What is the proper answer to this math problem (from youtube).

The problem is 9 - 3 ÷ 1/3 + 1 = ?

I say that the answer is 9.

Using the proper order of operations:
[ul]
[li]3 ÷ 1 = 3[/li][li]3 / 3 = 1[/li][li]9 - 1 + 1 = 9[/li][/ul]

The video talks about the proper order of operations and explains it correctly but then makes the mistake (at least I think it’s a mistake) of dividing 3 by 1/3, as follows:
[ul]
[li]3 ÷ (1/3) = 9[/li][li]9 - 9 + 1 = 1[/li][/ul]He seems to think that / and ÷ are two different operations and that / takes precedence over ÷. I say he’s wrong about that. What say you?

I didn’t watch the video. I agree with you.

In the video, is 1/3 represented the way it is here, with three keystrokes, or do they use the actual “one-third” symbol? If the latter, then 3 divided by (one-third) is 9. Otherwise, yeah, you’re right.

Would your answer be the same if the calculation were laid out thus:

9 - 3 ÷ ⅓ + 1 = ?

The fact that he uses both ÷ and / suggests that he does in fact intend to make a distinction between them . I haven’t watched the video either, but I think he intends his use of / to invite us to treat ⅓ as a quantity, not an operation. He could (and perhaps should, for greater clarity) instead have written 0.3 with a little dot over the ‘3’. (My technical skills don’t allow me to form this character, and his use of 1/3 where ⅓ is meant may have a similar explanation.)

It depends on whether you consider the / to be a division operator, or 1/3 to be one number (fraction).

Without watching the video I agreed with the video, that is I treated 1/3 as “a third” or the same as 0.333… So 3 / .33… = 9, everything else follows from there.

Now that I’ve watched the video I see that it is written as a 1 over a horizontal line over a 3. Obviously intended to be read as a fraction but very ambiguous, even more ambiguous than 1/3.

I didn’t watch either, but I think you wrong. See here. Multiplication and division are inverse operations which means:

3 ÷ 1/3 must equal 3 x 3/1

Your method doesn’t preserve that fact. It doesn’t really make sense to divide by a fraction which is why the above must be true.

Intentionally vague math questions to cause arguments over order of operations are a dumb internet trend. The only way to win is not to play.

That’s my question. In the video he shows 1/3 like this:
1
_
3

He even discusses that vs 1 / 3 and seems to say what I’m saying about it, but then goes on to divide 3 by one third and give the answer of 1. Maybe I’m misunderstanding him.

But only if you think of 1/3 as a value. If you look at it as 1 divided by 3 then you go left to right:

3 divided by 1 = 3
3 divided by 3 = 1.

It makes perfect sense to divide by a fraction. But it’s ambiguous as to whether we consider it a fraction or an operation.

I think the answer in the video would be correct if 1/3 were expressed as a decimal, or if a set of parens was used around “1/3”. Since neither happened, then I agree with the OP.

Exactly. I suppose it’s an argument about notation. If you have two numbers, one on top of the other, with a horizontal line between them, is that a single value or is it a division operation between two numbers?

I think where, in the same equation, you use the ÷ symbol to denote the division operation, there’s a fairly strong inference that the horizontal line is intended to denote a fractional value rather than a division operation.

But, I agree, it would be better to remove any ambiguity with the use of brackets.

One other factor that we should consider is that the typographical conventions used for mathematics do vary from place to place - e.g. the use of a comma rather than a point to separate integer from decimals. It may well be that, for the audience for which this equation was first written out, they would all understand the horizontal line used in this context to indicate a fractional value, and they wouldn’t think this was ambiguous at all.

Correct that these weird ambiguous (or questionable) order-of-operations problems have
been trendy lately – we’ve had a number of similar threads on this very message board.

Palooka, above, has the right answer: "Intentionally vague math questions
to cause arguments over order of operations are a dumb internet trend.
The only way to win is not to play. "

HOWSOMEVER: Without having viewed the video, but just reading the above comments,
I can see that this one is a little different. The dispute here seems to be whether 1/3
(or however it was presented in the video) is to be viewed as a single fraction, one-third,
or as an operation between two operands: 1 divided by 3.

If the division symbol was in fact a horizontal line, with the 1 above and the 3 below,
then it should certainly have been treated as a single fraction – or, equivalently,
treated as if enclosed in parentheses: (1/3).

A horizontal bar, as in a fraction, serves also as a symbol of grouping.

Example:



a + b
-----
c + d

Here, everyone should agree that the additions should be done first, and the division last.
No need for parentheses to force that. The horizontal bar serves as a grouping symbol.

But write the same as: a + b / c + d
and now, the division is done first. To force the additions first, this format requires parentheses.

What about this:



       b
a  +  ---  +  d
       c

Here, again, the horizontal bar is a grouping symbol, but this time tells us to do the division first.

In the UK it is standard, when writing a fraction, to use a horizontal bar rather than a /

So for me I’d definitely treat it as 0.3333

We’ve done variations on this thread several times in the past:

Example here or here or several other threads in the past.

Usually, my response is that an ambiguous math problem is just poorly written and you should send it back to the writer. I’m tempted to put this one in that category.

But, in this case, it’s not especially poorly written (though certainly deliberately ambiguous). It is nearly universally understood that the 1/3 written as it is here means “one-third” rather than the operation “one divided by three”. In this case, you should be dividing three by one-third.

That said, whoever posted this problem is a jerk trying to pull a gotcha. That’s why it’s “nearly” universally understood and not universally understood. Thete’s still that bit of wriggle room that makes the OPs interpretation acceptable (if perhaps idiosyncratic). There’s no reason to write this expression in such a way, except to make other people feel bad. It serves no pedagogical purpose. It’s the math equivalent of somebody deliberately committing a typo to berate other people for missing it.

I understand the point the OP is making. However, I tend to disagree with him.

The issue is whether we should treat a / operation the same way we treat a ÷ operation. If so then the multiplication and division operations should carried out from left to right: three divided by one divided by three.

The other possibility is that the / operation is distinct from the ÷ operation. And I feel this is the case because two different symbols were used. This, to me, indicates that the person who wrote the sequence made a conscious decision to use two different symbols in order for them to have two different meanings.

To me, the intent behind using the / symbol is to indicate that this operation is not normal division. It indicates the creation of a fraction and these fractions have an implied precedence ahead of the normal divisions. So one divided by three should be solved first and then three should be divided by the result. In other words, x/y = (x ÷ y).

Yes, anyone who has watched the movie Wargames has learned this. :wink:

I watched the video and saw how the 1/3 was formatted within the equation. It is unambiguousy intended as a fraction, and not an operator on the term (3 ÷ 1).

In fact, I’ll go so far as to state that no one WOULD insert a term such as (x ÷ 1) into an equation when attempting to teach order of operations.

You know, unless they were so incompetent in basic arithmetic that their grasp of the correct order of operations was questionable.