A lot of people are having trouble with this math problem that requires some basic algebra

[sarisataka]

Nothing like a viral mind boggling math problem to make you feel like you should have paid better attention in high school algebra.

The equation in question? 9 – 3 ÷ 1/3 + 1 = ? The deceptively simply math problem recently blew up in Japan, as people quickly found they were routinely getting the wrong answer. According to PopSugar, only 60 percent of 20-somethings solve it correctly, a study showed.

So what is the remaining 40 percent doing right? YouTube user MindYourDecisions shows you, bringing in mathematician Presh Talwalkar, the man behind the problem, to explain how it’s done.

https://www.yahoo.com/news/lot-people-having-trouble-math-161950574.html

[scscholar]

1. I don't understand why this is supposed to be hard?

What am I missing? It's just division then subtraction then addition.

[seabeckind]

2. Quite a few people's algebra never included operator precedence

The situation wasn't an issue until the emergence of computers.

[PoliticAverse]

7. Which is why in my computer programs I just use lots of parentheses. n/t

[Aerows]

23. Operator precedence

is kind of a key portion of algebra and very significant in physics calculations.

[BobTheSubgenius]

28. Very significant in all calculations.

Calculations, or miscalculations, in physics often have bigger consequences.

[scscholar]

37. I disagree with the before computers part

You had to know it to solve equations like "2x + 1 = 5". You had to know to multiply before adding.

[Android3.14]

106. Incorrect. The order of operations has been around since algebraic notation, and probably before

Computer languages, while certainly influencing the teaching of this concept, often have a specialized order of operations depending on the notation and language.

[oberliner]

5. It's a clickbait article

Originally from the "Pop Sugar" website.

People see the headline, are curious about the problem, click on the link, money generated for ads.

[seabeckind]

9. The statistics are interesting, you have to admit.

[oberliner]

14. Click bait articles often provide interesting information

But they still tease you with the headline so you will click through.

If they had in the headline, "Most people don't know how to multiply by fractions and are unclear about the order of operation rules" you'd get less clicks than "People are having trouble with this math problem".

Not that there is anything wrong with click bait headlines. In this day and age, I suppose they are necessary.

[1939]

26. Dividing by a fraction

From my 1948-1950 arithmetic teacher, Miss Richmond, "invert the divisor and proceed as in multiplication". Miss Richmond also gave me "Unsatisfactory" in twelve consecutive grading periods for handwriting (with a straight pen). I did learn arithmetic from her (always got "Excellent&quot , so she was pretty good at that.

Since engineering pays better than calligraphy, it all turned out well.

[yeoman6987]

71. I agree

But any American who can't figure out this easy math problem has bigger problems.

[PoliticAverse]

6. It's hard if you don't know that you are suppose to do the division first...

or you get confused by dividing 3 by 1/3.

[TexasMommaWithAHat]

123. Because we've forgotten the order of operations!

Most of us who need the answer to that question wouldn't set up the problem that way in real life (I would most likely do it my head); hence, we've forgotten the order of operations since we don't use it.

[scscholar]

185. Thanks

It didn't occur to me that someone would add or subtract before dividing!

[hollysmom]

184. I used to code for actuaries - and made them give me everything with parenthesis

they would make mistakes in order all the time - they had calculations in production that gave wrong results because of coder confusion or actuary confusion. I told them I wanted to discuss every single equation their programs and they thought I was being annoying until we found so many did not reflect what they wanted.

[Kingofalldems]

4. I actually got it right.

[TexasBushwhacker]

169. I did too, but I think many forget about the order of operations

and just start solving it from left to right straight away.

[seabeckind]

8. I might suggest that it's a teacher/education problem

more than a student problem.

It's likely a teacher who learned their algebra before around the mid-70s never encountered the need for operator precedence rules.

Unless their continuing education emphasized it, they don't have it.

What was that thing about Core again?

[SheilaT]

60. Hmmm. I took algebra in the mid-60's. I do not recall

order of operations being learned, although it's possible I've merely forgotten. However, my high school math was UICSM, University of Illinois Committee on School Mathematics and it was the most amazing math program ever. I'm yet to stumble across someone I didn't go to high school with who's even heard about it.

In any case, in this amazing math program we proved EVERYTHING. It was rigorous, and apparently taught us stuff that normally never is taught in high school. Thirty-five years after taking UICSM, without another math class inbetween, I tested into algebra 2 at my local junior college. And I'd sit in that class, and in the subsequent college algebra, and phrases from that old math class would bubble up to my consciousness. Things like something is true if and only if something else is true. And that if and only if statement never came up in my jr. college class. When I asked the teachers about it, they said that such language simply wasn't commonly used at that level, only at higher levels. And so on.

The problem with most high school math is that it is predicated on rote memorization of various things, and then plugging them in blindly to the problems. My UICSM program taught me to think, to question, to understand what I was doing. Which is why I find that problems like these that assume parentheses to be incredibly dishonest and misleading. Also, the audio explanation talks about the problem that arises if you input that problem into the calculator. Well, duh! While calculators are truly wonderful (OMG! How I LOVE graphing calculators, having solved far too many quadratic equations before they came about) they still have their flaws. And the problem with total reliance on those calculators is that you may never understand that those flaws exist.

[Android3.14]

113. As a former high school math teacher teacher educator, you're half right

We do rely on calculators way too much. We should limit their use until a student has mastered algebra 1, pre-algebra at the earliest, and then introduce calculators in order to avoid stupidity like this OP illustrates.

As far as rote memorization in math, this is untrue, and has been for 40 years at least. I taught Mathematics for Elementary School teachers for several years at the local college until about five years ago, and I've given a basic arithmetic test at the beginning of each semester. Typically 60-80 percent of students are incapable of multiplying single digit factors with any consistency. Many are incapable of adding single digit numerals without counting on their hands.

Manipulating fractions or performing division (long or otherwise) has a failure rate of about 95 percent.

These are adults.

There is far too little rote memorization in mathematics these days.

[seabeckind]

171. You didn't forget.

Its introduction was in line with the "new" math that came in right around that time.

I was fortunate (or unfortunate as the case might be) to have the new math introduced in my high school one year behind me.

IOW, I learned the old way (rote) but those one year behind me learned the new way where the emphasis was more on the why.

For various reasons my high school decided that some of the students needed to have the new math background. So at the same time I was taking junior year advanced algebra I was also taking freshman algebra, then the following year it was senior calculus and junior advanced algebra.

I'm glad they did. Worked out well for me.

[SheilaT]

173. My UICSM course was a sort of "New Math",

but it was taught in a way that promoted a deep understanding of what we were learning. At the very beginning it went more slowly than the traditional math program, but soon forged on ahead. We only spent one semester on geometry, in which we did not memorize theorems and then work problems, but we derived all the theorems from what we learned. Our proofs wound up being quite rigorous, it seems, although don't ask me to recall details. By the third year we were doing finite math (and I only know that term because years later when I told college math teachers some of what we did, they were very surprised, and said that stuff normally wasn't taught until well into college) and calculus. Which normally comes after at least three, maybe four years of h.s. math.

My point is, beyond basic arithmetic, I did very little rote memorization. Well, we memorized the quadratic equation, but first we'd derived it. Just don't ask me how we did. I suspect the UICSM program was harder to teach because of the way the teacher had to bring the students along. But oh, boy, did we learn a lot.

[Nye Bevan]

10. It does not "require some basic algebra".

It's just arithmetic.

[LongtimeAZDem]

13. +x

[auntpurl]

25. lol

[auntpurl]

76. I am sorry, but you deserved more than one "lol" for this

lol

[Hoyt]

22. Exactly. Further, you'd seldom run across something like that in real life.

Algebra, yes. I've used it a bunch for work. But the puzzle is guaranteed to produce different answers without parentheses.

[1939]

29. I have seen division by a fraction quite often in real life (and you don't always get parenthses)

Especially before computers came along.

How many 3/4 inch dowels will fit in a box 9 inches wide?

9 divided by 3/4 is equal to 9 times 4/3 = 36/3 = 12

[Hoyt]

35. Division by fraction, yes. This: 9 – 3 ÷ 1/3 + 1 = ? Not likely.

[longship]

77. For Christ sakes. Division is the inverse of multiplication and 1 is the multiplicative identity!

Plus multiply (division) have priority over addition (subtraction).

That's all one needs to know here.

9 - 3 / 1/3 + 1 = 9 - 3 * 3 + 1
= 9 - 9 + 1 = 1

Pretty elementary algebra. And I did not have to click through to solve it even though I took algebra in 1961 (but thankfully have used it for decades, and taught it for years).

R&K

[Lucky Luciano]

111. As written in the OP it is not division by a fraction.

As written in the OP it is straight left to right division. In the video, it is more clearly division by a fraction.

http://www.democraticunderground.com/?com=view_post&forum=1002&pid=7957890

[Warpy]

151. I was a nurse and I'm still a weaver

so this sort of thing pops up all the time.

[Hoyt]

152. Could you give me an example of something like 9 – 3 ÷ 1/3 + 1 = ? that you would run into as a

nurse or weaver. If you are calculating dosages, no wonder there are so many medication errors if that is the kind of ambiguous equation used.

[NobodyHere]

68. I would consider order of operations basic algebra

[Gabi Hayes]

159. exactly. it's pre-algebra, if you consider that as math that includes variables.

that's how it's approached in our district.

long before any algebra is explicitly taught (it's a magic word in my school, where we tantalize kids with 'missing' numbers and warn them that they'll be doing a lot of stuff like that in a few years when they start to learn ALGEBRA!), we teach order of operation:

PEMDAS

can you figure out the acronym? should be easy, if you know your.....PEMDAS

[WinkyDink]

11. I've seen enough of these to know this: IT DEPENDS UPON WHEN ONE WENT TO SCHOOL. I, for one,

was in 8th Grade ca. 1962.

We would've done each step as is, because THERE ARE NO BRACKETS.

We weren't taught to IMAGINE the placement of brackets, or that any particular function took precedence over another.

Bah and Feh.

[longship]

79. You don't need brackets to solve this!

Simple algebra is all one needs.

1. Hierarchy of operators: ^ then * then +

2. Inverse operations: * and /, + and -.
x * 1/y = x/y
x / a/b = x * b/a <== important here
Division is the inverse operation of multiplication.

3. Identities: 1 is the multiplicative identity
1 * x = x. x / 1 = x.
0 is the additive identity 0 + x = x <== irrelevant here.

The rest falls right into your lap.

This is elementary algebra.

[stone space]

83. The problem here is the ambiguity.

x / a/b = x * b/a <== important here

Does x/a/b=x/(a/b)?

Or does x/a/b=(x/a)/b?

This ambiguity is quite easily resolved by the proper use of parentheses.

You are using the right to left convention.

Others might use the left to right convention.



The moral of this little exercise is, "parentheses are your friends".

[longship]

86. No fucking need for them. There is no ambiguity here.

And any of my first semester high school algebra students would have made mince meat of your argument.

I will stand by my posts here.

There is only one mathematically correct interpretation of this algebraic expression.

The answer is unequivocally, and unambiguously: 1

[stone space]

95. Some folks insist emphatically that 27/9/3 = 9.

Others insist just as emphatically that 27/9/3 = 1.

And some of us just sit back and chuckle at how emphatically insistent folks on both sides of this momentous issue are.

[immoderate]

139. Equivalent operations go from left to right.

--imm

[Lucky Luciano]

112. Sorry mate - stone space is correct.

You wanted to treat 1/3 as a fraction you are dividing by, but

3/1/3 is indeed 1.

[WinkyDink]

170. I was speaking in general terms of all these alleged "brain-teasers." Get a grip.

[LanternWaste]

175. Or 4-3.

"The answer is unequivocally, and unambiguously: 1"

Or 4-3.

[AntiBank]

183. absolutely correct

[Gormy Cuss]

118. Ugh.

I sometimes forget when exponents are solved but the rest of the order was internalized through years of rote learning in late elementary school. The important thing though is that I know the solving order matters and will check a reference when I'm in doubt.

[lumberjack_jeff]

130. PEMDAS isn't a newfangled thing. n/t

[Kilgore]

12. Obviously someone was sleeping during the order of operations lecture

Otherwise known as "please excuse my dear aunt sally"

[WinkyDink]

15. Whippersnappers think the way they were taught is somehow superior to 1950's-1970's methods.

But then, we learned cursive and diagramming, too.

[Kilgore]

18. That's one I have not come across

Details please

[Donald Ian Rankin]

16. I'm a professional mathematician, and this would give me pause as you presented it.

In the form in the video, it's easy.

But as you've written it in ASCII, it's horribly ambiguous. Brackets Over Divide Multiply Add Subtract, but does a / with a thing on the left and a thing on the right count as an "over" or a "divide"?

If I had to, I would have plumped for the interpretation that matches the video, taken it as an "over", and evaluated the / before the ÷, but I don't think a mathematician would ever write the formula that way - they'd either actually put the 1 over the 3, or put brackets round it for clarity.

The people who *do* regularly write formulae looking like that are computer programmers. In programming languages you typically don't have a ÷, so / is explicitly used as "divide", and you evaluate left-to-right, so 3 ÷ 1/3 would come out to 1, not 9.

[1939]

30. The problem is that

it is difficult to write a fraction with the horizontal bar on a computer, so we are forced to use the "slash".

Fractions in the textbooks are always written with the horizontal dash separating the upper and lower number.

[athena]

49. That is not correct.

Open any undergraduate or graduate-level physics textbook, or look at any physics paper. You will see "/" used all over the place inline. The horizontal line is usually used only when the equation is set out by itself. When you use it inline, the numbers tend to get too small to read.

[Igel]

67. Yes, / is used inline.

The way it's written in line in the OP doesn't have as the obvious choice that 1/3 is a fraction.

You have to stop and realize two things: One, there's a bit of a space so the writer's trying to set off 1/3 as its own entity. Two, the writer is distinguishing between the obelus and the solidus as division operators, and the only reason that he's likely to do this is if they're not being used in the same sort of context. The solidus is pretty much de rigueur in fractions.

(And I *never* thought I'd need to ever use the word 'obelus', but my keyboard doesn't have that symbol and I'm too tired and lazy right now to go to the trouble of inserting one.)

[1939]

89. I don't have my Physics textbook anymore, but do have my "Theory of Simple Structures" textbook

Spring semester 1960 and fall semester 1960 which was an application course of the mechanics branch of Physics. All fractions are printed with the horizontal bar in formulae. Cost of book covering two semesters and 7 credit hours (4 and 3) was $9.50.

[Godhumor]

34. Yup, exactly what I did

But for me, I over-thought it and decided the formula in the OP was supposed to be a tricky "gotcha" problem using two different divide signs.

When I clicked on the link and saw how it was written in the video, I switched to the "right" answer.

And this is speaking as a professional statistician/analyst (Bet you can't guess which program I spend most of my day in).

[Lucky Luciano]

116. I hope you use a lot of R or Python for data analysis!

Last edited Sun Jun 26, 2016, 06:28 PM - Edit history (1)

The open source stats for Python along with pandas is awesome unless you need extremely high performance machine learning types of stats problems...in which case you better use C/C++

[struggle4progress]

54. +

[bemildred]

93. In programming you always parenthesize so as of avoid future problems.

And as a mathematician I had to look at it a while too, to see what the trick was. I would never code it or write it like that, and I used to write expression parsers etc.

Programmers do do that sort of thing all the time, but they are bad programmers, just like they would be bad mathematicians.

[stone space]

102. Folks may disagree on the value of 3/1/3, but I hope that we can all agree that 3/1/3=27/9/3.

That much seems clear, whichever side of this momentous issue folks may fall on.

There is common ground to be had!

[Donald Ian Rankin]

120. Wouldn't Americans write that as 1/3/3 = 9/27/3? N.T.

[stone space]

126. You win!

Wouldn't Americans write that as 1/3/3 = 9/27/3? N.T.

[Ex Lurker]

17. It's not even algebra, it's basic arithmetic

If you've forgotten it, it's not surprising, since most people don't do this kind of stuff in every day life. But if, as another poster said, some people were never taught it, that's a glaring deficiency in their mathematical education.

[alarimer]

19. Some people probably forget that dividing by 1/3 is the same as multiplying by 3.

But the order of operations confuses more people, I'd guess.

[brush]

40. Is dividing by 1/4 the same as multiplying by 4, also? Forgive me, I'm a fine arts major.

[PoliticAverse]

48. Yes. n/t

[JustinL]

74. dividing by the fraction a/b is always the same as multiplying by the fraction b/a

[NutmegYankee]

20. Probably forgot "Please Excuse My Dear Aunt Sally".

[Electric Monk]

21. BEMDAS is high level math now? whoa. nt

[teach1st]

24. It's poorly written

The equation should include parentheses. We're asking students to evaluate poorly written equations. I guess that PEMDAS is useful for encounters with poorly written equations, but to me it seems more useful to teach how to formulate correctly.

[athena]

27. You only need parentheses when the order of calculation is different from the usual rule.

There is nothing ambiguous about 9 – 3 ÷ 1/3 + 1 = 1. It's not "poorly written." This is not English class; it's math, where there are actual rules you have to follow.

I have a Ph.D. in physics, in case you feel tempted to attack me for not knowing math. (It wouldn't be the first time.)

[BobTheSubgenius]

32. Thank you!

I was beginning to have doubts, but then, I learned the order of operations over 50 years ago, so I was going to give myself a pass, anyway.

[teach1st]

33. I'm terrible at math

I rarely attack. There are mathematicians who agree with me and those who agree with you. I don't know which is the majority opinion, but I like things perfectly clear.

If you take PEMDAS literally, simplifying the following might be weird:



That said, it's important to teach the orders, if only to prepare students for the ambiguous math they encounter on tests designed to make sure we've taught the orders.

[athena]

43. This is not a matter of opinion, and it's not just about tests.

In physics research, for example, this stuff comes up all the time. You need to know how to interpret something like:

E^2 = p^2 c^2 + m^2 c^4

I can't imagine that any real mathematician would agree with you. This is really basic stuff. It's not a matter of opinion. If scientists, engineers, and mathematicians were arguing about stuff like this, they could never get any real work done. Bridges would be collapsing, and no one would have walked on the moon. In math or science, you have to have the basic rules clearly defined if you want to be able to build up to something where the results are not obvious.

As for the expression you've posted, I see nothing weird about it. (6+2)/(3+1) = 8/4 = 2.

[teach1st]

45. Here's one

H Wu, University of California, “Order of operations” and other oddities in school mathematics," 2007

https://math.berkeley.edu/~wu/order5.pdf

One of the flaws of the school mathematics curriculum is that it wastes time in fruitless exercises in notation, definitions, and conventions, when it should be spending the time on mathematics of substance. Such flaws manifest themselves in assessment items which assess, not whether students know real mathematics, but whether they could memorize arcane rules or senseless conventions whose raison-d’ˆetre they know nothing about. An example is the convention known as the Rules for the Order of Operations, introduced into the school curriculum in the fifth or sixth grade:

[athena]

47. That says nothing about the rules of math themselves being ambiguous.

It's a statement about math education and what teachers should spend time on. It is not a statement about the rules of math.

ETA: Even if teachers in elementary school spent less time on this stuff, students who wanted to go on to study math and science in college would still have to learn these rules. The rules are not ambiguous. When you know the rules, you can then, much later, at the postgraduate level, do stuff like leave out c (the speed of light) and leave it to the reader to figure out what is meant, since the units always have to work out correctly. If there were any ambiguity about the basic rules, such sophisticated thinking would not be possible. No one would ever know what anyone was talking about.

[Person 2713]

55. +1

[teach1st]

58. I wrote from an elementary teacher's perspective

I posted from an elementary teacher's perspective. I should have realized that and noted it. My bad. The orders are important, especially in advanced math, but PEMDAS is not the way to introduce it in elementary. In fifth grade at least, it confuses kids who aren't mathematically sophisticated (which most fifth graders in my neck of the woods). It's more important to help them understand how the commutative and associative properties help. From my perspective as an elementary school teacher, complex equations should include parentheses.

From a higher math perspective, the equation in the OP is not written poorly. I think it is too much for fledgling mathematicians (and as we see on tons of like Facebook posts, many adults have difficulties with the orders). Understanding the orders is important and certainly prevents heavily bracketed and parenthesized sentences. But asking young students to translate without understanding why we're using the orders is not cool.

Thanks for a lively discussion! You are correct that the orders are important.

[athena]

66. Thanks for clarifying. We agree on the fundamentals.

It's been a long time since I was in elementary school, and that's not my area of expertise, but I believe that at that level, it's probably just as important to show students how cool and important math is as it is to make sure they learn the rules. I don't agree with teaching to the test, and I believe teachers should have a lot of leeway to teach things the way they want to teach them, as long as the students are learning. Too many people go off into life thinking math is boring. When you focus just on the rules and on nothing else, the whole thing can seem pointless indeed. If someone thinks math is cool and worthwhile, they will learn the rules. But if they just learn the rules and never see the point, they will not pursue it further and will eventually forget the rules as well.

The question posed in the OP does not do anything to show the coolness or importance of math. It seems designed to make people feel stupid or smart depending on whether they got the right answer. Overall, a total waste of time for everyone.

[stone space]

82. Parsing conventions are just conventions, not mathematics.

If scientists, engineers, and mathematicians were arguing about stuff like this, they could never get any real work done.

That's why mathematicians don't waste time arguing about parsing conventions.

If I'm reading Caley's Eliptic Functions, for example, I don't waste time trying to argue with the author about how to parse expressions like

1 - x^2 . 1 - k^2 x^2

I just accept his parsing conventions as given and move on.

(see page 2)

https://books.google.com/books?id=pj0LAAAAYAAJ&pg=PA1&source=gbs_toc_r&cad=3#v=onepage&q&f=false



It parses as ( 1 - x^2) . ( 1 - k^2 x^2), btw.

After all, we aren't really talking about mathematics here. We're just talking about parsing conventions.

[athena]

87. In that case, you know what is meant from the context.

You know from context and experience, and simple logic, that the writer does not mean 1 - x^2 - k^2 x^2, since he would then have written that, or 1 - (1 + k^2) x^2. It is not technically correct, just as it is not technically correct to omit c in physics expressions, but the experienced reader will know what is meant from the context. That doesn't mean the rules have been thrown out the window.

Furthermore, it is not a "parsing convention" to write 1 - x^2 x 1 - k^2 x^2 to mean (1 - x^2) . (1 - k^2 x^2). If it were a parsing convention, the writer would not have used - (1 + m x^2) (1 + n x^2) and other similar expressions on the previous page.

Now, when you see an expression like
9 – 3 ÷ 1/3 + 1,
there is no context, so you go back to using the rules you were taught in elementary school. The fact that two different symbols for division have been used should not cause any distraction, since you instinctively interpret 1/3 as a number and divide 3 by that, which gives you 9, so that the final result is 1.

Since my gender apparently makes it difficult for some people to agree with an obvious point I made, let me say that I showed the expression to my husband, who is also a physicist, and he also came up with 1 on the spot.

[stone space]

94. I've always thought that physicists omit c in multiplicative expressions because c=1.

I'd be pretty nervous about omitting c in additive expressions like c+5, a situation which I presume physicists seldom face, even if one views c as a dimensionless quantity.

[cemaphonic]

172. I suspect that the fact that the problem uses two different symbols for notating division...

is exactly the source of the ambiguity and confusion. Like you, I evaluated 1/3 as a unit first, and thus got the "correct" answer. But if you treat "÷" and "/" as equivalent symbols for division, and apply PEMDAS left-to-right conventions, then the expression will evaluate to 9.

IOW, the spacing (grouping 1/3 together) and the use of two different division symbols implies a parenthetical grouping, but the fact that it isn't made explicit makes for an ambiguous and poorly formed expression.

Certainly, if you were going to write that in any C-family programming language (which has only one division operator and doesn't care about spacing) you would need to write 9 - 3 /(1/3) + 1 to get the correct answer.

[Godhumor]

39. It depends on how and where you do most of your math

I agree, the formula in the video is not poorly written. However, it is ambiguous in the OP for one reason--the / means divide in programs like Excel.

Before glancing at the video, I got the problem wrong, because I thought the point of it was too say "Hur, Hur people don't know that / is the same as the divide sign".

Using a / to write a math problem does need a bit more explanation. (1/3) would make it clear that it is a fraction.

The video, on the other hand, makes the problem unambiguous.

[athena]

41. Math is math. Excel does not make up the rules of math.

I'm sorry, but getting this problem wrong means you've forgotten the math you learned in elementary school. Anything else is no more than justification to make yourself feel better.

[Godhumor]

42. Quick, which is the way you'd write this problem using /

3/1/3 or 3/(1/3).

One guess as to which returns the right answer.

[athena]

46. Again, this is not about Excel.

Excel has its own rules, just as every other program. The question of which of the two examples you gave works in Excel has nothing whatsoever to do with the rules of math.

In case you are about to attack me for not knowing about programming, I also happen to have worked as a professional programmer and know C++, Fortran, Perl, Python, and JAVA, as well as HTML, CSS, and JavaScript. Programming languages have their own rules. For example, a = 1 and a == 1 are completely different statements in C++. That has no bearing whatsoever on the rules of math. If someone gets the question in the OP wrong because they're a programmer, it doesn't mean there is anything ambiguous about the question; it means they have forgotten the math they learned in elementary school.

[Godhumor]

50. I don't attack people on their math or programming knowledge

I'm saying that the example in the OP, as it is written, can and will be interpreted differently by people who do different kinds of work. Math is math. The interpretation of symbols is, well open to interpretation. To me, / means divide and is not a notation for fraction. You'll have to trust me if you've never read my posts before, I understand order of operations just fine.

A better way to think of this is like the term 1M. To most people that means 1 million. To bankers it means a thousand. Everyone looking at it understands math, but, in this case, a certain subset of people use the same symbol to mean something different.

Any guess as to how bankers notate a million?

[athena]

52. The question as it is posed in the OP is obviously a math problem

from elementary school. It is not posed as a problem in Excel, nor as a problem in banking.

What annoys me, as someone trained as a physicist, is that people are so insecure -- so much more focused on themselves than on truth -- that they would argue that there is something wrong or ambiguous about the basic rules of math. I suppose this is to be expected, but it is sad.

[Godhumor]

57. You are obviously missing the point that several people in this thread are raising

None of us are arguing that it is a math problem. All of us are arguing that / does not represent a fraction in how problems are typed using symbols on a computer.

All of us when we saw the fraction in the video got the problem right.

Why is it so hard to understand that / doesn't mean the same thing to everyone?

It is 1MM for a million, by the way.

[Chathamization]

69. Indeed, as far as I can tell "/" is considered an acceptable symbol to denote division

Interesting to note that the video says as much, and even says that it would be a mistake to write the problem the way the Yahoo article in the OP wrote it [Edit: also the way it's written in the title of the video] (since the ÷ and / could both mean division, in which case the answer is different). I can't really find anything backing up the claim that "/" is an inappropriate way to denote division, and some brief research indicates that there are mathematicians use it as such.

Not that it matters so much either way, since this is focusing on mathematical convention rather than math itself.

[Lucky Luciano]

109. Ok I have a phd in math and I disagree with you.

It is not clear from the equation that you are dividing by a fraction. I read 3 ÷ 1 / 3 as 3/1/3=3/3=1

The video makes it much more clear that you are dividing by a fraction.

See
http://www.democraticunderground.com/?com=view_post&forum=1002&pid=7957890

[stone space]

114. My PhD in math tells me that 9 – 3 ÷ 1/3 + 1 = 9 – 27 ÷ 9/3 + 1.

I'm still not sure of the exact numerical value of either side of this equation, but I'm trying to find a little common ground here, on which we can all agree.

[Lucky Luciano]

117. Well they certainly are equal! Just a roundabout way to get there!

...and no phd required of course!!

[stone space]

122. I think we can all agree on the value of 2^2^2 as well.

[Lucky Luciano]

124. Mind blown!!! LOL. nt

[Duppers]

135. A person with a PhD in physics, b.a. in math

Just told me, "Professional papers would have written it as...

9-(3/(1/3))+1


Makes no difference to me; not trying to argue but just throw another thought in. With no math or science background myself, I can see all sides to this.

[Lucky Luciano]

137. I agree with what you wrote except the outer parentheses are not needed.

The inner one on the 1/3 would be used and would be unambiguously clear.

Professional papers certainly never ever use the horizontal line with the two dots on either side. That symbol goes away after elementary school.

[Gormy Cuss]

121. Exactly.

Parens indicate "solve this first."

[Stinky The Clown]

31. 1

I had the answer in about 4 seconds

[LWolf]

36. It's a simple problem

with an easily reached answer.

People who are routinely getting it wrong probably don't remember order of operations, or how to divide fractions; both simple procedures that can be forgotten if not used regularly.

[forgotmylogin]

38. I got it right.

I did remember that division/multiplication is done first in absence of brackets.

[OnDoutside]

51. I got the right answer, but we learnt it as BOMDAS rather than BODMAS, and where he mentions it, he

has multiplication first then division.

[demwing]

174. To be accurate, PEMDAS could also be PEDMAS, or PEMDSA, or PEDMSA

but then creating a mnemonic would be awkward. Think of the term like this: PE(MD)(AS)

Multiplication and division hold the same position in the order of operations, as do addition and subtraction. After working the parentheticals and the exponents, you wouldn't do all multiplication before any division. You would do all multiplication and division before any addition and subtraction, and you would work those operations from left to right.

[3catwoman3]

53. I took algebra as a high school freshman in 1965-66.

I did very poorly, and thought I just didn't "get it." I still remember the teacher's name, but I don't remember her ever saying anything about PEMDAS or orders of operation. She was a first year teacher. After her second year, she got canned, because apparently I wasn't the only one who didn't "get it." My last quarter of senior year math, our teacher handed us a small black book, and said, "Here. This is all self explanatory." It was calculus. It was NOT self-explanatory.

[apcalc]

61. We teach PEMDAS.

[DCBob]

56. The BEDMAS rule is dumb.. it simply causes confusion and errors.

Just use parentheses.. eliminates any ambiguity.

[SheilaT]

59. As several others have already pointed out,

without parentheses you do each step in order. Anything else is assuming parentheses. Where none exist.

So: nine minus three is six, divided by one third (and I might be wrong here) is the same as times three, so eighteen, plus one is nineteen. I actually haven't looked at any other answers.

[SickOfTheOnePct]

63. No, you don't do it in order when there are no parentheses

And the answer is not 19.

[Nye Bevan]

70. Enter 2 + 4 x 5 on your calculator. You will get 22, not 30, as the result,

because you do the multiplication before the addition, even with no parantheses.

[Ms. Toad]

72. On cheap calculators, you will get 30.

Having taught calculator-based math for several years, I can tell you that there are calculators set up both ways. (If you are working on a Windows based computer, pull up the calculator app and try it. Then try it here: http://web2.0calc.com/)

That's one of the reasons it is important to understand the correct order of operations.

(You are correct that you should get 22, not 30. Unfortunately, some calculators "know" math as well as some people, apparently.)

[Recursion]

62. Where does this require algebra?

It's 1, no algebra involved.

[Enrique]

64. there should be parentheses

even if not strictly required. Or even break it up into two or three steps, if it makes it more clear what is being done.

[backscatter712]

73. Order of operations. Learn them.

I'm a trained software developer, among other talents. Of course, I got the answer right the first time.

Alternatively, you could switch to doing math using postfix notation...

[Donald Ian Rankin]

91. Software developers are the people I'd expect to get it "wrong" most often.

Brackets Over Divide Multiply etc. But is the / an "over" or a "divide"?

On balance I'd plump for "over", and looking at the video makes it clear that that's what's intended.

But in every programming language I know (which admittedly isn't many), / is explicitly used as "divide", because there isn't an obelus symbol, and so 3 divided by 1 / 3 would evaluate to 1, not 9 because you work left to right.

[stone space]

75. Who the hell knows? Parentheses are your friends.

9 – 3 ÷ 1/3 + 1 =

Use them!

Either that, or state your parsing conventions up front.

[SickOfTheOnePct]

88. Parsing conventions?

Hardly. It's basic math, and the rules don't change.

[stone space]

92. Here's one parsing, which seems perfectly reasonable to me.

9 – 3 / 1 / 3 + 1 =

(9 – ((3 / 1) / 3)) + 1 =

(9 – (3 / 3)) + 1 =

(9 - 1) + 1 =

8 + 1 =

9

Others prefer the following parsing:

9 – 3 / 1 / 3 + 1 =

(9 - (3 / (1 / 3 ))) + 1 =

(9 - (3 / 0.33333...)) + 1 =

(9 - 9) + 1 =

0 + 1 =

1

Personally, as a mathematician, I don't really care, since nobody in their right mind would write such a thing.

If somebody really wanted to be understood, they'd use parentheses.

[SickOfTheOnePct]

96. If you're really a mathematician

then you know that what you're claiming is ridiculous.

Brackets or no brackets, there is an order of operations...we don't get to choose which way we would prefer to do the math.

Well, I take that back - we can choose, so long as we don't mind getting the wrong answer.

[stone space]

97. Which parsing did you choose? I assume that you got either 9 or 1 as an answer.

[SickOfTheOnePct]

98. I chose the only correct method

and the answer is 1.

[stone space]

99. So, does that mean that 27/9/3=9?

I chose the only correct method

That's what you get with your "correct" method.

The "incorrect" method gives 1.

[SickOfTheOnePct]

100. Wrong again

From the link:

The trick is to remember PEMDAS, the order of operations formula, which stands for parentheses, exponents, multiplication, division, addition, subtraction. As Talwalkar explains, this means first tackling 3 ÷ 1/3. Three divided by 1/3 is nine, and then carrying along with the equation from left to right you end up with the correct answer of — drum role, please: 1!

[stone space]

101. If you claim that 3/1/3=3/(1/3)=9, you should also claim that 27/9/3=27/(9/3)=9.

Otherwise you are using a right to left convention for one problem and left to right for the other.

I don't care which convention that you use, but for consistency, you should use the same convention for both problems.

[SickOfTheOnePct]

107. I'm not claiming anything

other than basic math, i.e., the order of operations.

As you purport to be a mathematician, I'm shocked that you don't appear to understand basic math.

[stone space]

108. Do you agree that 3/1/3=27/9/3?

If not, why not?

As you purport to be a mathematician, I'm shocked that you don't appear to understand basic math.

By the way, personal insults are really not needed in a silly thread like this one.

[SickOfTheOnePct]

115. Sure

3/1/3 = 1 because as all operations are the same (division), you calculate left to right

27/9/3 = 1 because as all operations are the same (division) you calculate left to right

But when there are mixed operations and no brackets or parentheses (addition, subtraction, multiplication, division) you work the innermost multiplication or division first, then the next multiplication or division, then once those calculations are done, you work left to right.

So, the first operation to calculate is 1 divided by 3, then three divided by 1/3, giving a result of 9. Then it's all left to right - 9 - 9 + 1.

Of course, had you bothered to watch the video in the OP, you would have already seen all of this and shown where you're wrong to believe there are options as to how to do the calculation.

[Lucky Luciano]

128. Nope

3 ÷ 1/3 = 3 / 1 / 3 = 3 / 3 = 1

As written you are implying division by fraction because of the ÷ symbol, but that symbol is equivalent to / just like video says in fact!!

[SickOfTheOnePct]

129. I suggest you go look at the video

where is says the answer is 1, not 9, as your calculation above would show.

You do the innermost multiplication/division first, not left to right as you're doing.

[Lucky Luciano]

131. Point is that the equation as written in the OP is

not the same as is written in the video. In fact, the equation as written in the OP is the one the video explicitly uses to say how the equation in the video is solved incorrectly!!

In fact, the video goes on to solve the equation as written in the OP and they get 9!

[SickOfTheOnePct]

133. It's written exactly the same

Assuming one knows that 1/3 is the same as one-third.

Obviously, that's a pretty bad assumption.

[Lucky Luciano]

134. Sigh...you are dead wrong...it is a division operation not

meant to be treated differently than any other.

The video clearly evaluates the equation as written in the OP as 9.

[SickOfTheOnePct]

155. And says that 9 is incorrect

Did you not learn to do the innermost division/multiplication first rather than from left to right, as you're saying it should be done?

[Lucky Luciano]

157. equivalent operations (i.e. division) happen left to right without parentheses

In other words, you would say that division is not associative.

so a/(b/c) does not equal (a/b)/c.

You are evaluating it as a/(b/c), whereas the lack of parentheses would mean to evaluate it as a/b/c which is equivalent to (a/b)/c.

Again - the point is that the way the equation is written in the OP. The way it is written in the video is different. The way it is written in the OP should be evaluated as 9.

The way it is written in the OP:

9 – 3 ÷ 1/3 + 1 = ?

The way it is written in the video:

________1
9 – 3 ÷ ---- + 1 = ?
________3

I consider these two different equations. The second one is more clearly dividing by a fraction. The first one is just two consecutive divisions which should go from left to right with the absence of parentheses. In the video, they say that the error is caused by first setting the second equation to the first - that is an error. However, given the second equation and only the second equation, the video goes on to show that that equation evaluates to 9. The text of the OP gave the second equation.


Multiplication and addition, on the other hand are associative which means that parentheses involving only these operations are never needed:

a(bc) = (ab)c

and

a + (b+c) = (a+b) + c

However, subtraction and division are not associative.

a - (b-c) ? (a-b) - c (try a,b,c all equal to 1)
and
a/(b/c) ? (a/b)/c (try a,b,c all equal to 2 - the first gives 2 and the second gives 1/2)

[SickOfTheOnePct]

160. Another person who doesn't understand

that 1/3 is one-third.

Sad.

[Lucky Luciano]

161. I officially give up - you would fail my class. nt

[SickOfTheOnePct]

162. You would fail people who got the correct answer - 1?

Not much of a teacher.

[Lucky Luciano]

165. Whatever dude.

Not much of a teacher - I'll give you that since I failed to educate you. I tried, but I'm done.

[SidDithers]

119. Nailed it...

since nobody in their right mind would write such a thing.

If somebody really wanted to be understood, they'd use parentheses.

That's absolutely the key. "If somebody really wanted to be understood".


The equation in the OP is nothing more than a mental exercise. It's not using mathematical language to convey an idea, or model a real-world situation. It's the equivalent of stringing a bunch of words together with improper punctuation. It's designed to lead to ambiguity and confusion.

If the author wanted to be clearly understood, which is the goal with any language, they would have written their equation so there would be no chance for misinterpretation.



Sid

[uponit7771]

168. +1, "with improper punctuation" I had to use order of precedence all the time in writing psuedo code

... and something like this would get caught quick in review

[uponit7771]

167. +1, order of precedence semantics

[Sam_Fields]

78. My 9th grade science teacher grilled us on the order of operation.

This was an easy problem.

[auntpurl]

80. This thread absolutely proves DUers will argue about anything

Anyone remember the corn flakes flamefest? Olive Garden?

[Orrex]

81. Here's how to determine whether it's ambiguous

Find the person who wrote the equation and place them in a machine that will cause their agonizing death if an average student fails to solve the problem correctly in one attempt.

You'll quickly find out if they think that it's adequately clear in its current format.

[Aerows]

85. I think the confusion comes in because

we are taught things by rote, but not what mathematical equations are actually stating.

3 ÷ ⅓ really means, how many times can you divide up three into chunks of ⅓? 3 divided into 1/3 pieces.

You have 3 apples. You divide each of them into ⅓ sections. How many people can you give a section to?

I've always had a hard time with algebraic rules until it was rendered into something concrete so that I understood the concept. I was awful at algebra until I took physics and calculus. Then it all made sense because you finally understand the concepts.

Yeah, I was the nerd that struggled with everything but the word problems. It was always a blessing to have some of those on a test, because those I could answer simply by using logic rather than rote memorization.

Mathematical expressions are their own language as surely as any computer language is - that's why they call them languages.

[Doremus]

90. Old math, new math, it's all pffffft to me lol

Graduated in the 70s (the 'good old days' haha) and wouldn't have been able to answer the question then anymore than I can now.

How in Hades does dividing a number give you a larger number??

I'm sure there's something lost in translation. Just like one of the questions on my high school geometry exam which I totally tanked. In the post-exam recap, the teacher used plain words to reframe the question: how many legs does it take for something to stand up on? Why didn't he just say that? lolol

[GreatCaesarsGhost]

103. Translating math to English, it clearly says

9 - 3 divided by 1/3 +1.

9 - 3
-----
1/3 +1

[Lucky Luciano]

104. I am a mathematician, but had to click through to get it right.

The reason? It is written incorrectly in the text of the OP!!

The OP has:

9 – 3 ÷ 1/3 + 1 = ?

Which is the same as

9 - 3/1/3 + 1 = 9 - 3/3 + 1 = 9-1+1 = 9!!

When I clicked through, it was written in a way that was more clearly dividing by the fraction 1/3:

________1
9 - 3 ÷ ----- + 1 = 9 - 3x3 + 1 = 1
________3

Apologies for the _s - it wouldn't render correctly with spaces.

Here the parentheses on the fraction are better implied.

[stone space]

141. The factorial of 9 factorial is YUGE.

= 9!!

OK, couldn't resist.

[Lucky Luciano]

142. You talking about Trump's credit card debt? nt

[1939]

164. Think of it as

9-3/0.333333333+1

[Lucky Luciano]

166. No I get that...but strictly speaking...

3/1/3=(3/1)/3=3/3=1

I think by using two symbols for division, the implication was that the / was for a fraction, but that is incorrect if the two symbols for division are to be considered equivalent. Also, some people might be assuming that the white spaces between terms imply the fraction, but white spaces should not imply anything from an order of operations perspective.

[Android3.14]

105. 4th grade arithmetic

Please Excuse My Dear Aunt Sally.

The only reason a "lot of people" have difficulty with this is the same reason we have people flocking to Trump. Profound ignorance, rampant anti-intellectualism, and a failed public education system.

In order to cultivate a progressive population, we must embrace a conservative approach to education.

[SidDithers]

110. An ambiguous question leads to different answers...

Math is a language, and the goal of any language is clear communication of ideas.

The problem presented in the video is the equivalent of an English sentence without proper punctuation.

I like cooking my family and my pets.

I like cooking, my family, and my pets.


Nobody would ever construct a mathematical equation like the one in the video, if they were genuinely trying to use mathematical language to represent a real-world situation. The fault for the variation in answers is entirely on the person who created an imprecise mathematical sentence.

My $0.02.

Sid

[KentuckyWoman]

146. Yes

If you use the math order rules I was taught the answer is 1.

9-3/ 1/3+1
9- (3/ 1/3) +1
9- (3x3) +1
9-9+1 = 1

If you use the math order rules my mother was taught the answer is either 19 or 19.18181818181818.......

9-3/ 1/3 +1
(9-3) / 1/3 +1
6/ 1/3 +1
(6x3) +1 or some learned (6/ .333333333) +1
18+1=19

If you use the math order rules my niece is currently being taught the answer is 4.511278195

9-3/ 1/3+1
(9-3) / (1/3 + 1)
6 / 1.33333333... = 4.511278195....

And actually an argument can be made for
(9-3) /1 / (3+1)
(6/1) / 4
6 / 4 = 1.5
It doesn't follow any order rules of course but the equation is not specified so the reader is left to decide.

[Adrahil]

125. It's a simple order of operations peoblem.

C'mon folks.... 5th grade (or less) arithmetic. No algebra involved.

[LongtimeAZDem]

127. Clearly, some here are in need of some remedial mathematics:

[edhopper]

132. I googled this

9 – 3 ÷ 1/3 + 1 =

and the results it gave were this:

(9-((3/1)/3))+1= 9

Google!

You still think the answer is obvious to anyone with 6th grade math?

[backscatter712]

136. Dividing by a fraction is the same as multiplying by its reciprocal.

3 ÷ 1/3 is equivalent to 3/1 ÷ 1/3 = 3*3 = 9

So the original equation is:

9 - 3 ÷ 1/3 + 1 =

9 - 3/1 ÷ 1/3 + 1 =

9 - 3/1 * 3/1 + 1 =

9 - 3 * 3 + 1 =

9 - 9 + 1 = 1

The correct answer is one.

[edhopper]

140. I know that

but Google obviously didn't do it in that order. (9-((3/1)/3))+1 gets you 9.

The equation is not clear, as many have mentioned.

[demwing]

178. That's a keyboarding issue

My IPad keyboard works like Google, and doesn't include or recognize different characters for division and fractions. When given an equation that uses both, you have to translate, and treat the "/" as a fraction, which means keying it in within parentheses.

That's not technically a math problem, that's a Google (or a keyboard) problem.

[LongtimeAZDem]

181. That's because the Google algorithm failed to recognize that the obelus indicated that the fraction

was a distinct quantity

[Binkie The Clown]

138. Just like Donald Trump, I have the best answer. The GREATEST answer!

You've never seen an answer like mine. All those losers who can't get the right answer and just pathetic. Believe me when I tell you my answer is so much better than anyone else's answer you're going to wonder how you ever got along without my answer.

[Adrahil]

148. Well, if you can't do basic math....

I guess you can't get a job where knowing how to do basic arithmetic might be required.

[Adrahil]

150. I'd disagree....

There are LOTS of jobs that require it. But if someone can't do it, they might not see them.

Order of operations is VERY basic arithmetic. Why are people so afraid of learning math?

[Buns_of_Fire]

144. The answer is 42. nt

[demwing]

180. Wrong equation

You solved for M(L+U+E).

[TheDebbieDee]

145. I thought the answer was 1...

9 - 9 + 1 = 1

[IgelJames4]

147. This doesn't come as a surprise, honestly

Our education system has been dumbed down over the years, thanks to GOP budget cuts.

[applegrove]

153. Spoiler:

1

[DemFromPittsburgh]

154. I was told there would be no math.

[LongtimeAZDem]

179. +1 for the SNL reference (nt)

[L. Coyote]

158. Meanwhile, on DU I'm betting about 10% got it correct.

American educated!

[rickford66]

163. Now test everyone with some boolean logic

[Orrex]

176. Logic 2: Electric Boolean

[Warpy]

177. It's called parentheses, people.

Either it's (9-3)/ 1/3 +1 = 19 or 9-(3/ 1/3) +1 = 1. People are having problems because the problem is not clearly written.

[AntiBank]

182. why is this ultra simple problem so hard for so many people?

scary