Maybe you learned it as BOMDAS but, used in any way, PEMDAS is an example of failed socialized education. Please Excuse My Dear Aunt Sally is a sad attempt to simplify something that doesn’t need simplification. It succeeds in confusing the issue of the order of operations for as many as it does clarify it. It is painful to watch, even worse to participate, in one of these social media math problems like the one in the picture. I want to get right to the math to cover as much of the bad math as I can. But, before we get there, I must make some clarifications.
8 / 2(2+2) = 1
The answer to this viral math question, what is the sum of the number eight divided by two times the quantity two plus two, is one. Now the question is, how do you get that answer? By following convention and not playing with the question. Do not rewrite the question. Do not add terms or operators to the problem. Follow the order of operations, if you really know them. Do not trust the internet or a phone calculator to solve this problem as written. Many of these devices are not programmed for complicated math. Some devices can’t even type the problem as written and, they try to assume things they shouldn’t. That search link up there will show you a device that rewrites your question no matter how you type it.
Math is Reduction
It may seem contrary especially when adding but, all math is reduction. You start with many terms and end with as few terms as possible and, hopefully, no lingering operators. Yes, the 2 in 1+1=2 is bigger than either of the 1’s but, we have reduced two terms and one operator to one term. Simple but, what happens when you have a more complicated problem? How do you speak that problem and get people to write it down correctly as you envision and follow the order of operations to solve it?
Traditionally PEMDAS has been the solution, it’s cute and people remember it. But there is a problem with memory devices. I had a link to a Harvard paper calling PEMDAS ambiguous but, I lost a ‘friend’ and the link at the same time when I called him out on his “appeal to authority” before ever giving his own answer. At any rate, the abstract, not a rule, says that PEMDAS is ambiguous. The memory device is ambiguous but, what it represents, the order of operations is not. PEMDAS could as easily be EPDMSA. The point is, PEMDAS is not literal. You do not skip over M to get to D. Once you have solved P and E, or E and P if you will, you go back to the beginning to look for M and D. You solve them from left to right where they have no other terms to consider.
PEMDAS is Soooo Cute!
So, PEMDAS is cute and helps people remember something about math but, if they don’t know the rest of the rules they err in their work and thus their answer. I don’t think this can be fixed so maybe it’s best to eliminate it entirely? But, that’s not what I’m getting at here, this is about the math. The first thing people forget is Distribution. It really amazes me because I will present this problem 2(2 + 2) and people will get it right but, add another term and operator and they’re thrown off.
The mathematical expression of this problem is a(b+c), verbally this is “A times the quantity B plus C”. This problem is solved using the distribution method. Both B and C are multiplied by A and then added together. A x B + A x C. Why do we have this convention? When you add numbers, this convention doesn’t matter but, when you multiply or divide, it makes all the difference in the world.
The problem 2(2 + 2) = 8 no matter how you do the math. If you solve inside first, 2(4), the answer is 8. If you solve by distribution 2(2) + 2(2), the answer is still 8. But, what if the operation of the quantity is multiplication or division? What if the numbers aren’t the same? Take the problem 2(2 x 3), “Two times the quantity two times three”. If you incorrectly “straight math it”, 2(2 x 3) is the same as 2 x 2 x 3 or even 2(6) which all equal 12. But, through correct distribution 2(2 x 3), becomes 2(2) x 2(3) or 2 x 2 x 2 x 3 and all equal 24.
Re-Writing the Problem
It will not help one bit to rewrite the problem. The problem is correct and solvable using conventional math. The most common error I’ve seen is re-writing the problem as 8 / 2 x (2 + 2). When you say this or write it out, it reads “Eight divided by two times two plus two”. That’s a different problem. It suggests that the first two in the equation from left to right is not dependent on any other terms or operators. It tells you something different than you learned in school. Remember, 8 / 2 x (2 + 2) is the same as 8 / 2 x 1(2+2). Let’s do some math, 8 / 2 x 1(2+2) is 8 / 2 x 4 which does equal 16 but, that’s not what the original question asks, you’ve rewritten the problem and that’s why you get the wrong answer.
I found a video, the same one you watched probably? In that video, the guy says he had to go back to 1917 to find an example of math done this way. I’m not 102 years old. I was taught the basics of math just 25 or so years ago. Math has not changed. The way it’s taught has changed. Like, little tricks and mnemonics now used to get people to solve a problem instead of just know the answer from memorization.
Wrong PEMDAS, Wrong Answer
His answer is wrong. The answer “16” is wrong. It does not matter how he justifies being wrong, he’s still wrong. The reason for the distribution method of math is clarification in written problems. It’s also how you show to multiply using fewer terms. For instance, 7 x 4 can be written as, 7 + 7 + 7 + 7 or, 4 + 4 + 4 + 4 + 4 + 4 + 4 or, 4 x 5 + 4 x 2 or, 4(5+2). These are all the same problem and they equal 28 respectively. Change the answer by adding a new term and operator and doing simple left to right math. Once again, keeping in mind that Multiplication and Division are equal, and you don’t skip D’s to get to M’s. For example;
8 / 7 x 4 = 4.6 (rounded)
Eight divided by seven times four equals four point six.
8 / 7 + 7 + 7 + 7 = 22.143 (rounded)
Eight divided by seven plus seven, plus seven, plus seven, equals twenty-two point one four three.
8 / 4 + 4 + 4 + 4 + 4 + 4 + 4 = 26
Eight divided by four plus four, plus four, plus four, plus four, plus four, plus four, equals twenty-six.
8 / 4 x 5 + 4 x 2= 18
Eight divided by four times five plus four times two equals eighteen.
8 / (7 x 4) = .29
Eight divided by the quantity seven times four equals point two nine.
Remember when the teacher said, 8 / (7 x 4) is the same as 8 / 1(7 x 4)? I do, clearly.
8 / 28 = .29
Eight divided by twenty-eight equals point two nine.
These are very different answers but, what proves which is/are right where our actual a(b+c) problem is concerned? Simple, keep the original thought in mind here. 7 x 4 = 28, this is a fact. So, if 8 / 28 = .29 then 8 / 4(5+2) also equals .29 as does 8 / 4(7). Parentheses around the seven are specific and indicate that multiplication is done before the division. You must get rid of the parentheses. Let me state it again, this is done so that you can READ a math problem and SAY a math problem to someone else and have them write it down how you mean it.
Say the Problem Out Loud
What is the sum of the number eight divided by two times the quantity two plus two? 8 / 2(2 + 2) = ? The answer is one, it has always been one, it will always be one. If you get an answer other than one, you did something wrong. Most likely, the order of operations and failure to understand why they have a specific order is your mistake. You must solve for the denominator, you must solve the quantity before you can divide because that’s what the problem is asking for. The number eight divided by a set of as yet unsolved numbers. The problem does NOT ask “What is the sum of the number eight divided by two times two plus two”?
Can you see the difference? It should be totally clear by now; it is the use of the word “quantity” and knowing how it’s written mathematically for the purpose of operational order. If you cannot say these problems using words, then you cannot communicate them to someone else without a visual representation to accompany and, they won’t know what you mean. If you can’t read the problem in terms of words you won’t know how to solve it correctly. In the end, it’s just a symptom of the usual socialist problem, one size fits all solutions for different size problems. Instead of learning to do the work socialism teaches what is easy and expedient. Socialism takes complication and makes it more complicated. Socialism creates problems where there are none. PEMDAS is a Great Example of Failed Socialized Education.