After posting Trending: Only for genius ?? 3 – 3 x 6 + 2 = ?? yesterday, I had to post a follow up article with the correct answer.

Even with the hint about remembering the order of operations, people still got it wrong on my poll. If you haven’t done the poll yet, click on the link above and respond.

The question was

The correct answer is as follows:

- Question: 3 – 3 x 6 + 2
- Multiplication first: 3 – 18 + 2
- Left to right: -15 + 2 (or Addition first: 3 – 16)
- Answer: -13

**[Start Edit]** I have been asked in the comments numerous times, why step three just calculates left to right? “Shouldn’t you do addition first?”. So I am adding this explanation section to the top of the article (it is similar to information in the linked related posts).

*Order of operations (excluding the first two parts of Parenthesis/Brackets and Exponents/Orders/Indices) is [DM] or [MD] followed by [AS] or [SA]. So, Multiplication and Division can be done at the same time and then Addition and Subtraction can be done at the same time.*

*Try calculating the following examples from left to right:*

*Multiplication & Division: 4 x 3 / 2 = 6 and 4 / 2 x 3 = 6 …. The order is irrelevant.*

*Addition and Subtraction: 10 – 5 + 2 = 7 and 10 + 2 – 5 = 7 …. The order is irrelevant.*

*In fact, Division is just Multiplication of the inverse number. For example: Dividing by 2 is the same as multiplying by 1/2. *

*Also, Subtraction is just Addition of a negative number. For example: Subtracting 10 is the same as adding -10.*

*Maths is exact, that is one of the beautiful things about it. As long as you do it correctly you will always get the same answer.*

*For those of you who still insist on doing addition before subtraction:*

*Question: 3 – 3 x 6 + 2**Multiplication first: 3 – 18 + 2**Addition next: 3 – 16**Answer: -13 … the same answer as before.*

*Please note that -18 + 2 is not -20, it is -16. You cannot ignore the minus sign in front of the 18. *

*Also, 3 – 18 + 2 is not equal to 3 – (18 + 2) it is equal to 3 – (18 -2) because there is a minus sign in front of the brackets to the sign of the 2 must be changed when moving it inside the brackets.*

*If you don’t believe me, try solving 3 + 2 – 3 x 6 or 3 + -3 x 6 + 2 which are equivalents to the original equation.*

*All numbers (except for zero) are either positive or negative. Negative numbers have a minus sign in front of them and positive numbers have a plus sign in front of them. But, because we are lazy, we normally don’t write the plus sign unless it is between two numbers.*

*So 3 – 3 x 6 + 2 becomes +3 -3 x +6 +2 = +3 -18 +2. Now you can use a Number Line to finish the calculation.*

*The Answer still is and will always be -13.*

**[End Edit]**

**So what do you get if you type it into a calculator (physical or app)?**

That depends on if the calculator is simple and calculates as you type, or if it is smart and waits for the entire equation to be entered and the equals button pressed.

On my Windows 10 Mobile phone, the calculator app can behave in both ways.

In Scientific mode, it allows the entire equation to be entered and so correctly calculates the answer as -13 using the “Order of Operations” rules (same as above).

In Standard mode, it calculates as you type and so incorrectly gives the answer as 2. The way to think of it is that it inherently presses equals each time you press an operation button. This is just left to right calculation and ignores the “Order of Operations” rules.

- 3 = 3
- 3 – 3 = 0
- 0 x 6 = 0
- 0 + 2 = 2

If your calculator behaves in this simple way, you would need to apply the rules yourself. Here are the buttons to press:

- 3 x 6 = MS
- 3 – MR + 2 =

If your calculator has brackets or parentheses buttons you can use them instead of the Memory buttons to achieve the same effect.

- 3 – ( 3 x 6 ) + 2

Looking at the poll results from the previous article, there are a number of incorrect answers being selected, so I thought I would try to identify the **actual** equations that would give those incorrect answers (changes highlighted in red):

- -17 = 3 – ( 3 x 6 + 2 )
- -17 = 3 – 3 x 6 – 2
- 0 = ( 3 – 3 ) x ( 6 + 2 )
- 2 = ( 3 – 3 ) x 6 + 2
- 8 = 3 – 3 + 6 + 2
- 13 = – 3 + 3 x 6 – 2
- 17 = – 3 + 3 x 6 + 2

Hope this all makes sense. Here endeth the lesson. 🙂

Have a look at the other posts in the series:

- Trending: Only for genius ?? 3 – 3 x 6 + 2 = ?? With Poll and Order of Operations
- Answer: Only for genius ?? 3 – 3 x 6 + 2 = ?? This article
- Conclusion: Only for genius ?? 3 – 3 x 6 + 2 = ?? Breaks down solution into steps
- The Last Word: Only for genius ?? 3 – 3 x 6 + 2 = ?? Explains ordering of DM and AS
- Viral: Only for genius ?? 3 – 3 x 6 + 2 = ?? Answer has gone viral and explanations

Enjoy

David

**03-Dec-2016:** Added additional explanations at the top of the article.

**05-Dec-2016:** Added more additional information on number lines.

This article was originally posted on http://www.winthropdc.com/blog.

So stupid

LikeLike

But did you get the correct answer?

David

LikeLike

Yes -13

LikeLike

I’m still getting confused by with answer, why -13 not 2

LikeLike

Hi Wisdom

There are rules in mathematics which specify the order of operations. These rules say that Multiplication and Division must be performed before Addition and Subtraction.

If you obey the rules you get -13. If you don’t obey the rules you get the incorrect answer of 2.

David

LikeLike

As mentioned above, maths is a precise science at most times, but leaving the door open for ambiguity and error by omitting brackets is NOT a precise way to offer a formula to calculate. It is not a matter of precision to state that an external fact – that an externally designated order of precedence – must prevail as it is a matter of convention. If I were to ask someone to calculate something I would maximise the chance of getting the right answer by supplying brackets. I was, when younger, a 1st class mathematician, so this is not some way to avoid calculations.

LikeLike

Hi Neil

Maths is precise ALL the time. People doing incorrect maths and getting wrong answers does not make maths any less precise.

Order of Precedence or Order of Operations is not some external guideline. It is part of the laws of mathematics. It cannot be applied optionally.

Yes, adding brackets can help make it easier and less ambiguous but they are not required in any way. Note that brackets/parenthesis are included the order of operations rules.

If you look through the comments, most people actually do correctly do the multiplication first, but then fail to add three numbers 3, -18 & 2 together because they don’t understand how to add a negative number.

I even had one comment that indicated that negative numbers can’t exist and that the equation was not possible to solve!

David

LikeLike

My point is that it jeopardises precision by omitting brackets. The practical offshoot of the omission is to expose the less capable to more errors. It is precise to give this calculation as the fare on a bus, but would the bus operator trust that all would get it right? Being explicit with non-mathematicians is the wiser path to take. Mathematicians can and should take all appropriate shortcuts of notation.

LikeLike

Hi Neil

You might be right, but where is the fun in that?

The challenge of the puzzle was to see who remembered what they were taught in their primary/elementary school maths classes.

Asking ((3 – (3 x 6)) + 2) = ? makes it too easy.

David

LikeLike

Please someone at the should simplify the equation

LikeLike

Hi Wisdom

The challenge is for you to solve the equation.

David

LikeLike

Hi,

I came across this 3-3*6+2=?? on my facebook timeline and was interested in the various answers that were put forward which piqued my interest so I looked a little deeper and I ended up on this page.

Now I am a pensioner so 50 plus years past schooled maths and my vaguely remembered priorities of actions (PEDMAS/BODMAS) started to explain just how people came to all these various answers.

So sorry to be contradictory but to quote you “Maths is exact, that is one of the beautiful things about it. As long as you do it correctly you will always get the same answer.” is really not true and should probably really read “as long as you do it this one way (apply the same rules) you will always get the same answer” which in reality probably holds true for all of the other methods employed.

The item in question 3-3*6+2=?? is devoid of periods and parenthesis or punctuation (unless you want to argue that the mathematical qualifiers are punctuation) which I think is quite deliberate from the perspective of the person that set it as they were exhibiting their arrogance in their mathematical superiority, so if I apply your “maths is exact” and read it exactly as it is I arrive at 2 and not -13.

So your opening statement of “The correct answer is” is in reality incorrect and should more like the correct answer can be, or the correct answers are.

You mention phones and calculators applying different methods of achieving the answer so does that not really mean BOTH answers are correct in their own way as no matter how many times you try, the answer will always be the same as long as you apply the same method?

LikeLiked by 1 person

Hi Neville

Thanks for your in-depth thoughts.

Mathematics has rules to follow. One of those rules is order of operations, which specifies that Multiplication and Division must be performed before Addition and Subtraction.

In the absence of brackets to change the order of operations, you must fall back to the rules and perform the multiplication first. If you follow the rules the answer will always be -13.

A simple calculator does not understand rules as it can only work with two numbers at a time. It relies on the operator to perform the calculations in the correct order (ie. do Multiplication first). If the operator does not obey the rules, you will get the incorrect answer of 2.

A more advanced calculator which can have the entire equation entered before calculating the solution, understands order of operations and will get the correct answer of -13.

I still stand by what I said, “Maths is exact”. You must obey the fundamental rules of maths including order of operations. If you don’t, you will likely get an incorrect answer.

There might be multiple answers (such as 2, -17 and -13), but only one is correct and that is -13.

Regards

David

LikeLike

Line 3 in your explanation is incorrect. You are treating 18 as a negative number before the subtraction actually makes the product a negative number 18 plus 2 is 20 not -16

LikeLiked by 1 person

Sorry Deb

Type 3 – 18 + 2 into any calculator and it will say -13 not -17.

Thus proving that -18 + 2 is -16 not -20.

Remember that subtracting 18 is exactly the same as adding -18.

See the number line examples in the added explanation in this article.

David

LikeLike

3-(18+2)

3-20

YOU MUST DO ADDITION BEFORE SUBTRACTION. I don’t care what your electronic device says.

LikeLiked by 1 person

Hi Deb

I am sorry, but you are still wrong. Please read the explanation in the article above.

Even if I do addition before subtraction the answer is still -13. However, addition and subtraction can be done at the same time.

You have made an error in your simplification.

3 – 18 + 2 becomes 3 – (18 – 2) = 3 – 16 = -13

When you move the 2 inside the bracket, you must change the sign because the brackets have a negative sign in front of them.

Without brackets you can do the same calculation:

3 – 18 + 2 becomes 3 + -18 + 2 = 3 + -16 = -13

David

LikeLike

Deb is correct. David you are wrong

LikeLike

If you believe that, then I hope you are not involved in any engineering works.

LikeLike

I’m dumbfounded by some of the comments. How some of you people get by in your day to day lives I’ll never understand.

LikeLike

Hi Sebastian

Did you get the correct answer then?

LikeLike

how did you come about -13

LikeLike

Hi

I came about -13 because that is the answer. Did you read the article before posting a comment?

David

LikeLike

You are wrong. The mathematical sequence is always BODMAS (brackets first, then OF, if it is a worded equation, then division, followed by multiplication, addition and lastly subtraction) In this equation you do the multiplication first which is 3×6 = 18, then you do the addition 18+2=20 and lastly you do the subtraction 3-20 = -17

LikeLike

Sadly, it is you who is wrong. You obviously did not read this article and the related articles. The rules are actually BO[DM][AS] as the order of the Division and Multiplication steps are not important and the order of Addition and Subtraction steps are not important.

Even if I do the calculation in the order you want:

3 – 3 x 6 + 2

Multiplication first: -3 x 6 = -18

Addition -18 + 2 = -16

Subtraction 3 – 16 = -13

You CANNOT ignore the minus sign on the -3 and only remember it at the end.

Google agrees with me: https://www.google.com.au/?gfe_rd=cr&ei=Hf-KWMHCA8Lr8Ae626DIDw#q=3+-+3+x+6+%2B+2

Bing agrees with me: http://www.bing.com/search?q=3+-+3+x+6+%2b+2&qs=AS&pq=3+-+&sc=8-4&cvid=446147D2D9FF453B979C92D251477C72&FORM=QBLH&sp=1

LikeLike

But remembering a rule of math is not IQ. To know this question does not require intelligence just a good memory from school. So though most didn’t get the answer correct it only proves lack of good education in math (failure of the school system) or poor memory skills. When was the last time as an adult at work someone asked you what is 27-6*11+3= is there a big call for this. Sorry I cant find my equals key I have a Japanese keyboard [Ed: I fixed it for you].

LikeLike

Hi Jeffrey

I agree that this is a test of how well you were taught basic maths or at least how well you remember it and is not an IQ test.

This blog article in the series includes a real world example where you need to solve 3 – 3 x 6 – 2:

https://winthropdc.wordpress.com/2016/02/29/the-last-word-only-for-genius-3-3-x-6-2/

As for your equation: 27 – 6 x 11 + 3 = 27 – 66 + 3 = 30 – 66 = -36

David

LikeLike

David:

what’s wrong with the dumb and stupid American?

LikeLike

Hi Keko

The issue here is not just with Americans. The lack of basic maths knowledge is worldwide.

David

LikeLike

I know the lack of basic maths knowledge is worldwide , only American argue even when they are wrong

LikeLike

I found this when searching for something else, a computer algebra package called genius, and this is just so stupid that I cannot help to comment on it. There are rules in mathematics that are taught at school (in an early (!) year of elementary school when pupils advance from addition to multiplication). Multiplication has priority, that’s just it. No way to get a wrong result, as there is no ambiguity. I do not understand why you publish such a scrap (sorry) in the internet.

LikeLike

Alexander,

I posted an entire series of articles based on this simple maths question after many people could not answer it correctly. I know that there is no ambiguity, but if you look at the comments (and the poll on the first post), you will see that most people get that multiplication should be done first but then they fail to add negative numbers. Then they argue with me to about their incorrect answers. My article is trying to help people get the correct answer and the information in it is accurate.

David

LikeLike

Hi David,

I think I know what you mean. This is incredible. Good luck.

Alex

PS: instead of -18+2 you may as well calculate 2-18, if there is no sign-changing key

on the simple computer

LikeLike

K sir i agree with your comment that -18+2=-16

But In simple calculator we don’t have options to write -18+2. Then how students are agree practically?

LikeLike

Hi Goudappa

On a simple calculator there is usually a sign changing key that looks like a plus and minus sign on it. You need to use this key to enter minus eighteen.

Press (1) (8) (+/-) (+) (2) (=) and it will show -16.

David

LikeLike

Since when did subtraction come before addition? -17

LikeLike

Hi Ethan

Did you read the article? The order of Subtraction and Addition is not important. Subtraction is just adding a negative number.

You will get the same answer whether you add or subtract first.

Since when does -18 + 2 not equal -16?

LikeLike