OK, so my daughter is doing an aptitude assessment for a job. A question came up on a 'practise test'.
What is $|-4| + |4| - 4 + 4 $?
Personally, I was taught $BODMAS$ where you do Addition before Subtraction.
So this would lead to :
$(4+4) - (4+4) = 0$
However the correct answer on the quiz appears to be $8$ which I would assume they have arrived at by doing this...
$4 + ( +4 -4 ) + 4$
So now I'm completely confused.
Can someone clear this up for me?
The mnemonic is convenient but inaccurate. Addition does not take precedence over subtraction, and multiplication does not take precedence over division.
The reason why is because addition and subtraction are equivalent operations: subtraction is simply the addition of a negated number. For example, $$4 - 4 = 4 + (-4) = 0.$$ Similarly, division is the multiplication of a reciprocal: $$4 \div 4 = 4 \times \frac{1}{4} = 1.$$
However, multiplication and division take precedence over addition and subtraction, so $$4 \times 4 + 4 = (4 \times 4) + 4 = 16 + 4 = 20.$$ Similarly, $$4 \div 4 - 4 = (4 \div 4) - 4 = 1 - 4 = -3.$$
If this usual order needs to be changed, then parentheses are used; e.g., $$4 \times (4 + 4) = 4 \times 8 = 32.$$
Mathematicians typically do not use brackets or braces as grouping symbols, only parentheses. We do not express nested groups, for example, like this: $$[4 \times (4 + 4)] \div (4 + 4).$$ Instead, we use nested parentheses: $$(4 \times (4 + 4)) \div (4 + 4).$$ The use of brackets or braces as grouping symbols is more commonly encountered at an elementary level as a visual aid in identifying matching pairs of grouping symbols. Similarly, the mnemonic "BODMAS" or "PEMDAS" or whatever is in vogue these days, is an often-used but inaccurate teaching tool.
In the absence of grouping symbols, expressions such as $$40 \div 8 \times 2$$ can be ambiguous; as such, it is encouraged in cases of potential ambiguity to use parentheses to establish the intended evaluation order. However, the rule in such cases is to evaluate from left to right: $$40 \div 8 \times 2 = 5 \times 2 = 10,$$ not $$40 \div 16 = 2.5.$$
Yours is a common misconception. Addition and subtraction have the same priority, and therefore they must be done in sequence. Obviously, $4+(4-4)+4=(4+(4-4))+4$ is the same as $((4+4)-4)+4$ because of the identity $a+(b-c)=(a+b)-c$.
The mnemonic is misleading. You do not do addition before subtraction; they are grouped together so you perform them left to right.
BODMAS is "Brackets $\prec$ Order $\prec$ Multiplication and Division $\prec$ Addition and Subtraction"
