Multiple divisions

Question

I was confused a bit with a little arithmetic here. For instance $1÷1÷2$ and $2÷3÷7$. BODMAS isn't effective in this case. My question is this: $2÷3÷7$ Am I to divide $2/3$ by $7$ or divide $2$ by $3/7$??

Answer 1

I agree with Bill. Mathematicians never write $2÷3÷7$.

But you can have the same question with subtraction, where the same ambiguity could be imagined. $7-3-2$, is it $7-(3-2) = 6$ NO, or $(7-3)-2=2$ YES. Even that problem goes away once we reach negative numbers, and interpret subtraction as adding the negative. $7-3-2$ means $$ 7 + {}^-3 + {}^-2 $$ and addition is assiciative, so both $$ \big(7 + {}^-3\big) + {}^-2 = 2\qquad\text{and}\qquad 7 + \big({}^-3 + {}^-2\big) = 2 $$ are correct.

Answer 2

You need to carry on first division first. $$\frac{\frac{2}{3}}{7}=\frac{\frac{2}{3}}{\frac{7}{1}}=\frac{2}{3}\cdot\frac{1}{7}=\frac{2}{21}$$
while $$\frac{2}{\frac{3}{7}}=\frac{\frac{2}{1}}{\frac{3}{7}}=\frac{2}{3}\cdot\frac{7}{1}=\frac{14}{3}$$

BODMAS is an acronym and it stands for Bracket, Of, Division, Multiplication, Addition and Subtraction .
It gives you the order of precedence of operations:

When there are similar operations, you can proceed with any one of them;
but when you have a mixture of operations:

• Bracket gets the highest priority
• proceed with multiplication and division before addition and subtraction
• For similar operations, you can proceed with the order in which they appear, or in any way you feel easier.

E.g. $$(4+2)-3=4+(2-3)$$ $$4\cdot \frac{9}{5} = \frac{4 \cdot 9}{5}$$ $$(4+2)\cdot3 \neq 4+(2\cdot3)$$ $$\frac{4-2}{3} \neq 4- \frac{2}{3}$$

E.g. $$4+(6-1)\cdot7-\frac{8}{2}$$ will be done like $$4+5\cdot7-4$$ $$=4+35-4$$ $$=35$$
And not like: $$4+6-1\cdot7-\frac{8}{2}$$ $$=10-7-4$$ $$=-1$$