MENSA IQ Test and rules of maths

Question

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?

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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.

Answer 1

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

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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.

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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.)

Answer 2

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)

Answer 3

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

Answer 4

I bet your work mates were amazed by your stunning mathematical ability, isn't the highest answer possible actually 61.5?

Answer 5

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

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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

Answer 10

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.

Answer 11

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?

Answer 12

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.