first step. When solving for the parentheses part of PEMDAS, you only have to sequentially solve the parts that are within parentheses and/or brackets. However, if you apply the multiplication outside of the parentheses first, then you'd be solving the problem like this:

62 ÷ 2(3) + 4 = 
62 ÷ 6 + 4 = 
36 ÷ 6 + 4 = 
6 + 4 = 10

However, this is actually the wrong way to solve this problem. If you apply a multiplication outside of the parentheses first, you're basically then solving the operations out of order (MPEMDAS, in this case). The parentheses step in PEMDAS doesn't apply to operations outside of parentheses or brackets.

Here's another example I can throw at you, say we have this problem:

6 × (5 + 3) + 4 = ?

We could take away the multiplication symbol since the parentheses will already tell us to multiply, yielding:

6(5 + 3) + 4 = ?

Now it should be clear that we solve the parentheses first and then multiply, giving us:

6(5 + 3) + 4 = 6(8) + 4 = 48 + 4 = 52

The same way that we removed the multiplication symbol in that problem could be applied to the original problem above. When you look at the problem and see it like this:

62 ÷ 2 × (3) + 4 = ?

Then it becomes far more apparent that the answer won't be 10. We can then see the right answer will be 58:

62 ÷ 2 × (3) + 4 = 58

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