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

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But did you get the correct answer?

David

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Yes -13

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

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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

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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.

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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

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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.

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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

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Please someone at the should simplify the equation

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Hi Wisdom

The challenge is for you to solve the equation.

David

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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?

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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

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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

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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

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3-(18+2)

3-20

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

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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

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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.

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Hi Sebastian

Did you get the correct answer then?

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how did you come about -13

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Hi

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

David

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