Order of Operations

I choose the former for both

Operations in parentheses take precedence over all else. Unary operators take precedence over binary operators. Exponentiation takes precedence over multiplication, which takes precedence over addition. Operations of equal precedence are executed from left to right. This is generally the way computer algebra systems and programming languages work, but I would go one step further and say that an expression such as $6 \div 2 \times 3$ is ill-formed. A mathematician would always write this in an unambiguous fashion, e.g., $\frac{6}{2 \times 3}$ or $\frac{6}{2} \cdot 3$.

Note that multiplication and division should be treated with equal precedence--they are both multiplicative operations on the field of complex numbers. That is to say, $a \div b$ for $b \ne 0$ is defined as $a \cdot b^{-1}$, where $b^{-1}$ is the (unique) multiplicative inverse of $b$ in the group structure. Similarly, addition and subtraction are additive operations and also have equal precedence.

In our country, we use the mmenomics PEMDAS, which means: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. We are provided that Parentheses we be in the order of parentheses ( ) , square brackets [], then braces {} (Sometimes, instead of using square brackets, and braces, we use parentheses several times to enclose the operation.). Multiplication and Division are of equal precedence, therefore a division can be used first than the multiplication. Also, addition and subtraction are of equal precedence.

For example $1 + 2 \times 3 \div 4$ = $1+ ((2 \times 3) \div 4)$ and $1 + 2 \div 3 \times 4$ = $1+ ((2 \div 3) \times 4)$

Also: $1 + 2 - 3$ = $(1+2)-3$ and $1-2+3$ = $(1-2)+3$

Answering your question: I choose the former of the both: $a + ( b \times c) = 100$ and $(6\div 2) \times 3$

In the first case, I'd always use the former. The second case is, like others also have said, expressed in a bad way. Mathematics is all about rigorousness and unambiguity (and i know there are cases when this is not entirely true, but in this case it is), and therefore, a mathematician would never write division using $\div$ without appropriate brackets.

I would agree that division goes before multiplication, and multiplication before addition. Most of the time I include parentheses anyways just for extra clarity.

In my teaching, and learning, I have found the easiest, and safest way to explain is via the 'left-to-right rule' or the (AS) then the (MD) operations respectively. In example 1, I think a+b x c must always be interpreted as the latter choice, multiplication must take place first. Example 2, perhaps the less clear one, should be interperated as the first choice, the first operation that comes is calculated first, 'no looking ahead' sometimes I will say to the younger students I have.

On a more conceptual level, I think understand the relationship of inverse operations helps to further clarify the rules of order of operations. Within all pairs of inverse (i.e mult/div, add/sub, log/exp) hold equal order of precedent in order of operations, because they are in essence the same operation, just "backwards".

An alternate strategy would be to go through a problem filled with addition and subtraction: for example 3 - 2 + 6 - 8 and rewrite each subtraction as an addition of the negative: 3 + (-2) + 6 + (-8) ( equals -1 I hope =). From here, it seems more clear that addition would just be done in order, as we read it, left to right. You could do something similar with multiplication and division, 5 * 6 / 2 is just 5 * 6 * 1/2 (=15).

Although I would choose the first way for examples 1 and 2; I know many students in the United Kingdom are taught BIDMAS, the fact that the D is before the M leads them to think that the division should occur before the multiplication. Presumably students who had been taught BIMDAS would think they have to do multiplication first, which would make them choose option 2 for the second example.

NO, BODMAS rule is followed. BODMAS- bracket, off, division, multiplication, addition, subtraction

The problem is that once students in India are thought BODMAS they presume Division always comes before multiplications and the same for addition.But as you suggested there should have been a bracket around D and M and also A and S.

for Example 1, I choose a + (b x c) = 100 instead (a + b) x c = 100 for a + b x c = 100, because multiplication and division must be done first before addition and substraction.

We use "BODMAS": B-Bracket, O-Of, D-Divide, M-Multiplication, A-Addition, S-Subtraction.

Hence by this, both former answers are correct. :)

I choose the latter for both

In example 1 the former is internationally (almost) accepted to be correct. However, there is a lot of confusion regarding the second one, which I believe to be the former. Some interpret the division symbol as a fraction bar. This is exactly why mathematicians omit multiplication symbols and use fraction bars instead of division symbols for clarification. The second one could be $\frac{6}{2}\times 3$ or $\frac{6}{2 \times 3}$

There is an easy way to remember the order of operations and that is BODMAS ( Brackets of Division, Multiplication, Addition, Subtraction).......... It is very clear that division comes first, multiplication 2nd, addition 3rd and subtraction 4th. So in both the cases the former one is right and don't worry, this is an internationally used system.

Another one would be $12 \div 2(2+1)$. By BEDMAS, some students argue on whether to compute it as $12 \div 2 \times (2+1)$ or compute $2(2+1)$ first before division(since brackets first in BEDMAS. Anyway I think 'B' mean inside the brackets, not this. But some teachers also compute $2(2+1)$ first). Different calculators, including scientific ones give different answers. So I would also like to know is it 18 or 2.

BODMAS RULE SO a+b \times c multyply first