Ambiguity on BODMAS, Multiplication and Division

Question

$24\div4(8\div 4)$ - This is the equation. We need to solve. Lets do it then

$=> 24\div4(2)$

$=> 24\div8$

$=> 3$

Division is also the multiplication of the inverse. Therefore $24\div4(8\div 4)$ can be written as

$24*\frac{1}{4}(8*\frac{1}{4})$. Let us solve this

$=> 24 *\frac{1}{4}(2)$

$=> 24 *\frac{1}{2}$

$=> 12$

My question is which of the answer is correct? 3 or 12. I have followed BODMAS rule in both representation of the equation.

The correct answer is $12$. In the first case you did $4 \times 2$ before doing $24:4$, but this is not correct. — Crostul, Nov 11 '20 at 09:42
But according to the bodmas rule, first we solve what is there inside parenthesis followed by multiplying the value of the parenthesis right? — Vigneshwaran, Nov 11 '20 at 09:44

score 2 · Accepted Answer · answered Nov 11 '20 at 09:52

The trouble with BOMDAS is that it's not usually implied.

It's like this
() brackets
close product (ie ab)
f  of   So cos ab = cos(a×b).
× and ÷  left to right
+ and -  left to right

What actually happened in your calculation, is that when you switched the division into multiplication, then $4(8÷4)$ should had went to $\frac1{4(8÷4)}$. This evaluates in both cases to 8, the division is thus $3$.

The fun really starts when you do something like $ad÷bc$ as $\frac{ad}{cd}$. Because BOMDAS does not involve close division, then the equation ought be $\frac{acd}b$.

Normally, BOMDAS is designed to reduce brackets. But the rule is faulty: The numeric appears at the front of any product, and any numerics further down are read some other way. This allows, eg $2\cos 2\pi\theta$.

score 0 · Answer 2 · answered Nov 11 '20 at 11:27

For my two cents, the reason for this ambiguity stems from mixing two methods of writing multiplications and divisions, which is never justified.

Method 1: Using $\times$ and $\div$

$$\begin{align} 24 \div 4 \times (8 \div 4) &= 24 \div 4 \times 2 \\ &= 6 \times 2 \\ &= 12 \end{align}$$

Method 2: Not showing multiplications, writing divisions as fractions

$$\frac{24}{4} \left( \frac{8}{4} \right) = 6 (2) = 12$$

Both calculations are unambiguous and yield the same, correct answer.

The two methods should simply never be mixed; whenever they are, there is inevitably the possibility of ambiguity.

Note that either method could yield the other answer with a little re-writing:

Method 1:

$$\begin{align} 24 \div \left[ 4 \times (8 \div 4) \right] &= 24 \div \left( 4 \times 2 \right) \\ &= 24 \div 8 \\ &= 3 \end{align}$$

Method 2:

$$\begin{align} \frac{24}{4 \left( \frac{8}{4} \right) } &= \frac{24}{4(2)}\\ &= \frac{24}{8} \\ &= 3 \end{align}$$

Ambiguity on BODMAS, Multiplication and Division

2 Answers2