TITLE: MENSA IQ Test and rules of maths
QUESTION [17 upvotes]: 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.

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

REPLY [11 votes]: 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.