MENSA IQ Test and rules of maths
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.)
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:
Parentheses or brackets. Perform the part of the expression enclosed in parentheses or brackets, first, working from innermost to outermost. $$50/[5 \times (2 + 3)] = 2$$
Exponents. Then take powers of a term. $(2+3)^2 / 3 = (5)^2/3 = \frac{25}{3}$.
multiplication and division (performed from left to right).
$8/4 \times 3 = 2\times 3$, $8\times 4/3 = \frac{32}{3}$addition and subtraction (performed from left to right)
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.