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In a Mensa calendar, A daily challenge - your daily brain workout. I got this and put a challenge up at work.

The Challenge starts with..

Assume you are using a basic calculator and press the numbers in the order shown,
replacing each question mark ...continues... What is the highest number you can 
possibly score?

Basically, only using $+,-, * ,\div$, once in place of a question mark.

$5 ? 4 ? 7 ? 3 ? 2 =$

We all worked out the operators to be

$5 + 4 $ x $ 7 - 3/2 =$

Except that I calculated the answer to be $31.5$ and the others argued $30$. THe answer sheet from MENSA says the calculated total is 30.

Nobody actually understood the first part about using a basic calculator. I initially thought the challenge was all about the rules of maths. And when I asked why nobody else applied the rules of maths, they all forgot about it, not because the challenge said to use some "basic calculation"

I emailed MENSA and queried them about the challenge and they replied,

Thank you for your email.

On a basic calculator it will be:
5 + 4 = 9
9 x 7 = 63
63 – 3 = 60
60 ÷ 2 = 30

Kind regards,
Puzzle Team

My Reply,

 Thank you for your reply. 

 Could you please define what a basic calculator is? I tried 4 pocket,
 £1 calculators, and all gave me 31.5.

And finally their definition.

I guess what the question really means, whether you do the sum manually or on a 
calculator,  is don’t change the order of the numbers. The Casio calculators we have in 
the office allow you to do the sum as it appears:

5 + 4 = 9
9 x 7 = 63
63 – 3 = 60
60 ÷ 2 = 30

Kind regards,
Puzzle Team

So they guess the challenge meant to do it that way. Why not just say ignore rules of maths. What is the point of this anyway?


My original question, on Maths Stack (this one) was why MENSA used 30 instead of 31.5. And initially I did not understand that using a basic calculator meant calculating left to right by pressing equals after each operation.

So what is going on here? If they wanted us the ignore rules of math they should of said taht. Because my basic calculator gives me 31.5 and not 30.0 (I dont have a special. Casio MENSA calculator though)

Windows standard calculator gives me 30. Why? None of my pocket, office, el cheapo calculators do this.

enter image description here

Google, or Windows Scientific give me 31.5 - As do ally my elelctornic calculators.

enter image description here

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Windows calculator gave me 11 though... –  ireallydonknow Feb 11 at 15:35
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Seriously. I type in 5 + 2 * 3 in that order and get 21. I change to Scientific Calc and i get 11???? WHat about the MENSA question. is it 30 or 31.5? –  ppumkin Feb 11 at 15:37
17  
Technically, the rule isn't "multiplication before division" and "subtraction before adding". It's "multiplication and division before addition and subtraction", performed left to right. But that's really just a convention we learn as kids. In any serious mathematical pursuit, parentheses are utilized to remove any ambiguity. –  Roger Feb 11 at 15:41
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@ppumkin - Before finishing reading your question or any of the answers I wanted to work out the problem myself. I got 31.5. So you aren't alone! Maybe it's due to my profession of being a developer/programmer, but I always have had it ingrained in my mind the rule of operations: left to right, multiplication and division before addition and subtraction. –  Code Maverick Feb 11 at 17:07
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I saw the facebook quiz results with basic math (order of operations) and thought "surely they must all be trolling..." The apocalypse really is drawing near! –  emragins Feb 11 at 19:56
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12 Answers

up vote 25 down vote accepted

In response to edit of initial post, then answer is clearly $30$. Basic calculators are assumed to evaluate in order from left to right.


Original post I responded to

In a Mensa calander, IQ dialy challenge I got this and put a challenge up at work.

Using +,-,time and divide only once. Use the math operator only once to get the highest answer.

5 ? 4 ? 7 ? 3 ? 2 =

We all worked out

5 + 4 x 7 - 3 / 2 = 30

Except that my result answer was 31.5 and not 30, like in the answers of the MENSA calendar.

Why was I the only one that applied the rules of maths on this? ANd when I asked why nobody else applied the rule of maths, I got the weirdest looks. Nobody knew about multiplication before division, subtraction before adding? I thought that was why the question was marked as the most difficutl to test if you knew this.


Response to original post

Sadly, many people forget the basic rules of arithmetic as they (a) don't view them as affecting their lives, (b) didn't like maths, and/or (c) know technology can handle the problem for them. The issue with the last point is that different technologies handle things differently. The Google calculator (much like most graphing calculators) will handle order of operations for you correctly. The standard Windows calculator appears to be operating like an old 4 function calculator which evaluates after every operation is completed as opposed to correct order of operations. Though this can also happen when users hit enter after every operation is finished as opposed to when the whole expression is finished. (Don't have access to a Windows calculator right now so can't tell which is the reason for the wrong answer.)

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@ppumkin Your expression certainly evaluates to $31.5$. –  John Habert Feb 11 at 15:43
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MENSA is full of people and people make mistakes. –  John Habert Feb 11 at 15:53
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Most printed material also goes through editors who may make changes to the "right" answer their calculator gave them. –  John Habert Feb 11 at 16:00
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@ppumkin Welcome to the club (of those who want to cry about this). As an educator, I see this way too often in the US. –  John Habert Feb 11 at 16:03
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I feel sorry for you. Statistically 1 in 10 (in my organization) people know about this... That 1 was me. But hey, now 10 in 10 know about and I hope they flippin remember! I might be getting underpaid... –  ppumkin Feb 11 at 16:04
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You are indeed correct in that we apply multiplication and division before addition and subtraction.

However, multiplication and division have the same precedence, as does addition and subtraction. When multiplication and division (or addition and subtraction) are both part of an expression, we evaluate which ever appears first (when reading from left to right).

"When do I apply what?"

The good old "order of operations, once again:

  1. Parentheses or brackets. Perform the part of the expression enclosed in parentheses or brackets, first, working from innermost to outermost. $$50/[5 \times (2 + 3)] = 2$$

  2. Exponents. Then take powers of a term. $(2+3)^2 / 3 = (5)^2/3 = \frac{25}{3}$.

  3. multiplication and division (performed from left to right).

    $8/4 \times 3 = 2\times 3$, $8\times 4/3 = \frac{32}{3}$

  4. addition and subtraction (performed from left to right)

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I don't understand. In school they taught me (Drilled me for 16 years) to always create virtual brackets (first multiplication, then division, then...) etc etc. Power come before multiplication, i learnt that in UNI. And that, ALWAYS use rules of maths. This is not grammar?!?! It makes no sense to me. –  ppumkin Feb 11 at 15:42
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@ppumkin For example, the expression $4/2*8 = 16 \neq \frac{4}{16}$. –  John Habert Feb 11 at 15:44
    
OK Thanks. I knew that. So there is absolutely no way that we should add before multiply, in no circumstance? Is MENSA wrong? –  ppumkin Feb 11 at 15:44
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Your computations, in your post, are correct! (It just so happens that in the post, multiplication appears before division, and addition before subtraction, when read from left to right.) –  amWhy Feb 11 at 15:44
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Sounds mad to me to, ppukin! Mensa, of all people! –  amWhy Feb 12 at 13:06
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With the edits you've made to your question, it's now clear — at least to the older generation — that MENSA's answer is correct and yours is not.

The point is that the challenge does not, as you seem to think, ask you to replace the question marks in:

5 ? 4 ? 7 ? 3 ? 2 =

with the operators +, , × and ÷ in any order, and then to evaluate the resulting string as a mathematical expression according to the rules of arithmetic. If that's what they'd meant, that's what they would've written. Rather, what the challenge literally tells you to do (emphasis mine) is:

"Assume you are using a basic calculator and press the numbers in the order shown, replacing each question mark..."

In context, it should be clear that, by "basic calculator" they mean a classic 4-function pocket calculator (as opposed to a fancy modern formula calculator) like this one:

These kinds of calculators traditionally had no memory to store complicated expressions (and no way to enter or, indeed, display anything like parentheses), so they used immediate execution: they would only have enough memory to store two numbers and an operator, and every time you pressed an operator button (+, , ×, ÷ or =), the previously chosen operator would be applied to combine the two stored numbers into one. (The function of the = button was simply to perform the last operation and show the result without queuing any new operation.)

Effectively, such calculators thus evaluated all expressions stricly from left to right, ignoring arithmetic precedence; the sequence of button presses 5 ? 4 ? 7 ? 3 ? 2 = would be evaluated as (((5 ? 4) ? 7) ? 3) ? 2 = regardless of the operators the ? marks stood for.

Of course, this is a pretty limited method of computation: for example, there's no way to directly calculate an expression like (2 * 3) + (4 * 5) on such a calculator. As a limited workaround, most pocket calculators did include an extra memory slot into which the user could store an intermediate result for later recall. (That's what the "M−" and "M+" buttons in the picture are for.) Of course, if you needed more than one such intermediate result, you'd better have either some paper and a pencil handy, or just a good memory. Still, it was the best you could do cheaply using 1970's technology, so people learned to live with the limitations.

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I cant even find one of those calculators. The only thing that emulates that is the standard windows calculator. I did not even know about this issue until now. I just cannot find one of those calculators anywhere, with immediate execution. So really, the user of those calculators had to input the numbers in the correct order. SO yes, that opening statements makes sense. BUt like you said, its age related. I had no idea that is what they meant. Wow. What a question –  ppumkin Feb 12 at 16:42
    
You could probably find some really cheap Chinese-made ones, possibly solar-powered like the one in the picture, still sold in places like dollar stores. They still have a marginal market niche with people who want something very small, very cheap and very low-powered, and don't mind skimping on functionality to achieve it. Oh, and of course older folks who are just used to them. –  Ilmari Karonen Feb 12 at 16:48
    
4-function calculators are definitely still around (though not as common as scientifics and graphing ones these days). I actually have one i bought from Wal-Mart last year (~$10), because they were allowed in calculus even where the fancier calculators weren't. It was one of maybe a half dozen different brands/sizes at the time -- and that's not even counting the adding machines, which work much the same way but can print as well. –  cHao Feb 12 at 16:58
    
Yea this is definitely the answer I was looking for. Taking into account that you knew about the 4 functional - calculator, you wouldn't bother applying the rules.I had no idea about this and straight away started to play with arrangements of the signals based on rules, ignorantly,ignoring the very first sentence because of lack of knowledge. What makes it even more interesting is that whether you apply the rules or not, the signs are valid in both ways- which was the essence of the challenge. It sure stirred up a storm... –  ppumkin Feb 12 at 17:09
    
The models we were allowed in high school worked that way. It had support for parentheses, though. (Yea, no auomated graphing and similar stuff for us.) –  Raphael Feb 12 at 17:52
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Just to play devil's advocate and give MENSA the benefit of the doubt since we don't have the actual question. Maybe for the following reasons, 30 can be THE valid answer:

  1. The actual sequence of operators does not change whether it is 30 or 31.5, which is the key for getting the right response.

  2. There is a difference between IQ and simply knowing rules. Figuring out how to get to 30 takes a keen eye and a bit of thought (calculators not allowed for MENSA test). e.g. imagine this a multiple choice question without the answer 31.5.

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Yes- I think you are right. THis question must have been developed by generations of mensa members, to make it correct and incorrect at the same time. But the truth is, maths doesn't lie- and there is only 1 answer to a maths equation, otherwise we would life a life of goo. I don't think I would get along with people at MENSA. Hahaha –  ppumkin Feb 12 at 9:36
    
A better multiple choice question would include the answer 31.5 just to throw you off. –  Dave Feb 12 at 20:20
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Our mnemonic was "MDAS" but we didn't interpret it as addition before subtraction. It was multiplication and division before addition and subtraction, as others have pointed out.

So, evaluating 5 + 4 x 7 − 3 / 2 would be:

  • 5 + 4 x 7 - 3 / 2
  • 5 + 28 - 3 / 2
  • 5 + 28 - 1.5
  • 33 - 1.5
  • 31.5
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It could as well be DMSA, you just couldn't spell it. If you have both a Multiplication and a Division, you calculate them from left to right. Just as John Harbert stated in a previous comment: $4/2*8 = 16 \neq \frac{4}{16}$ –  karatedog Feb 11 at 21:01
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The correct answer is neither 30 nor 31.5 but +×−÷.

I have never seen MENSA calendar but you stated “like in the answers of the MENSA calendar.” Did you mean readers' or publisher's answers? The first would be amusing, the second would be appalling.

Anyway, being smarter than the average MENSA enthusiast is something one welcomes rather than complaining about.

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Are you ready. The publishers answer actually says 5+4*7-3/2=30 That is the answer. I answered 5+4*7-3/2=31.5 which makes me incorrect. Are you crying yet, laughter or from grief? –  ppumkin Feb 12 at 12:36
    
The question is biased so that cleverness is challenged by Casio calculators that some people have in the office. Something is rotten in the state of MENSA. –  ran Feb 12 at 14:50
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I have just seen your update to the question with the reply from Mensa.

The question setters are trying too hard. It's the case that you need to be able to think like them to get the right answer rather than being intelligent. If they meant that the calculator ignores rules of precedence then they should state that explicitly as a basic calculator can mean many things and doesn't have to exclude rules of precedence as computing is very cheap compared to the 80's when a basic calculator might be expected to ignore precedence.

Do you really want to join such a club?

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Yea exactly y thoughts. both on the calculator bit and whether I want to join such a club. Besides this is not an IQ test calendar...? Its a public challenge based on your intelligence. This is what is annoying me off. :( –  ppumkin Feb 12 at 12:35
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I bet your work mates were amazed by your stunning mathematical ability, isn't the highest answer possible actually 61.5?

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It would be better if you said how you think that result would be achieved. –  Daniel Fischer Feb 11 at 20:55
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He obviously solved (5 + 4) x 7 − 3/2 = 61.5 If you are going there, you should probably go all the way to: (5 + 4) x 7 / (3 - 2) = 63 –  MrWonderful Feb 11 at 21:08
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5+4*7-3/2 The Google agreed with me.

Yes, order of operations is Multiple and Divide first:

5+(4*7)-(3/2) = 31.5

So then, you want to multiply the two biggest numbers you can and have the largest denominator as possible when subtracting or smallest denominator as possible when adding. You could do an proof by exhaustion since the dataset is small:

5+4-7*3/2 = -1.5
5+4-7/3*2 = 4.33
5+4*7-3/2 = 31.5
5*4+7-3/2 = 25.5
... etc
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But parentheses weren't allowed in the original question so you cannot apply them selectively. You either work with mdas rules (implied parentheses) or the primitive left to right calc. Thus neither 61.5 nor 63 are valid.

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Where did you get 61 from? Yes- It seems like they wanted us to do left to right... using a basic calculator. but they cant actually tell me what a basic calculator is. It must be what you're saying. Its so clever it makes me want to cry. –  ppumkin Feb 12 at 10:37
    
Presumably, this should've been a comment to user128049's answer. –  Ilmari Karonen Feb 12 at 16:03
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It seems like a fair assumption that the rules of math should be applied to a mathematical expression.

Btw, it is not stated (but I guess it's implied) that the operators must be put in place of the question marks, or else

-5/4+7*32 = 222.75

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By that logic the max is 54732. Just don't use any operators. –  ChrisC Feb 12 at 10:32
    
Yes, if "Using +,−,∗,÷ only once each" is taken to mean "at most once" rather than "exactly once". I used the latter interpretation but now that you bring it up it's not entirely clear to me which is correct (I'm not a native speaker). –  madings Feb 12 at 11:46
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the answer is 31.5. you know it but you messed up the math. with subtraction and addition it shouldnt matter which is carried out first; they can both be considered to be identical functions (addition of a positive and a negative no or vice versa)

the answer is therefore 28+5-1.5; whichever way you look at it.

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