A fairly simple problem causing a major headache

[Wackone1]

a = 7
b = 6
3a ÷ 3b = x
x = 7/6 or 42?

From what I've heard, you can't separate a variable and it's coefficient, which means you can't just solve the problem by using the old PEMDAS method and going left to right such as (3*a)÷(3*b) but I have heard conflicting reports on this.

Anyone want to give some insight?

 

[mathmike]

how did you get 42?

you need to explain y u can't separate varibles

 

[TD]

It depends on what is really asked, more specifically: how it was noted.
I would interpret this as (3a)/(3b) but as it is written; that's not neccesarily so.

$$\begin{array}{l} 3a \div 3b \to 3 \times a \div 3 \times b = 21 \div 3 \times b = 7 \times b = 42 \\ 3a \div 3b \to \left( {3a} \right)/\left( {3b} \right) = \frac{{3a}}{{3b}} = \frac{a}{b} = \frac{7}{6} \\ \end{array}$$

 

[uart]

a = 7
b = 6
3a ÷ 3b = x
x = 7/6 or 42?
From what I've heard, you can't separate a variable and it's coefficient, which means you can't just solve the problem by using the old PEMDAS method and going left to right such as (3*a)÷(3*b) but I have heard conflicting reports on this.

Anyone want to give some insight?

Yes, if an expression is written like 3a ÷ 3b then in most every instance I've seen the author has meant it to be interpreted as (3*a) ÷ (3*b) .

Now on the other-hand if you type something like 3*7 ÷ 3*6 into any calculator or maths software that is "order of operation" aware then it will give the answer 42.

This potential ambiguity also annoys me. Now you might think it's resolved with that "you can't separate a variable and it's coefficient" clause you stated above, but that leads to other problems. For example in the expression $$3a^2$$ if you apply that "cant separate a var..." then you'll end up interpreting this as $$(3a)^2$$, which is definitely not in accordance with it's usual meaning.

One thing to remember however is that in any "real world" problem an expression such as "3a ÷ 3b" will almost never arise other than on a line that follows a previous line in your own working. In other words you'll always know the meaning of the equation because you wrote it yourself! It's only in school algebra type questions that you get something like "3a ÷ 3b" as a starting point.

Anyway, on occasions that I have had to teach this sort of stuff to high school algebra students I resolve it by teaching them "BIIDMAS" for order of operations. Normally in our school system students are taught "BIDMAS", (Brackets, Indices, Division, Multiplication, Addition, Subtraction), for precedence of operation. I just modify it to "BIIDMAS" (Brackets, Indices, Implied_multiplication, Division, Multiplication, Addition, Subtraction) and it resolves the issue.

 

[homology]

I've recently been exposed to teaching elementary math to students (long division, fractions and the like). I was looking through workbooks to see what I ought to teach next and came across PEMDAS and the like. I have to say that I wonder if such couldn't be done away with by using parantheses correctly.

I'd have to agree with Uart when he says that these ambiguous statements don't often come up. In fact I don't remember ever looking at an expression and thinking "mmmm PEMDAS.."

In fact the more I reflect on it (over the past 30 seconds) the more ridiculous it seems to even consider PEMDAS at all, why not just write unambiguous expressions?

Kevin

 

[uart]

In fact the more I reflect on it (over the past 30 seconds) the more ridiculous it seems to even consider PEMDAS at all, why not just write unambiguous expressions?

Hmmm, I don't think I'd like to have to deal with the large amount of unnecessary parenthesis which that may result in. There's nothing wrong with having a well defined default order of operation, it's a good thing. Sure if there really is a circumstance where you think there is some potential for ambiguity throw in a set of parenthesis to make it clear, that's what I do, even if in a strict sense they may be unnecessary. But to have to put in parenthesis on every occasion could be both tedious to write and difficult to read.

 

[homology]

I only say that because after five years of physics and mathematics, I don't think I've ever used PEMDAS or its cousins. Everything is clear from context. Of course I don't pepper my equations with parantheses, but they take care of the occaisional ambiguity.