So, how's your dear Aunt Sally these days?

When you evaluate the expression \( 5 \times 5 \div 5(1+4) \),
​which of the following is the correct answer?
A)   \(1\)
B)  \( 25\)
C)  \(125\)
D)  That's a poorly written math expression.
E)  A and D
F)  B and D
​G)  C and D

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Believe it or not, the answer you give is determined by how well you know your Dear Aunt Sally!!  Remember her?  She's that stickler for rules that you probably met in third or fourth grade.  Does PEMDAS ring a bell?  When I was in school, we were taught to remember PEMDAS by thinking of it as an acronym for "Please Excuse My Dear Aunt Sally".  

The whole point of learning PEMDAS was to permanently stamp upon our brains the order of operations for arithmetic.  Without rules for a precise ordering, any arithmetic expression becomes subject to dispute.  There is no longer a certainty of meaning.  Mathematical communication breaks down.  Anarchy ensues.  And then fights will break out, chalk and erasers will be thrown, etc.

​So, our Dear Aunt Sally is supposed to help us remember PEMDAS.  And PEMDAS is supposed to help us remember...well...do you remember?  Let's see, the "P" stands for "Parentheses".  You always start with Parentheses.  Then Exponents.  Then Multiplication and Division, in order from left to right.  And finally Addition and Subtraction, also in order from left to right. 

Now, take note.  Because most people forget about "in order from left to right", and that causes problems.  Lots of problems!!  Some folks think that since "M" is before "D" in PEMDAS, then Multiplication is always performed before Division, wherever it occurs within the expression.  But, NO!  That's not correct.  Multiplication and Division share the same hierarchy, and they are performed on a "first come, first served" basis, as you travel from left to right across the expression.  Same goes for Addition and Subtraction.

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Let's look at the expression in our problem again:  \(5 \times 5 \div 5(1+4) \).  
First we are going to work what is inside the parentheses.  This gives us \( 5 \times 5 \div 5(5) \).  Now, this is where many folks will mess up.  They will automatically multiply those last two 5's together first, before doing anything else.  Why will they do this?  Well, often folks will do it because the multiplication is written using parentheses, and they're still thinking "Parentheses go first!".  But the "P" in PEMDAS only says that what's inside parentheses goes first, not whatever is touching parentheses.  Remember that \(5(5) \) really means \( 5 \times 5 \), so we could rewrite the expression as \( 5 \times 5 \div 5 \times 5 \). 

And now, here's where others will mess up.  They're stuck thinking Multiplication always comes before Division, because "M" comes before "D" in PEMDAS.  But recall what we said before.  Multiplication and Division have the same hierarchy, so do them in order form left to right. 

So....let's do them in order from left to right.  This gives
\[ \begin{array}{c} 25 \div 5 \times 5 &=&  5 \times 5\\
 &=& 25 \end{array} \]

Okay, we see the expression evaluates to 25.  But the answer is not (B).  No, it's (F).  Why?  Because the expression really is poorly written!  It's written as if...well, almost as if it was designed to trip people up.  (Probably because it was!)    In general, it's a bad idea to use the symbol \( \div \) in a math problem involving more than two numbers, unless you're prepared to make generous use of parentheses to remove any doubt as to intention.  Writing mathematical expressions is just like writing sentences - the purpose is to convey ideas accurately and efficiently.  Nine out of 10 people reading the expression above are going to breeze right through it and evaluate it incorrectly - unless they suspect it's a test, or some type of puzzle, and then slow down.   

So, yeah, most complex math problems don't involve the \( \div \) symbol.  Instead they represent division using fractions.  It's just clearer that way, at first sight.  Hey, they're being nice to the speed readers!    

And we wouldn't be good math professors if we didn't assign some homework, of course.  Here are some practice problems to help you remember your dear Aunt Sally.  We will post answers tomorrow.

1. \(5 \div 5(1+4) \)
2. \(5(1+4) \div 5 \times 5 \)
3. \(5 \times 5 \div (1+4) \)
4. \((5 \times 5) \div [5(1+4)] \)
5. \(5(5 \div 5) (1+4)\)
6. \(5 \times 5 \div[5(1+4)] \)

Solutions to the homework problems can be found here.

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