60÷5(7-5) = ? Mathematician Explains The Correct Answer
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- เผยแพร่เมื่อ 19 มิ.ย. 2024
- What is 60÷5(7-5) = ? Everyone is arguing about this problem, so let's figure out the correct answer! The problem involves the order of operations, historical math notation, and binary expression trees. Glad to see the world is passionate about mathematics!
I studied Mathematics and Economics at Stanford. Press coverage of my work:
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0:00 Order of operations
2:06 Historical usage
4:40 Binary trees
*I get many, many emails about this problem and am unable to reply to them.
Here is a 1917 article from "The American Mathematical Monthly" that explains the usage of the division symbol as an exception to the order of operations
www.jstor.org/stable/2972726?s...
Google evaluation
www.google.com/#q=60÷5%287−5%29
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Good to know I haven't lost any of my math skills after all these years. I was wrong then and I'm wrong now.
This video is wrong. The anwer is 6.
The correct answer is 24.
60÷5(7-5)=12(7-5)=12(2)=24. You are still correct, the other person is wrong.
You weren't wrong, he is.
@@CarlMCole you are wrong, including about your false claim of genius.
I'll stick to the old way. So 24
When I entered the formula into Excel I got the message, “there is a problem with this formula.” In order to get Excel to make the calculation, the user must add parentheses to clarify the order of calculation. So yes, the formula as written is ambiguous and the person needs to clarify how the problem should be solved. In other words, don’t leave all of the decision making to a calculator.
there's a difference between how a computer calculator reads and how mathematics is done by hand by a person. When we are writing a division sign by hand, we use the traditional division sign ÷, but for a computer calculator we end up using the /..which can also be interpreted by a computer calculator as a division sign separating the numerator and the denominator..so for this question, it was written as 60 ÷ 5(7-4) which is the correct way to write this question for the answer to be 36. If you want the answer to be 4, it should be written by hand like this: 60 ÷ (5(7-4)). This has always been how mathematicians do math by hand. Using the / sign on computers, laptops, phones is what is causing the confusion. People are so used to seeing the / on digital devices, they think it's creating a fraction and start thinking numerator and denominator.
I asked my father who was an engineer for 45 years and literally helped build parts for the space program and the nuclear programs, and he said the answer is 6. He explained that there are 2 elements. 60 and 5(7-5), these values represent something and are not just numbers. So, there are only 2 expressions. The equation should be 60 / (5(7-5)). This shows how setting an equation up correctly is most important. Given the fact that these guys sent several capsules to the moon and back, I'm going to go with his answer.
I agree with your conclusion. I was taught that Mathematicians, Engineers and Physicists preferred where possible to rewrite an equation without the division ÷ sign to avoid ambiguity.
If mathematical conventions are being changed to suit Calculators preferences, surely an honest person would consider that a very dangerous precedent.
I am willing to be corrected.
@@malcolmbrewis5582 One other thing I did to test my dad's conclusion was I googled pictures of famous mathematic problems and equations. Secretly I was hoping to prove the old man wrong, lol. But I could not.
I could not find a ÷ symbols on any of those blackboards. I took that to mean this issue of confusing how to write an equation had come up before, so to be clear and accurate, they did not use them. It makes since that they would not want to have their proofs interrupted in different ways. The same issue could easily surface in grammar as well by including or omitting punctuation like comma's.
In physics I haven't seen a single time the ÷ symbol being used, it's always a fraction, like V=∆s/∆t
Almost anyone who works in STEM or has higher education will give the answer of 6. Japanese calculators also give the answer of 6. Anyone who only did high school, American high school teachers, and newer American calculators, will give the answer of 24. Make of that what you will.
The reason I treat the answer as 6 is simple - if I see an equation like "x/2y = 1" then I don't think it should ever be interpreted to actually mean "xy/2 = 1", which is basically the same question. Nobody who ever said an equation like that would mean for it to be interpreted that way (unless they're deliberately trying to trick you), and having rules that make it function differently will only ever make things more convoluted than they need to be for no practical benefit.
If you wanted to write 60/5(2) to mean you're dividing by 5, then instead write it the sane way as 60(2)/5 instead.
The kicker is the "divided by" operator in its presented form. (At school in Germany in the 70s we used : for division).
This sign however suggests a fraction with 60 in the numerator and everything that follows the division sign, hence 5(7-5), in the denominator. That would be 6 then.
In our school we were encouraged to express divisions in fractions because they are visually easier to resolve when they become large and contain many variables.
It seems like the sequential PEMDAS convention is the generally accepted one mainly because of computers.
That symbol has meant divide from before the 70's.
In true math, one does not use fractions...ever. One uses decimals. 1/2 in a math is indicated as 0.5 in order to be absolutely clear. It leads to less problems, and in programming it leads to a lot less problems.
Sorry but nothing about a division sign suggests a fraction at all. It sounds more like you were given a bad suggestion by someone trying to make things seem easier.
This is also not a restriction from computers. They could just as easily have been programmed to solve it following the second pattern but they weren't because that has been wrong for more than a century now (predating computers).
@@Nempo13 What do you mean, no fractions ever? How do you write x/y in decimals?
@@Cdaragorn That's not true. A division 5÷3 (or 5:3 as we did in my school) can be written as a fraction 5/3 or in words five over three.
@@R2BMusicCH Yes of course you can convert it to a fraction. The fact that you can convert it does not mean it's implied to be that at all.
Your original conversion was wrong. The fact that you did it wrong does not mean it was implied that it should be that way. It just means you don't understand how to convert between those representations.
A correct conversion would be 60/5*(7-5)
People aren't passionate about mathematics - they're passionate about arguing.
I'll dispute that. Passionately.
Buckhorn Cortez NO, I’m not!
I know I'm passionate about arguing.💗 But I'm always drawn to problems that require solving.
Passionate about correcting people that i know are incorrect*
I am passionate about truth
This problem is the reason why you should use fractions instead of the "divide"symbol: makes it completely unambiguous. Either the (7-5) is in the lower part of the fraction (denominator in English?), making the answer 6, or the (7-5) is completely outside of the fraction, making the answer 24.
you would need to write as 60:(5(7-5)) to make the whole thing the lower part of fraction. Otherwise, modern way to calculating will give u 24.
agreed entirely. written with a division symbol introduces ambiguity
There is no reason you can't switch the position of the 5 and the (7-5). Then you would have 60 in the numerator, and the 7-5 in the denominator. What next you would do is calculate 60 divided by 2, which is 30. Then you would multiply 30 times 5, to get 150.
Fractions are not interchangeable with division.
1 vinculum 2 is the fraction one-half
1 solidus 2 is the division 1 divided by 2
You cannot just replace one with the other willy nilly.
@@sinub801
60
-----------
5(7-5)
The correct ways to phrase the questions (depending on what you want to ask) would be:
60/[5(7-5)] for which the answer is 6.
Or (60/5)(7-5) for which the answer is 24.
The question as originally phrased makes no sense. The division sign is never used beyond grade school nowadays (it is not there even in a computer keyboard), but it was there in the question but without the multiplication sign. It was not only confusing but sloppy. One set of parentheses would have eliminated all ambiguity.
Assuming the question was originally an algebra question for which you then substitute in the actual numbers, then "6" as the answer actually makes more sense.
If you trust Texas Instruments' calculators, then the rule changed between 1993 and 1996. My TI-83Plus user's manual (page 1-24) says implied multiplication has the same priority as regular multiplication and division, so 1/2x is evaluated as (1/2)x, *but* the TI-82 gives a higher priority to implied multiplication so 1/2x is evaluated as 1/(2x). According to Wikipedia, the TI-82 was released in 1993 while the TI-83 in 1996.
Modern TI-85Plus also has same precedence for implied and explicit multiplications, so they give answer 24. But modern Casio (at least my fx-CG50) work like old TI-82 and gives answer 6.
Don't key expressions unthinkingly, verbatim. Electronic calculators are not to be trusted that much. That is learned very early. The insertion of brackets is often needed. Rewriting with or without a fractional exponent can be useful. Sometimes, as in 1° 1', a "+" must be inserted to show addition. Juxtaposition can mean different things. 3pi indicates multiplication. 31 indicates the addition (of 3 × 10 plus 1 × 0). An electronic calculator frequently need to be told how to operate.
I’m 70, and I’m just thrilled to find out I still remember being taught this! And no, I’m not a math geek. I’m a little old lady who has stayed motivated to keep learning all my life!
So true Lynn. We may never need this particular equation in our everyday life, but it's nice to know the method anyway.
Hi Lynn, you and me, both.
I'm 66 and went to one of top 20 High Schools and the then top Accounting and business University in the nation. The answer is 6.
You are awesome!!!
Ditto. Great isn't it!
If I owe you, my calculation is 6.
If you owe me my calculation is 24.
Yeah its 6. Isnt it?? I dont want to watch the whole thing. It should be 6.
@@thetruth3828
Must be a quadratic, because the claim is, if you are old school it's 6.
But if you are new school it's 24.
Don't know how it can be either or either, as there must be an intended definite outcome.
Good answer! Ha ha!
@@onlythetruth883 You clearly don't know what a quadratic is--but without knowing what you're doing or saying you've accidentally hit on the problem with many of the arguments in this thread: the given task is to evaluate a simple ARITHMETIC EXPRESSION using the generally-accepted rules for doing that.
Attempting to translate it to an ALGEBRAIC EXPRESSION and applying rules useful there, then translati9ng back to do the arithmetic, does not work.
If it involves just numbers and arithmetic operators and brackets, so one could evaluate it on a calculator or calculator app, then use the usual rules for evaluating arithmetic expressions and don't try to remember your high-school algebra and misuse that very foggy recollection to confuse yourself and others.
@@waynebrehaut7183
Of course I was being sarcastic when I said must be a quadratic. And you did get the point-->. There is no point until the rules are firmly established.
Although we have modern PEMDAS to adjudicate how to interpret such expressions, this is really an inherent language flaw, as you pointed out mid video. It is rooted in the idea that you can omit the multiplication symbol between and number and an opening parenthesis. If you write it as 60 ÷ 5 * (7 - 5), you still need PEMDAS to interpret it, but it is much less tempting to get it wrong.
I am old as dirt.
I always distinguished a difference between
N*(a-b) and N(a-b)
With N(a-b)
== (N(a-b))==(N*f(x))
Just my shorthand.
As written, for me, the correct answer is 6 because there is no "x" between 5 and (7-5). To get 24, it should be written 60÷5x(7-5).
When using excel, i always “over use parentheses “ to force excel to evaluate exactly what i want. I can’t afford surprises.
I so the same thing.
When using Excel, all you need to do is be sure of your maths. I know your "parentheses syndrome" because I suffer from it, too. But the truth is that I'm just not good enough at intuitively simplifying fractions, so I force the program to jump through all the hoops I need to be sure I got it right ;)
I see the need to do that too to ensure I get the correct answer. Although it might be a bit more complicated, it is worth to to avoid later headaches.
Plug this into Excel =60/5*(7-5) answer =24
ABSOLUTELY agree.
24, didn’t even watch the video.
I watched the video, just to find out what else answer might be correct. And i laughed so hard as 6 was given as a possible solution. Yeah, with outdated, medieval math, 6 is a solution. But we live in 2018 :D
Exactly...
Same
IKR same
Same
Part of the issue is whether one considers the number parked outside the parentheses to be a common factor of the terms within the parentheses, or just another number in the sequence. I was taught that the number just outside the parentheses (in this case 5) is a part of the terms inside the parentheses ((a-b), with in this case a=7, b=5) unless separated by a multiplication sign. So (5a-5b) is the same as 5(a-b), but not the same as 5*(a-b). This would lead to a result of 6, which I would consider to be the proper result. Also, look at the division sign itself. The top dot is the stuff to the left, the bottom dot is the stuff to the right. Which would also yield 6.
I learned back in the 1980s and 90s that you have to interpret equations for computers and calculators to get the proper results. So I would input the above equation as =60/(5(7-5)) when using a calculator or computer. Which would again yield 6.
It's honestly feeling like a bunch of people were the subject of teachers trying bad ideas in an attempt to make things easier.
That's not what "common factor" means at all. And as someone has already pointed out having the * explicitly changes absolutely nothing. It wouldn't make sense to have it change anything.
“So I would input the above equation as 60/[5(7-5)].”
You completely changed the equation the way you wrote it. You can’t just add an extra set of brackets in the middle of the equation. Had it been presented in that form, then yes, the answer would be 6. People are getting confused with what “brackets first” actually means. They think if they see brackets, that means everything touching the brackets gets done first. Brackets first means you solve the inside of the brackets first. Once you do that, the brackets part is done. 5(2) is 5X2 is 5*2. It doesn’t matter what form you use, they’re all the same thing. Since it’s now just a straight up multiplication and division equation because the brackets have been solved, you move from left to right.
And the above commenter is correct that 5(a-b) is the exact same thing as 5*(a-b) is the exact same thing as 5a-5b.
If a=4 and b=2
5(4-2)= 5(2)
5(2)=10
Also
5*4-5*2=20-10
20-10=10
I put this in a calculator on a computer and it came out 24 so you're wrong
Kudos! The expression on the RHS must be evaluated first before the division. What the RHS says is that there is a common factor of 5 and so the full expression on the RHS is 5(7-5) = 35-25 =10. And so the answer is 6. I don't care what Google says!
@@mohasat01 You also don't care how math works. That's not what a common factor is. And even if it were common factors is just an interesting fact of the numbers and has nothing to do with how or when you evaluate them.
Per order of operations 60 / 5 must be evaluated before 5 * (7-5) because multiplication and division are to be evaluated left to right.
The problem I have with this kind of problem is that it is not really math. It’s grammar. Just write the darn expression in unambiguous way so we can do actual math. We have more interesting concepts to learn in geometry, trig, calculus, etc.
100%
I wholeheartedly agree
@@jakemccoy Yea, I agree.
@@UniversalS757 Don’t worry. Math grammar has correct answers too, but math grammar is different than math concepts. I have been an engineer working in the real world for 30 years. Not once have I debated stuff like this on the job. I will just put parentheses in there and keep it moving. This is a discussion that may be fun, but it needs to stay on academia.
@@jakemccoy ok
I remember being taught that parenthesis was calculated first, multiplication came next, then division, then addition & lastly subtraction. This gave me 6.
I graduated from high school in 1987 and that's the way I was taught. The answer would be 6.
Peace.
That's what they taught for most in our country's basic education too, literal PEMDAS in strict order (as the letters). Only in college that both math and computer science professors agree on the real correct method. I'm mildly infuriated that they always teach children outdated or plainly wrong things like this (the four taste regions also comes to mind, so wrong)
BODMAS????
@@davidevans8858 B-Braces/Brackets, O-Orders
@@mk_rexx well, pemdas (or pedmas as I know it) isnt wrong though. But the order of division or multiplication doesnt matter, and the order of addition and subtraction doesnt matter, as in both cases they are effectively the same operation. So everything in brackets first. Then all multiplication and division. Then all addition and subtraction.
Based on the education I received in the USSR in the 1980's the answer is 6.
I guess we now know why your economy and government crumbled when someone over there found MTV.
Your teaching was correct and equivalent to the teaching in the US in the 1970s, which is when I did my primary education.
Yes, and 6 is the correct answer worldwide.
Let me summarize the positions as I see them:
> for folks who are followers of the PEMDAS philosophy and believe such things as
x/3x is equal to x squared divided by 3
the answer is 24.
> for folks like me who believe that PEMDAS is BS and screwing up the teaching of math in America and believe in such things as
x/3x = 1/3
the answer is 6.
Now I do recognize that this is America and one is free to choose, but
from my viewpoint it does appear that the PEMDAS philosophy falls into the category of metaphysics; - - - you know, that abstract theory with no basis in reality.
@@Borvo1 its A/BC = AB/C?
True or False?
All of these type example are due to someone writing mathematical statements in the most confusing way; in REAL mathematics, physics and computer programming we choose the write mathematical statements so as to prevent confusion. These example-makers lift a few excerpts from journal (or written text) articles where one is forced to use only a single line of text space; however most likely elsewhere equations are presented in an correct format.
As a computer scientist I have worked quite some times with machine interpretation of such expressions. Therefore I can see both sides, since the rules I often implement are modifiable - and in more abstract math, the operators are defineable almost as you please. However, left-to-right associativity on same precedence operators (multiplication and division above addition and subtraction) is probably seen as the norm.
It's not just a norm. It's the strict rule. If it wasn't it would literally be impossible write mathematical equations because you couldn't guarantee everyone would get the same answers out of them.
If the computer is not doing left-to-right associativity on same precedence operators, then the computer is programmed wrong.
@@MrGreensweightHist Then all programming languages (100+) I know of are wrong... My guess is that those guys having made the programming languages actually studied precedence rules a bit.
@@xian8531 "Then all programming languages (100+) I know of are wrong"
Or you don't understand them as well as you think.
60÷5(2). Why so much of fuss. Some are multiply 2and 5. They don't seem to understand it is 60×2÷5. My simple method is 60×1/5 ×2. I am not a mathematician or a rocket scientist but I am merely applying mathematical rules. Whichever way you look at the answer is 24. If that's not simple then stop the world rotating for 1 sec so that I can jump of
Order of operations is just a way to guess at what to do when someone writes an expression poorly. Be clear when you write.
Correct is you/ gob jood
Exactly. The whole reason we're taught order of operations is in case some jackass writes an expression like the one in the video. It's amazing to me how many people are so bent on disagreeing about it.
@@frocat5163 ok I understand what you are saying but the cornerstone to my math is BODMAS, how else can you do it ?
If an expression is badly written the order of operations can't fix it.
How could it?
The question needs to be written properly by the person who wrote it. That's the only way the ambiguity can be resolved.
It's like writing Sin60/2.
It's terrible notation and is ambiguous as a result.
No order of operations will fix this.
Only appropriate brackets will.
E.g. Sin(60/2) or (Sin60)/2
The order of operations is a tool to simplify and cannot fix anything.
PEMDAS itself is a poorly taught and explained acronym... M does not come before D and A does not come before S.. M is equal to D and A is equal to S, so you calculate them from left to right in the order they appear.. PEMDAS should be taught as P, E, M or D, A or S
Math and CS teacher here. I think everyone misses the most important part here: spacing. A common practice in CS is to use spaces to display precedence, so for example you would write a*b + c*d. It helps readability and can be really usefull for less known operators precedences like and/or. And also not all languages follow the exact same precedence rules, especially for bitwise operators. So in the ambiguous expression shown here, the modern precedence rules would give 24 but the spacing indicates that it's actually 6. For the same reason, when I see 1 / 2x, I tend to understand it as 1 / (2x).
you can add all the spaces you want. The result is still 24..
60 / 5 (7-5)
= 60 * 1/5 * (7-5)
@@ChespiritoChavo322 There's no "the result is ...", it's all about conventions. Don't take conventions as rules written in marble, they change over time, they change from a country to another, from a book to another, from a calculator to another, etc. We don't know the context of this expression, maybe it's from an old book for example, so we cannot know for sure that modern precedence rules apply. But the spacing clearly shows the intention, and that's something we can rely on.
@@pierreardouin6441 i didn't follow any rule. Just used the formal definition of division.
@@ChespiritoChavo322 There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not.
That's what's causing the different answers. Division is used in either case.
Why in the hell would you put parentheses around 2x when it did not have parentheses to begin with???? Spacing or no spacing, the answer is 24.
As a physicist. I went with 6.
5(2) counts the same as (2) ex x. the modern method generates 60/5x1(7-5) . the main problem with the modern method is the improper disposing of (). as soon as you make it 5x1() your disposing it the same as 5() without changing the number inside. no matter what () must be removed before proceeding even if exponents or multiplication takes place. this was know as completing operations between signs in the 90's.
an example would be 2 x abc= vs. 2 x a x b x c . abc is a complete expression of 1 number to mutipply 2 by. no it may turn out a shortcut is it's all multiplied together so order doesn't mean much. but only if the shortcut doesn't ater the answer.
in this case the shortcut 5x1(2) alters the correct answer so you have to follow the 5(2) = 10. since rewriteing 5(2) as 5x1(2) allows the interpation 60/5 x2 this is a basic half step to proper order of exponents and such.
probably one of the most disturbing parts of teaching maths in a system
I was helping my 13 year old with his math homework 15 years ago and learned something that I was never taught in school. Not even in College. "Please Excuse My Dear Aunt Sally".
My 7th grade math teacher used her name in it: "Pretty Please, Mrs. Dovers Always Says".
try to apply it to : 3*47-1/4398473+10-8/33 without parenthesis
@@aligator7181 That's (3*47)-(1/4398473)+(10)-(8/33) = 150 + (25/33) = 150.7575757575 . . .
Most/all decent calculators will get that without using ( )s
@@Chris_5318
Yes, but the trick is to get the order right. I have never used an expensive scientific calculator, I am assuming they probably sort out the order automatically?
@@haroldprice1030 Different, but almost identical, models from the same manufacture can give 6 or 24.
The correct answer is the one found by using the same convention that the author used. We have not bee given tha info. However, the author would have to be crazy if he was expecting anyone to get 24.
My Sharp EL-520W gives an answer of 6 for the expression "60÷5(7-5)", while it gives an answer of 24 for the expression "60÷5×(7-5)". This is also the way I was taught it in school. Implied multiplication with no operation symbol as in expressions like "xy or 3(5)" takes precedence over division indicated by the ÷ sign, while multiplication indicated with a × symbol has the same precedence as ÷, evaluated left to right. I didn't even realize this was controversial till I saw this mentioned in some of your videos. When did this other convention become popular?
Exactly. No one in their right mind would evaluate 1/xy as y/x.
I'm not sure it's the convention becoming popular so much as a simplified set of rules being widely taught in some places.
Goodness, you're right. The syntax changes depending on whether you use / or ÷
Same. My first answer is 6 cause the first thing I do is multiple 5*(7-5) wich is be come (35-25) and decrease the number at parenthesis, so it will be 60÷10 and is 6. Sorry for my bad grammar...
Same for me on Sharp EL-531W
5(7-5) is a single term because there is no operator. This expression is telling you to divide two terms - 60 by 10.
If the intent was to multiply terms, then it makes no sense to write it this way. In that case it should either be written (60÷5)(7-5) or 60÷5x(7-5).
See, when you get to "60 / 5(2)", to my mind, the 5(2) is an outer parenthesis+bracket expression (which should be evaluated after the inner parentheses+bracket) and should be evaluated before the typical multiplication+division. I was taught pre-PEMDAS, however, but I think that approach clarifies a lot of these "ambiguous" problems.
No in that case it would be [5(7-5)]. That’s where you would use inner brackets and then outer and that’s where you’d multiply by 5 before moving left to right from the beginning. In this case there are no outer brackets so once you’ve solved what’s inside of them, you move left to right from the beginning. 5(2) is the same as 5X2.
PEMDAS has been around for centuries, modern PEMDAS has been around for more than a century (as the video showed), so no you weren't taught pre-PEMDAS. You were just unfortunately taught wrong.
PEMDAS is not used by mathematicians, physicists or engineers:
th-cam.com/video/lLCDca6dYpA/w-d-xo.htmlsi=Rzfnvk4hUtqL6ZVq
@@trickortrump3292 This is why I get dinged when I write essays in school - I genuinely use *too* many parentheses and think parenthetically. Not so much that I'm particularly proficient in LISP, though.
@@trickortrump3292 No, the outer brackets are implied and aren't necessary. 5 is the coefficient of 2, so 5x2 must be done first.
Seems i was thought the 1917 version. My result was 6 too. Maybe you could do a follow up video on why the modern version is now used. What advantage does that interpretation bring?
In large part because expressions cannot be presented to computers by use of a divide bar that clearly shows what is in the numerator and what is in the denominator thereby showing grouping. Computer languages demand expressions all be in-line and there is no way to group subexpressions other than with explicit use of parenthesis.
it is manipulating the mathematics as they do it with everything this days. All depends who is calculating and for whom. If that was you assessed by tax office it would be 24 but if that tax would be calculated for Bill G. it would be 6. - 😏
the sentence when be written as a fraction with 60 on the top and the rest in the bottom and the result is obvious.
@@lubanskigornik282 And I just know if I buy Bitcoin, somewhere along the line my payout is going to use the New Math and end up dividing my payout by 24 instead of 6.
Correct!! But then I was in school in 1917!!! I think it is used to save space and characters in computer. 5(7-5) uses 1 less character than 5x(7-5). New Math!! You know, 2+2=5.
Left to right, what a nonsense. The fact there is no multiplication sign between bracket and the 5 is a clear indicator, that this is just one term, that the 5 and the bracket belong together, period. Anything else is sophism. 6 is the solution, period.
Ok, I figured out.the equation…
It’s (9min video)+(wrong answer)+(huge comments engagement)+(3,000,000 views) = $6000
You have the winning answer
The guy even _said_ at the beginning of the video why he was making it. Cha-ching!
Good for him and I am glad that he is uncovering something that is making us say 🤔
@@sandragrant327 I agree, it’s quite an undertaking to make math controversial
Yep, telling people that 5(2) is (5+5) .. first order operation or 5(2) is a scalar .. second order operation, wouldn't have given him my 2 cents.
5 is linked to its brackets therefore must first be solved before being devided into 60
To me 6 is correct because you can write that equation 60 over 5(7-5) to get 24 you would need parenthesis around the 6 divided by 5 to show you work it out first. After all that is what parenthesis are for. You have them use them.
The correct answer is to use proper consistent notation.
You want the answer to be 24?
60 / 5 * (7-5)
60 / 5 * 2
12 * 2
24
You want the answer 6?
60 / (5 * (7-5))
60 / (5 * 2)
60 / 10
6
Exactly. Notation is key
24 not 12 but yes ^^
@@thairorecordsamv1040 lol yes. Ill fix that
So, everything you've written is correct but I'd like to add:
5*(7-5) vs 5(7-5)
There isn't a clear-cut difference but I'd lite to think that the latter represents factorization whilst the other is normal multiplication. If this was the case 6 would be the correct answer. Considering how unclear the notation is you wouldn't know the difference but this would simplify your second calculation.
@@peckapuder The multiplication sign separates terms in the expression. The coefficient is part of the term. Order of operations applies to each separate term in the expression. What people are calling "implied multiplication" is simply using the number as a coefficient of the parenthetical expression as a term within the complete expression.
"Why would they change Math?? Math is Math!!!"
Well said Bob/Mr. Incredible , well said
Math did not change in this case. Writing and glyph interpretation changed.
Oh Contraire, math must now be expected to include critical race theory.
Math developed from the human ability to conceptualize, there is no inherent law of nature behind math.
I read a while ago that it was 4% of Mathematicians who use it this way. The rest of the population didn't. Probably someone in a wee office somewhere decided.
Not in "1984"... "He" is "she" and "She" is "He"... or whatever they say it is...
I guarantee if you answer 6 / 6x = 1x you will be marked wrong in any math class beyond elementary school.
When was PEMDAS implemented in Mathematics? Would the result not be; 60 divided by the result of everything following the division sign, meaning 6 would be the answer?
÷ is terrible notation
True. I learned in 5th class to use fractions and no terrible Division Symbol ...
And no one genuine has used the obelus symbol in the same expression as parenthetical multiplication. It just isn't done, except during these social media "math experiments" that offer no insight into how mathematics works. If anything, these problems just confuse math students (particularly young or inexperienced ones) trying to figure out order-of-operations rules in a realistic setting.
inline division signs deserve a painful death.
Not to mention that the ISO for mathematical notation has (for quite some time) said that the obelus should never be used for division. Math classes and mathematical exercises are not supposed to use the symbol, so teaching it is only a way to confuse younger students.
So ÷ =! /... plus I hated it because it sometimes looks like a minus sign if your dots are too small or a plus sign if your dots are too big
I prefer the ”special rule” version from 1917
I like writing my division as a fraction. That way there is no doubt as to what is numerator and denominator.
The special rule seems to follow this process.
I you use Google TH-cam to post this then you should stick to Google way of evaluating math expression. Google is the best guide.
You can just write it as a fraction and not division
i dont get why the separate division and fraction, isn't 1 over 2 0.5? Isn't 1 divided by 2 0.5? Then why are they so FKN different when they are the SAME?!
true, that's why don't use parentheses for multiplication in these situations, use * or the dot instead
@Anika Anjum That's why writing everything on a single line is ambiguous. The school I was taught is the division is a grouping operator so that everything to the right of it comes under the operator IE in the denominator. You were taught in a different school of thought. These different schools of thought are why equations need to be clearly written out.
See, if you tried this in Desmos for example which you input division as fraction as they should be, then you will receive an answer of 6 since division has a grouping element of its own
The answer is 24. 60/5*2 = 12*2 = 24. Multiplication and division have the same priority, so when no parentheses are present, perform the operations as they are encountered from left to right.
wrong
This shows that I am getting old, I came up with the answer of 6
One can get forgetful with age but plenty of young people fail to get the correct answer as well. The correct answer is 24
@Michael Stocker WRONG. There is absolutely nothing wrong with this expression except for the ignorance people have about parenthetical implicit multiplication.... The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly is 24
sorry for not knowing all the correct english terms
So do I the paranthesis is broken down for easy of handeling and shopuld be multiplied as it stated 5(7-5) -> (35-25), of the five should be diveded down to a 1 by devidind all groups by 5 to clear it out (60 / 5(7-4) -> (60/5)/((5(7-5))/5 ---> 12/(1(7-5) --.> 12/(2)
The 5(7-5) is a part of the paranthese operations and even in pedmas paranthese has priority
@@RS-fg5mf This has nothing to do with being forgetful and everything to do with what method an individual is taught on precedence.
@@ronhan9 Wrong...
60/5(7-5) does NOT equal 60/(35-25)
Easy handling is to simplify what is inside the parentheses. 5(2) is not a parenthetical priority and is exactly the same as 5×2...
The TERM 60/5 is to be multiplied by the value of the parentheses 2 and the only correct answer is 24
This is interesting and the reason for the change is that in the old interpretation the division symbol was actually a fraction symbol. The point above the bar represented all of the equation to the left and the point below the bar represented all of the equation to the right. Now however the division symbol is simply that, a symbol to divide the order of operations to the left by the order of operations to the right.
It's somewhat akin to English changing from archaic to modern English. The meaning of words has changed and if you keep up with the current meaning, you will understand what is being said.
For example, If I said, your room is in shambles. Currently that would mean your room is a mess, however it would have meant that your room is in a meat market. What fun.
Prior to 1917 SOME text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math that were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses. So the ERROR was corrected post 1917...
This was an ERROR brought about by the text book printing industry in regards to the misuse of the obelus. This is not why most people evaluate this expression incorrectly. They get the wrong answer 6 because they incorrectly believe that parenthetical implicit multiplication has priority over division.
Im not sure if y'all do it too but here in italy we use : without the fraction symbol
I'm 59 years old, for what it's worth. I was taught the fractional representation method in school and it still makes sense to me. Draw a line and solve for the numerator, then the denominator, then divide. That is how it was done then. If it is incorrect then how did we ever get to the Moon? LOL
@@BeerIndependence4All fractions and divisions are 2 different things
@@SmashingCapital how?
The fact that this continues to come up is evidence that there are two very different interpretations that have been taught to various people depending on when and where they were taught; and because this is the Internet, people are more than happy to boldly proclaim the other side to be wrong. FWIW, I was taught in school that the 5(7-5) is resolved completely before the division.
The answer seems really to be more explicit, brackets are cheap.
Think about this. This situation and debate about the correct way to solve an equation has come up before, I am sure. Engineers and physicists need their proofs to be interrupted accurately for peer review. There is no room for misunderstanding. I googled pictures of famous equations and I found no ÷ signs. They don't use them. Maybe we should abandon them entirely.
"I was taught in school that the 5(7-5) is resolved completely before the division. "
You were taught wrong.
That isn't a valid interpretation.
That is just you having been given false information.
There is nothing in math saying to include the 5 as part of the ()
Sorry
@MrGreensweightHist actually there is. Saying 6÷5 is the same thing as 6/5. The division symbol replaces a fraction, quite literally showing this to a trig professor and an a math calc professor, both have said 5(7-5) is the denominator. There is a reason they don't use the division symbol anymore and just use fractions.
@@Dogsparkster I am sorry you have bad teachers.
"There is a reason they don't use the division symbol anymore and just use fractions."
The division symbol IS a fraction bar
3÷4 is 3/4 is ¾
The reason ÷ isn't used anymore is simply because / is one line while ÷ is a line and two dots.
/ is faster to write.
that's the ONLY reason it changed.
X however, became * because X is too easy to confuse with the variable x.
using X instead of * can cause confusion.
Using ÷ instead of / alters nothing.
@@MrGreensweightHist yes that is the type of cocksure reply I expect from TH-cam comments, thank you. Juxtaposition having higher precedence than explicit multiplication or division is a long accepted notational convention that doesn’t appear to be universally accepted because it contradicts the sacred PEMDAS rule children are taught in elementary school, hence these internet controversies that continue to spring up. Since we aren’t in 1890 and trying to minimize characters when printing equations in books, we can all just be more clear for everybody’s sake and use more brackets.
also it could be written as 60/(5(7-5))=6 or (60/5)(7-5)=24 to be less ambiguous. I've done a significant amount of coding over the years and I like the use of parenthesis to reduce confusion.
24 isn't ambiguous from the get-go, though. To get a different answer just assumes grouping around 5(7-5) which does not exist in the original problem.
I've done a lot of coding over the years and I hate it when people overuse parenthesis trying to reduce confusion because it just makes the statements harder to read. Now I have to parse a bunch of parenthesis to figure out if you actually changed the PEMDAS order at all with them only to find out you didn't, you just wasted my time.
A fan as well. Being explicit is the way to go.
"also it could be written as 60/(5(7-5))=6"
No, it couldn't.
That would be wrong for this problem.
There is no ambiguity.
Extra () are not required when the operations are already in order.
You CAN use them, but the problem has clear meaning without them
It definitely doesn't make it harder to read, quite the opposite imo
I've used HP calculators with Reverse Polish notation from the start when they hit the market! In that system you start calculating the content of parenthesis and then go outward. With this logic, the result is definitely 6. During the whole time of my physics studies (that means dozens of textbooks in physics and applied mathematics), I haven't found a single case being confronted with any ambiguity of a mathematical term!!! If someone gives me such an ambiguous expression to calculate, I simply refuse to calculate! I will tell him to study mathematical semantics first! (This has already happened)
yeah, but our text books were kick ass.
notated,indexed and bibliographied with special symbols etc.
RPN rules!
The answer for us is 6.
I memorized times tables in the mid 60's; PEMDAS wasn't a thing when I went to school; I never took physics or calculus, only went as far as trig; the answer I got is 6.
I went to school in the 70s and 80s. Was always taught the method that gives the answer 6.
According to this 12x÷6x = 2x²
You don't see that often...
(I would go for the ambiguous)
according to my 1989 public USA education the answer is 6.
60÷5(7-5) =
60÷5(2) = And here is where the fight begins. Technically, according to the 1989 USA public education I received, the
PARENTHESES still exist that this point, and therefore has to be resolved first by Order of Operations
60÷10 =
6
Parentheses (inside first, then anything dealing with the Parentheses), Exponent, multiply/divide, add/subtract.
Even the distribution rule give the same answer
60÷5(7-5) =
60÷(35-25) =
60÷10 =
6
Your memory must be faulty or you had a bad teacher... I have at least 5 different math books from 1907 to the present and they all state the same thing... You evaluate what's (WITHIN) the grouping symbol not outside. And ALL multiplication and division can be evaluated equally from left to right..... When there are no (OPERATIONS INSIDE) the brackets/parentheses left to evaluate you can remove the parentheses and replace with an explicit multiplication sign or leave them to represent implicit multiplication and nothing more....
When you have a single value inside the parentheses that step is done... (7-5) is a parenthetical priority 5(2) is NOT a parenthetical priority and is exactly the same as 5*2
As for distribution, the whole point of distribution is to eliminate the need for parentheses by pulling what's inside to the outside not the other way around... Distribution requires that you multiply all the terms inside the parentheses with the TERM outside the parentheses. Terms are seoerated by addition and subtraction....60÷5 is one term to be multiplied by the two terms 7 and 5
60÷5(7-5)=
60÷5*7-60÷5*5=
12*7-12*5=
84-60=
24
60÷(5 (7-5))=
60÷(5*7-5*5)=
60÷(35-25)=
60÷10=
6
2+3+4+5 is 4 terms
10-9-8-7 is 4 terms
10÷2×6÷3 is 1 term
10÷2+5×3 is 2 terms
I hope that helps you understand better....
Richard S Again, that is how I was taught and I noted when and the type of education. That's why I explained it the way I did. It was so everyone can see 1) the logic I used because 2) it was the logic I was taught by educators 3.) using math books they provided.
So, with the correct answer being 24, you now have to ask the question; why are so many people like myself getting the answer 6?
Because we were educated wrong!
@@ComputerGarageLLC unfortunately a lot of people swear that they were never taught to multiply and divide before they add and subtract. Are we to believe this as well?LOL
I graduated in 1985 and was not taught in that manner. I have never seen a math book that supports your argument. I would be very interested in seeing a math book that supports your argument?
It is very concerning that so many people do get this wrong considering that the order of operations supports 24 as well as the commutative property and distributive property support 24 and the multiplicative inverse of division supports 24 as well as the majority of online math engines and scientific calculators support 24.
I guess this just goes to show that most people don't have to use math other than basic addition and subtraction on a regular basis.
Thank you for your input. Have a great day
You are free to not believe me. That is your choice. But it was how I was taught through the public education system. Clearly I was taught wrong, and it appears that many others were taught wrong too. we, those who are wrong, are a reflection of what we were taught.
And you are correct. a majority of people never use more than adding and subtracting most of their lives. Perfect example. Today a shirt cost $11.99, but tomorrow that shirt is on sale for 25% off. How much will you save by purchasing the shirt tomorrow? The answer that most people will give you......25%.
Another example I use. Mary has $10, but she need 2 gallons of milk @ $1.98/gallon and at least $5 in fuel. Does Mary have enough money. Doesnt matter, as mary will go buy the 2 gallons of milks at the gas station, and tell the clerk to put the rest in fuel.
So now, most of us never use more than very basic math most of our life.
And you have a wonderful day also.
that's probably faulty education as Richard S said. 5(2) is a multiplication, and you can't split 60÷5 in half with distribution of lower priority.
when I was in school we were taught to distribute the 5 to the numbers in the parenthesis first. thereby the resulting answer would be 6. At some point in time we changed the way we did math in order to confuse our children...I mean make math easier lol. I still enjoy the videos; keeps the mind working.
So when you say "modern" what time frame does this refer to? I feel like I was taught the left-by-right order, so I'm wondering when this changed. I'm not that old, so I can't figure out if I was taught incorrectly or has it changed since I was in school?
Left to right is correct.
This rule is over 500 years old.
Three is a myth it was different in the "old days"
This myth is spread by people who learned it wrong, but assume everyone in their generation was learning it the same way they did
The reason I came up with 6 was the fact I was taught that the order of operations was in the actual order of the letters. Parenthesis first then exponents, Math then Division, Addition then subtraction.
VERY EYE-OPENING AND EDUCATIONAL. GREAT VIDEO!!!
That is how I also learned it. I was taught to remember - (P)lease (E)xcuse (M)y (D)ear (A)unt (S)ally. (P)arenthesis, (E)xponents, (M)ultiplication, (D)ivision, (A)ddition, and (S)ubtraction. Please note...I went to a public school. LOL!
Although it wasn't mentioned in the video, the reason multiplication/division are not given a specific importance is because they are the same operation, so you perform them in the order as written.
Division is really just multiplying by a fraction. Ex: 60÷5 = 60 x (1/5). The same holds true for addition/subtraction. Subtraction is really just adding a negative number. Ex: 23 - 8 = 23 + (-8)
If you change all division operations to the equivalent multiplication operation, and then multiply straight across, you would see the answer will always be 24 to the equation presented in this video.
No you’re right it’s 6, cause multiplication is before division. The creator is just trying to cause division
Y'all learned wrong or were taught wrong. The correct translation of the acronym is "...Multiplication AND Division..." (equal rank performed left to right), "...Addition AND Subtraction.. " (equal rank performed left to right).
@@Paul-yb8pf no. Multiplication and division are equal. You solve left to right.
The expression typed into my Casio calculator exactly as shown returns the result 6. Which is exactly what I calculated as I was always taught that if there was no operator between a number and an expression in parentheses, then they were linked and to be calculated together. I.e. 5(7-5) = 10
Old Casio calculators do not handle the order of operations correctly.
@@Harmonic14 My TI-85 also states 6. I was always of the school of thought that when in doubt, use more parenthesis.
Exactly so. And his sentence is not ambiguous. The verb saw separates the subject (I) from the direct object (man) and any modifiers of the object (binoculars). So if you wanted to say you saw the man by using binoculars, the binoculars would have to modify the verb saw.
So did I. And it confirmed my "old fashioned way" to interpret that unclear calculation. It is from 1981.
@@wacholder5690 The order of operations has the answer
24.
One way that was taught to calculate expressions like 5(7-5) is to multiply the value outside of the parentheses by each of the values within. If it were an algebraic expression like 5(a-7), no one would argue that it work out to (5a-25). Right?
So, 5(7-5) turns into (35-25). Still within parentheses. That then calculates to 10. Then the rest of it becomes 60 divided by 10 which equals 6.
Depends. Some people apply the idea that there is implied multiplication or multiplication by juxtaposition. That would result in the solution being 6. Others do what you did, which is called the Distributive Property, and that also gives 6.
I find PEMDAS/BODMAS counterintuitive. For example, parentheses are a grouping symbol, and multiplication is often explained as "X groups of Y". From that angle, if we apply the problem to something real, such as money, we see the issue. $60 divided among 5 groups of 2 people (because 7-5=2). How much money does each person receive? Not $24.
" If it were an algebraic expression like 5(a-7), no one would argue that it work out to (5a-25). Right? "
Correct, but it isn't 5(a-7)
It is more like
60/5(a-7)
Which is
12(a-7)
And works out to
12a-84
Though keeping with the original problem you should have used
60/5(7-a)
instead of reversing the minuend and the subtrahend 7-5 and 5-7 are two very different things.
So
60/5(7-a)
12(7-a)
84-12a
For the actual equivalent of
60/5(7-5)
I think everybody is missing the point. The fact is that a mathematical expression like this is derived to calculate an aswer to a problem in the real world. Before we can know which binary tree to follow, we have to know the real-world problem.
What does 60 represent - it it people, who are being divided by ... what?
We also need to know what the 7 and the 5 represent, and why they are bound trogether in the bracket.
Mathematics is a tool - not an entity in itself.
You said : The "MODERN" interpretation. A lot of people, including myself, have been taught the one that gives 6 for result. I love math and was always at the top of my class. 24 would never have been the answer.
6 is absolutely the correct answer. 24 is result of a different equation. The video is wrong.
i think my math training also results in 6. so we just changed the definitions. no right or wrong.
ABSOLUTELY......me too !!!
The issue is that the video author doesn't understand BODMAS correctly... "brackets" means you grab the brackets first and solve them themselves using BODMAS. So 60/5(7-5) the bracketed term is 5(7-5) which expands to 35-25 which makes 10. 60/10 = 6.
Now, if you add a multiplication sign then it changes the precedence because you are actually changing the equation significantly. 5 * (7-5) the bracketed term becomes only (7-5) which is of course 2.
A deliberate nuance used to create a video I think. Fair play.
You were taught incorrectly.
If TH-cam recommended this to you, it knows too much about you.
Probably
I mean, not necessarily. I never answer these questions because I know they're made to start arguments
The list of people with the wrong answer will help scammers rip these people off.
I feel called out.
I sometimes refer to myself as a recovering math head (I aced the math ACT test and was second in my high school in the MAA competition at the age of 16) but I am not recovering all that well. When I see this mistake being shown as the correct answer I cringe. Evaluate the numerator, evaluate the denominator, then divide.
The author has completely misunderstood the issue. It's got nothing to do with any historical interpretation of ÷ as he claims at 2:09 and everything to do with the priority of implied multiplication, which he fails to even mention. In formulae, implied multiplication takes priority over division. For example, on the Casio website, it states "A radian is 1/2πr of the circumference of a circle." This is the standard definition and it does NOT, repeat NOT mean (1/2) * π * r. No, it means 1 / (2 * π * r). The implied multiplication is done BEFORE the division. And remember Casio makes calculators, so they should understand this point.
The problem arises when folk blindly substitute numerical values into a formula and enter the result into a calculator. Calculators don't know the difference between implied and explicit multiplication, so the answer comes out wrong. So, returning to the original equation, 60÷5(7-5), the question I would ask before calculating the answer is "where did this come from?" If it is the result of blindly substituting values into a formula such as a/b(c-d), then the correct answer is probably 6 rather than 24.
Also, you are more likely to see division represented by / rather than ÷ in such formulae, so the formula 1/2πr really means:
1
_____
2 π r
The order of operation jeft to right is the result of the limitations and acceptance of calculators with a reluctance to using notes to set up order of operation as was defined by the strength of an operation symbol. Those people are line up in whether you type with your fingers or thumbs and verbally as the difference of "Well yeah" or "Wull yea-uh".
Scientific calculators (Casio & Sharp) give answer *6.*
The rest answer 24.
Pick your side.
And even half of Casio calculators give the correct answer 24
@@RS-fg5mf
I test it on fx-570EX
What do you use?
@@rrsharizam I don't use a CASIO calculator I use Wolfram Alpha a math engine and I dbl check with Mathway another math engine and if the two don't agree I find out why. But for basic arithmetic I only use them to validate my answer not to give me the answer...
CASIO fx-82es will give 24
CASIO fx-570es will give 24
CASIO fx-50fh will give 24
CASIO fx-991es will give 24
CASIO fx-570ms will give 24
My response to anyone who says the answer is 6 is to evaluate 60a(7-5)=24......a =?
Well a= 0.2 or 1/5 and the divisional reciprocal of 60*(1/5) is 60÷5
Soooo
60*(1/5)(7-5)=60÷5(7-5)=24
@@RS-fg5mf
"will give" ???
So, you don't even use Casio, yet you say it will answer 24?
I don't care whether the answer is 6 or 24.
I just wanna say that Casio & Sharp answer 6.
That's all
@@rrsharizam I have a pic of these model CASIOS giving the answet 9 to the expression 6÷2(1+2) So if it will give 9 to that expression it will give 24 to this expression....
I dug out my TI-85 from the early 90's, yes it still works and yes I still use it from time to time, and the answer it gave me was 6. the same answer that I first came up with when I saw the problem before I remembered that they changed the rules of mathematics because of lazy computer programmers.
Same here
Then you entered it wrong. The rules you're talking about haven't changed in over a century (as the video pointed out). And computers had nothing to do with it. It doesn't even matter to them. They could easily be programmed to follow a different rule.
You typed it in wrong then, simple as that.
@@Cdaragorn but the answer is still 6
"It used to be 6, but now it's 24."
No!
lol 😂
The correct answer is 6. Multiplication is done before division.
@@Slowburn726 Multiplication and division are the same level. Addition and subtraction are the same level. The mnemonic should really be PE(MD)(AS).
@@briant7265 I think you mean PEM/D(A-S)
@@Amblin80s PƏM/D(A-S). Stacked exponents are evaluated right to left.
x^y^z = x^(y^z).
I was taught In school to do it the “historical way” because it’s still in parentheses so you multiply it first
vikpun XD thats not how it works. You do whats IN the parenthese first not mulitplying or dividing the parenthesis
@@kevinsanderson4112 as written i would think the 5 was factored out 60/((5× 7) -(5×5))i know some teachers who teach it this way and my calc class was like that so my immediate thought was 60/10 =6
@@jmanwild87 Your expression correct and that is the way I learn maths. How 24 become the unambiguous answer.
right????
I also came to the historical way, although I think part of it for me was how I viewed the question. I saw it similar to 60/5x where x is (7-5) being 2.
PEMDAS- order of operations, parentheses first, than exponents, than multiplication or division, lastly add or subtract. Still remember that acronym from elementary school
Answer is 6
Not arguing but talking from my schooling 50 years ago. I got 6. I subtracted 5 from 7 resulting in 60 / 5 * 2. I then multiplied five by two. Then divided 60 by 10 = 6.
I just typed into excel =60/5(7-5). Excel insisted in inserting the multiplication symbol in between 5 & (7-2). It then produced the answer 24.
I think excel did the problem in the following order, 7 - 5 = 2, 60 / 5 = 12, 12 * 2 = 24. Clearly I should now treat multiplication an division problems as equal actions and operate from the left.
Fun presentation.
It's surprising how some modern calculators like CASIO, which are recommended by math teachers, also give 6 as the answer! (tested with models fx-82ES PLUS and fx-82SPXII Iberia)
Thanks for the info! CASIO's calculators were a thing for 6÷2(1+2) as well. I found one video, for example, that shows 9 on one calculator (fx-50FH) and 1 on another (fx-3650P), both which are marked in the video as "H.K.E.A.A. approved" (Hong Kong examinations and assessment authority). th-cam.com/video/IXUBepvylQg/w-d-xo.html
I would love to speak to someone at CASIO about this--would make for a great video!
SHARP Scientific Calculator EL-531LH , gives 6 as well
The calculators tend to put in a bracket (in this case before the 5 and at the end) before displaying the result.
it is something called syntax. it is not as much math as it is programming. it is the programming of how to READ math in a single line. LIKE A TRANSLATOR FOR THE CALCULATOR.(it works in binary data) you do not.
When writing an expression parser, you may want to capture the intent of the user input. As I mentioned in another comment, the expression 1/2a is most likely meant to be interpreted as 1/(2*a), not (1/2)*a. The intent is generally to raise the precedence of implied multiplication above that of explicit division.
Why would you use a calculator as the way to measure what interpretation to use. A calculator is just a computer and a computer only does what a human programmed it to do.
Read my comment above....I think you will agree with me...
Because a calculator follows rules laid out by its human programmers instead of the unqualified presumptions of youtubers
Why would you use a calculator for such a simple task? However when my daughter was 13 in 1989 I bought 13 candles at the local stationery shop. I gave the girl 13pence but she said I'd better check its correct & rang up 1penny 13 times. No it was an old till not computerised connected to stock control. She then said "Yes you are right 13 pence" & put out her hand for the money.
We couldn’t even use calculators in high school (they weren’t available in grade school) in an effort to prevent the inevitable, the DDOA (the dumbing down of America).
but computers are programmed to do math same way as us...7-5=2. 60/5=12. 12/2=6. In that order.
It's always interesting seeing these equations on social media. I recall hearing a mathematician explain PEMDAS once by saying that MD and AS were not of equal value and that PEMDAS is PEMDAS because multiplication should take priority over division and addition should take priority over subtraction. I'm not sure if that was once the case or not, be he was adamant that the Order of Operations was explicit and not to be loosely interpreted. It's funny how something like math can be so controversial, but I guess it's the same for all intellectual conventions.
Just curious why does the linear equation proof work if math is read like this, another way you can interpret 6 is if you
60 ÷ 5(7-5)
60 ÷ (35 - 25)
60 ÷ 10
6
Why does the proof for linear equation allow it to work if it is an exponent in the parentheses but not the case if there is no number in the parentheses?
You're assuming that the 5 is part of the brackets. This is the most common definition when doing higher level maths, but some consider it incorrect. If you don't consider the 5 as part of the brackets, than you need to remember that the division only belongs to the 5. So you could rewrite is as:
60*1/5*(7-5)
=60*(7/5-1)
=12*7-60
=24
If you consider the 5 as part of the bracket, you'll get:
60*1/(5(7-5))
=60*1/10
=6
I remember being taught that when there is an “understood” multiplication because no “x” sign is there, then this calculation would be done before the preceding division sign. The 5 and the solution to the calculation in the parentheses are linked together, like the expression 5y are linked. If y=2, then 5y=10. Then divide what is on the other side of the division sign by 10. If they wanted me to do the division before the multiplication, they would have used a multiplication symbol in place between the 5 and the parentheses.
I wasn’t taught this but I’ve always followed it as it seems more intuitive
correct because the 5 is the coefficient of the parentheses. whenever you have a parentheses, you have a coefficient, and whenever you have a coefficient, you have to utilize the distributive property.
Ditto. No times sign between the 5 and the 2, just parentheses, was to be calculated first with how I was taught. I see it both ways but unless the order of operations changed in the last 25 years and it was not made public knowledge, then my math teachers would tell me I’m wrong to give 24 as the answer.
I'm old school (65) and we were taught the same thing. 6
agreed
There's a reason why most math teachers have rarely used the '÷' symbol in decades. Almost every teacher will teach division in fraction form because the division symbol is very ambiguous. If written with a '/' or in fraction form, there would be no question what the right answer is.
60/5(7-5)=6
Reason is, everything multiplied on the right of the '/' is part of the denominator. Which is the reason most people are tripped up using the archaic '÷' symbol. The rules are slightly different. In order to get 24 with the '/', you would have to write it as:
(60/5)(7-5)
Easy. Thats why nobody who actually works with math uses '÷'. And in higher level math, such as calculus in fluid mechanics or thermodynamics, the order of operations is practically useless. You're stuck developing your own equations by following your units of measure to get from one place to another. No real need for PEMDAS when you have a force in Newtons or pounds, and you need to solve for pressure in kPa or psi. Or maybe you need max power output in Watts or horsepower. Then again, if it wasnt for archaic symbols used to confuse people who dont do math in this respect regularly, this channel would probably have died out long ago
WRONG.... Prior to the 1900's that's how the obelus ÷ was being misused. The solidus was never used in this manner
60÷5(7-5) and 60/5(7-5) are exactly the same and both equal 24
The solidus is NOT a grouping symbol only the vinculum (horizontal fraction bar) has grouping power....
60
------(7-5) = 60/5(7-5)=24
5
60
-------- = 60/(5 (7-5))=6
5(7-5)
Extra brackets required to keep the grouping of operations together that the vinculum provided when written in a linear format with infix notation....
That is not why most people get this wrong. They incorrectly believe that implicit multiplication has priority over division. It doesn't...
Not necessarily, Richard. If it was written properly, the (7-5) is part of the numerator. So, without parentheses, you would have to write it
60(7-5)/5=24
Everything multiplied on the left of the slash is numerator, everything on the right is denominator. You're welcome to disagree. That's cool. However my college professor would mark my answer wrong if I wrote it
60/5(7-5)=24
As I said, nobody writes equations or mathematical phrases like this for good reason. There are simple programs to write and paste complex formula as they should appear, not like this with the intent to befuddle. Best of luck to you, bud.
@@boredbales12345 WRONG again. Multiplication is Commutative.
60÷5(7-5)=
60 (7-5)÷5=
(7-5)÷5*60=
24
All 3 expressions are equal to 24..
Evaluate this equation
60a(7-5)=24...... a= ?
@@boredbales12345Your professor would be wrong for counting 24 wrong... LMAO
neither ÷ or / have the special treatment of taking a photo of content to the left, to the right and using the operation afterward. None of that is in the order of operations.
In 60/5(7-5) the / is a division symbol, and the order of operations says 24...
You must be thinking of the fraction slash but that requires (7-5) to be subscript, ⁶⁰/₅ₓ₍₇₋₅₎, to equal 6.
On a FX-100AU you get two different answers dependent on the way the statement is written: 60÷5(7-5) =6 therefore 60/(5(7-5))=6 or 60÷5x(7-5)=24 therefore 60/5x(7-5). In this case the multiplier between the 5 and the bracket is the key. Therefore what is the correct way of writing the statement? To me 60÷5(7-5) =6 is correct because 5 and (7-5) are actually (5(7-5)). Happy to be corrected.
You have it correct. Most of us who did science / engineering / high-level math will interpret 5(7-5) to be a singular expression (just like 5x is a singular expression) because that is how the rule goes. If the integer is right next to the parenthesis with no operator symbol in between, it is considered to be a singular expression; therefore, 5(7-5) == (5(7-5)).
I know science / engineering students who are still being taught this rule right now today.
So how do you work around the fact 5(7-5)=5x7-5x5?
I have always struggled with maths having left school at age 14 but I am passionate in trying to figure out any maths problem. Don't often get the correct answer but I enjoy trying. Thanks for the exercise. Blessings.
There's disagreement on this answer because the notation is very sloppy.
If you are uncertain about this particular math question, you might actually be better at math than you think. Your math might be very good when the notation is clear.
Maureen Mallett.....it's math. No plural
@@joc8092 Depends on where you live.
In England, it is pronounced Maths plural.
@@MrGreensweightHist k, I stand corrected
@@joc8092 I only know from watching Doctor who, and thinking, "That sounds so bizarre" until i got used to it :D
That's why we use fractions instead of the division symbol
I agree. It completely avoids the issue
That division symbol is called an obelus. Just fyi.
? Thats division still.
@@Ok-th2gd of course it's division, but using the fraction instead of the obelus it eliminates confusion like from this problem
@@Ok-th2gd 60÷5(7-5) can be changed to 60/5(7-5). From that 60 is the numerator and 5(7-5) is the denominator. 5(7-5) becomes 5(2) = 10 so 60/5(7-5) changes to 60/10 which is 6.
Looking at the bigger picture, how does this works in practice where a correct answer is required or is it purely theoretical?
As per pemdas rule bracket is to be removed before division so 5 (2) is done first to make it 10 and then 60 /10= 6, please explain
I was always taught that anything that touches the parentheses / brackets was next after evaluating what was inside the parentheses/ brackets.
That's how I was taught to calculate too and I absolutely stand by it even if the rocket crashes. :-P
Anything touching them simply implies multiplication if they wanted it to be the 5*2 first then they should have done this
60*(5(7-5))
@@starlordz6111 True.
@Chris Travers when I typed it into my ti-83 I got 24. And that was after I solved it without a calculator. Anything touching but not in parenthesis only means multiplication nothing else.
In "60 ÷ 5(2)", the bracket "(2)" has higher precedence than " ÷ ".
I'm not going to rely on a calculator's "judgement" on what is ambiguous. The calculator is merely following rules programmed by a human that could have interpreted an ambiguous statement one way or another.
There's nothing ambiguous there. It's plain and simple, unless you were born in 1910 or something. Rules change, so people need to adapt and forget the old ones.
Flat earther
@@BypassOne I agree the problem is not ambiguous but I'm merely pointing out that a calculator result is not proof of the answer to the problem but merely the result of human programming, which is not infallible.
@@wildasiandude432 Recognizing the fallibility of humans and technology is not the same as Luddism.
@@percyfaith11 Human programming that is based on mathematical rules. Calculators were invented to easen and speed up calculations, exactly because people tend to forget them. So, believe me, the expression is not ambiguous just because YOU forgot the rules.
2:35 suggest re-writing as 60 / 5 * 2. The 5(2) = 5*2
Also some people read "multiplication - division" and forget about same precedence = so read left to right.
Solve Parenthesis means to remove the parenthesis from the equation first. Before left to right the parenthesis marks need to have been solved (removed from the equation). You can't move on to the next step in either PEMDAS or BODMAS until the parenthesis have been solved and thus removed. 60/5(2) still contains parenthesis (brackets) and it would be outside of BOTH methods to continue while the presence of a parenthesis/brackets still exist. Regardless of the fact that it is a multiplication it is still a bracketed expression. 60/5(7-2) and 60/5*(7-2) are not the same expression. This is also why it is not ambiguous.
Groucho Marx - "One morning I shot an elephant in my pajamas. How he got in my pajamas, I don't know."
autophyte,
Do you think, he possibly bored through with his erect trunk.
And did it, sorry, did he survive?
@@onlythetruth883 who? the elephant or Groucho?
@@NatandGeorge
Unfortunately it seems as if they are one and the same. That's why I was wondering about its fate. Sorry, his fate.
Fun fact, all of the Marx brothers loved go to Alabama to shoot elephants for their tusks! Why Alabama, you ask?
Cuz everyone know that in Alabama, the Tuscaloosa.
@@lightningmacqueen4097 Boom -tsssshhhh
If written as a fraction, and expand the brackets you get 60/35-25, which is also equal to 6.
I’m surprised that this method wasn’t mentioned.
It wasn't mentioned as over in America they seem to teach that multiplication by juxtaposition does not imply grouping so to them this is generally the taught method.
You also need a bracket there with what you wrote:
60/(35-25) and 60/35-25 are not the same answer when written on one line.
However, it seems that multiplication by juxtaposition, ab or a(b) etc., may impliy grouping though, or it may not, so the notation is ambiguous making both answers valid. It depends on context (academic or programming).
Modern international standards, ISO-80000-1, mention that brackets are required to remove ambiguity if you use division on one line with multiplication or division directly after it.
The American Mathematical Society's official spokesperson literally says "the way it's written, it's ambiguous" even though they use the explicit interpretation.
Wolfram Alpha's Solidus article mentions this ambiguity also.
Microsoft Math gives both answers.
Many calculators, even from the same manufacturer, don't agree on how to interpret multiplication by juxtaposition. No consensus.
Entry 242 in Florian Cajori's book "A History of Mathematical Notation (1928)" (page 274)
"If an arithmetic or algebraic term contains both ÷ and ×, there is at present no agreement as to which sign shall be used first..."
It then goes on to say that brackets should be used to "avoid ambiguity in such cases"
"The American Mathematical Monthly, Vol 24, No. 2 pp 93-95" mentions there was multiplication by juxtaposition ambiguity even in 1917 (and not the ÷ issue)
"Common Core Math For Parents For Dummies" p109-110 addresses this problem, states it is ambiguous and says, "shame on that person for writing an ambiguous expression".
"Twenty Years Before the Blackboard" (1998) p115 footnote says "note that implied multiplication is done before division".
"Research on technology and teaching and learning of Mathematics: Volume 2: Cases and Perspectives" (2008) p335 mentions about implicit and explicit multiplication and the different interpretations they cause.
Other credible sources are:
- The PEMDAS Paradox (a paper by a PhD student on this ambiguity)
- The Failure of PEMDAS (the writer has a PhD in maths)
- Harvard Math Ambiguity (Cajori's book above is talked about here)
- Berkeley Arithmetic Operations Ambiguity
- PopularMechanics Viral Ambiguity (AMS's statement is here)
- Slate Maths Ambiguity
- Education Week Maths Ambiguity
- The Math Doctors - Implicit Multiplication
- YSU Viral Question (Highly decorated maths professor says it's ambiguous)
- hmmdaily viral maths (Another maths professor says it's ambiguous)
The volume of evidence highly suggests it's ambiguous.
But now you must follow the order of operations. 60/35=1.7143 then subtract 25 giving -23.2857.
That is because they dont actually solve the equation, they input it into a calculator from left to right without any thought to how calculators operate. this gives them an answer of 24 which they then seek ways to justify the answer given by the calculator solving 60/5*1(7-5)=x instead of solving 60/5(7-5)=x.
That correct.
@@BabySuzunaTechnically, solving left to right would still equal 24. 60/5=12. So 12(7-5) is distributed as 12*7=84 minus 12*5=60. 84-60=24.
I had a question about the order. In your example you said apply PEDMAS to solve so (7-2) would be solved first. Once that is done wouldnt the problem look like this? 60/5(2)? Why do you ignore the parenthesis that is still there? I am super curious. My answer was 6 because I assumed you needed to solve all parenthesis first before moving on.
"Why do you ignore the parenthesis that is still there? "
In absence of another operator () simply count as *
You can go....
60/5(7-5)
60/5(2) and just remember the () are multiplication.
or you can change
60/5(2) to 60/5*2 if leaving them there bothers you.
Once everything INSIDE the () are done, they are no longer needed.
You're somewhat correct although not for the reason you cited.
In advanced math, any term where the operator is omitted is a singular term (e.g. 2x is singular, 5(x-y) is singular). In this case, 5(7-5) is singular and must be calculated first because of the parenthesis.
Now, had it been written as: 5 x (7-5), it would be two separate terms where only the (7-5) would be calculated first, and then everything else is calculated from left to right.
I got both answers because of the ambiguity of the expression. This is a superb video. The rationale of both answers is so well set. 24 might be the most popular answer according to calculators, but it is not the only correct answer, says I.
I'm just glad I'm not involved in mathematics anymore if it's now okay to just change the rules to get the answer you want.
I would say 24 is the correct answer that I had in my knowledge.
If it's not the only correct answer then it's not math.
If we don't have hard rules on how to evaluate equations that are able to guarantee that everyone will produce the same answer then we cannot engage in mathematics.
@@Cdaragorn Yes but no. The rules for the order of operations are so simple to follow and evaluate. If you don't follow and evaluate the rules of math, you'll end up having a wrong answer and you're making too hard for trying to get a correct answer. If you follow and evaluate the rules of math in a correct way, you will have a correct answer. I did the correct way of doing order of operations and a lot of people were having problems solving order of operations, which is my point. Except me because I took it serious evaluating order of operations and I had correct answers on my knowledge.
@@bryanburgos7452 In this case I don't think seeing lots of people struggle to apply the correct rules is enough evidence against requiring them. IMHO the problem here isn't that the rules are difficult at all to understand. It's that too many people haven't actually tried to learn them.
Most of them are going off of vague memories from 10,20, or more years ago. I honestly don't think that's good evidence that trusting them is a bad idea.
Unfortunately, the answer is wrong, because MindYourDecisions focused on a completely irrelevant aspect of the problem and wasted 8 minutes discussing the calculators and stuff, rather than the notation.
No one reasonable argues about the PEMDAS; and no one reasonable would say in 2018 that the regular division is higher in priority than the regular multiplication.
So, *if the expression was* 60÷5×(7−5), the result would clearly be 24.
Except it isn’t. This is a different expression, and there is no explicit multiplication sign here.
Such a notation is called “multiplication denoted by juxtaposition” (see en.wikipedia.org/wiki/Multiplication_denoted_by_juxtaposition), and it is rather used on next levels of math *after* when the PEMDAS is taught at school. In expressions like 7x÷5y.
And, if you try to find the information on the implied (by juxtaposition) multiplication, you’ll see that _it is known_ to cause ambiguity; but still, very often in academic literature, this juxtaposition-defined multiplication is treated as *having higher priority* over the regular multiplication and division.
So, the outcome is:
According to PEMDAS, 60÷5×(7−5) = 24.
And, according to the academic traditions past the PEMDAS, 60÷5(7−5) ≠ 60÷5×(7−5); 60÷5(7−5) = 6.
And the reason of that is *not* that the division operator for historical reasons could have the lower priority than PEMDAS defines; the reason *is* that the multiplication operator, *when defined by juxtaposition,* normally considered having the higher priority than PEMDAS defines.
Ikr? Finally someone who agreed with me about the expression. I just wrote a comment before, showing simple calculations of the two different expressions.
I see what you're saying and agree, except that I think the antiquated division symbol *also* matters here. Nevertheless the juxtaposition also matters and so like you stated makes how we interpret the division symbol becomes mostly irrelevant. However, were the juxtaposition not present, then we would have to decide upon the meaning of the division symbol.
Your "no one reasonable" and "clearly" statements are hyperbole or juxtapo-exaggeration. You are seriously citing Wikipedia as your credible source? Come on. man!
Except you forgot that multiplication by juxtaposition is never used in conventional mathematics and only finds occasional use in programming applications.
Try actually researching things instead of copy/pasting articles you don't understand.
@@Harmonic14 polynomials, its not just programming languages but even with excel and standard scientific calculators care must be taken to know where and when you need yo modify your inputs to get the device to output the correct answer.
Early on after scientific calculators became popular in doing this type of equation, math teachers told us not to use a calculator because it would give the wrong answer. When learning how to solve complex equations written in fraction form, the math teachers taught us to do the math above and below the line separately, then do the division. Engineers and physicists will use the old school method, which is called juxtaposition. This method accounts for the equation written in fraction form. The divided sign or a "/" use in the equation is just syntax. It replaces the horizontal line in fraction form. When written in one line using the arithmetic symbols and parentheses, some of these symbols are implied. So, when converting an equation from fraction to line form if the person writing the equation doesn't include a parentheses or bracket after the division symbol according the to PEDMAS, it changes the equation and the answer given. However, the rule for converting the equation from the signal line expression is to put everything left of the division symbols in the numerator and everything right of the division symbol in the numerator. This indicates that there is an implied bracket, or parentheses, in the equation. Which method really is correct? Having worked in the engineering field where my calculations had to have the correct answer to make what we were designing to work, I used the juxtaposition method and always got the correct answer. When using a calculator, I inserted the implied parentheses in the calculation. It is my opinion that in order of operations, multiplication should take presidence over division. I challenge a math teacher to prove which is the correct method to use on an ambiguous written equation.
I was on the math team in school and was taught that either side of the / was implied parenthesis and the ÷ was not used at all. That was in the 90s so my memory might be wrong now, but i think you are right.
I don't remember being taught that and would argue against it, because then you get to pretty iffy territory. That seems like a special, jargon-like usage convention: If everything is always of that form in some field, it makes sense to omit superfluous parentheses for readability, but it is problematic for general usage. Of course, a lot of it is *visual:* Are you using/imagining a large slash extending a character height above and below the rest of the expression with room around it or a small one packed tightly in one of multiple separated addition terms?
To me, however, it is obvious that you can't just break an expression at a point, where there is no operator to break at (that matches the implied operation) for the sake of binding a part of that grouping to some another operator (with the same or lower preference). If you do that, you are just willy-nilly chopping the term in half at a completely unmarked place. The purpose of notation isn't to mislead.
I've used examples of 3x/xy and xy/3x elsewhere.
The minute you start "implying" something that isn't there in math you're wrong.
We could certainly have decided that multiplication has some precedence over division, but that would require us to change how we write our equations.
That's the point of all of this. The rules have to be fixed in order to do math at all. In theory we could make order of operations anything we wanted to. What we choose dictates how we construct the equations though. And some ways make creating equations much more complicated than others.
The simple fact is there is nothing ambiguous about this equation. You just can't invent things that aren't from the established rules for how math is to be evaluated and then complain when you get a different answer than the one the writer of the equation wrote it to produce.
@@Cdaragorn huh? we were taught that multply goes before division and what ever number is outside of the brackets it getting multplied by the inside number then division comes. you cant just change math and thing youre going to get the right result.
@@keenanvanaalst9865 My entire comment was explaining why you can't just change math so you're right. The problem is multiply doesn't go before divide and it hasn't for more than 100 years. Multiply and divide are equal in the order of operations. You do them together.
I'm sorry if you were taught wrong. Seems like a lot of people were given that misconception.
When I enter this equation into the iOS calculator (in scientific mode, with parenthesis and ÷) I get 30 and I can’t find a reason for that!
When and why did the 1917 historical usage of PEMDAS/BODMAS change?
That relies on everyone using calculators that have been programmed the same way. I know when I was at school, in our second year of using graphing calculators the school specified a specific make and model purely down to different ones giving different answers to equations such as this. Sometimes different models made by the same company would have differing behaviour. As you pointed out, PEMDAS is relatively new to the scene. EDIT and as pointed out elsewhere, it used to be PEJMDAS. PEMDAS was how you taught young kids and as stuff got more complicated it switched to PEJMDAS. Seems some people never learnt the more advanced version.
Exactly. The calculators aren't sentient.
the only way I know to use safely a small calculator is using the reverse polish notation. My HP calculator never failed me.
Me: when will I use this in real life
Math teacher: 13 years later on TH-cam
BRILLIAN
All the math we have is not meant for real life. It exercises and trains your brain to make it sharp so that your brain works instantaneously and perfectly to find a solution to your real life problems and also to help in your decision making..!!!!
😂
LOL!
@@RajaBabu-oe4be This was a JOKE......relax.....
The fact it takes 9 minutes to describe a math problem, is a problem🤷🏼♂️
@@polarblue7468 No, I think the man with binoculars saw him...
@@blackcosmos No, but I stayed at a Holiday Inn Express last week and it took me like 3 seconds to complete in my head😂
No, it makes it cool.
I was doing this stuff in Grade 7. People are making this much harder than it actually is.
Quite literally elementary school math equation that people are arguing over?? This is disturbing... This should take one seconds to figure out that the answer is 24. Math really wasn't a lot of people's cup of tea.
To get the answer 24 ypu have to assume a set of parenthesis that is not included in the given equation. As the equation is written you would have to settle the () first that gives 2 then according to PEMDAS multiplication is next and that gives 10 as the denominator under 60. So the correct answer as the equation is written is 6.
PEMDAS based on order of operations, the M (multiplication) is before D (division), so if encountered with no parentheses, then you multiply before dividing. The order has always been based on the utilization and presence of P E M D A S not the written out equation. To justify dividing before multiplying insinuates that you would calculate exponents before parentheses.
I got 6, whenever I see a ➗ I automatically turn that in a fraction /. So I simplify the top and the bottom independently before finishing the division.
You failed to turn it into the correct fraction.
60
-------(7-5) ÷ 60÷5(7-5)= 24
5
60
---------- = 60÷(5(7-5))= 6
5(7-5)
@@RS-fg5mf that was exactly what I did! Thank you for explaining!
@@KrogTharr so you understand now that the correct answer is 24, right?
and that's why you never use ➗ after the 4 grade... except when you want to make a semi trap video. Math is supposed to be clear, not interpreted. If you would have seen the correct fraction you whould have given the correct answer. It's not a matter of age, they just made this purposefully confusing. You won't find an engineer use this kind of writing.
@@grigturcescu6190 what this demonstrates is that people can't follow a few simple rules and that they need to be hand held all the way to the correct answer...
When you actually understand and apply the Order of Operations and the various properties and axioms of math you get the ONLY correct answer 9
It doesn't help that on average 70% of adults incorrectly believe that 5+2×10=70....
You have people under educated who fail to understand the Order of Operations AND yoy have people who are over educated and try to make more out of a basic 4th grade arithmetic expression than it is...
I'm not even that old and I was thought by all my math teachers that you would solve the multiplication next to parenthesis first regardless of from left to right, so I came up with 6. Blame my math teachers.
So I went directly to comments -> and found Alpharex Rex! You are my kinda guy 🙋 Saved me from even Watching the video. Clearly we made it this far in life, paying bills, so there must be Alternative Math that also works 😉
Me also
I am right there with you. I remember doing parens by distributing the 5 to multiply it by the numbers in the parens, so 35-25=10, so 60/10 was 6.
Me too. I was taught that 5(2) was a single term and should be simplified. Changing math rules is self destructive.
I made it 6 too .
I got it, I get it, the ambiguity:
So, for the "left to right precedence" (in quotes), are we supposed to do all the Multiplying _before_ Dividing, and all the Adding _before_ Subtracting?
Or are we supposed to rank M & D as equal, but work from left to right, and rank A & S as equals, but work from left to right?
Is it: P>E>M>D>A>S, or is it: P>E>(M or D, & L to R)>(A or S, & L to R)?
THAT is the Question.
The Binary Trees explained it well enough, to expose the ambiguity.
I was taught the way to get to the answer 6. Since 5(7-5) has a distribution function, and Multiply is before Divide, so it (the Multiplying) must be solved first.
Most calculators with straight 60÷5(7-5) put in, turns out 24. But if 60÷(5(7-5)) is how we're taught to read the original equation (and put it in a calculator), it turns out 6 - because Algebra 2 and Pre Calculus.
Neither are wrong when input correctly - if, on a test, multiple choices happen to have both choices as answers, it goes to show either the student or the teacher doesn't realize the ambiguity hasn't been understood or just hasn't been explained well enough, respectively.
If you view 5(7-5) as 5 being factored out of (35-25) you end up with the second binary tree. This is what we were still taught in the 60's and 70s, the parenthetical surrounding 5(7-5) is inferred, and this is NOT the same as 5*(7-5) where the parenthetical is not inferred.
Truth ≠ wisdom. The answer is: whatever my wife says.
You are the wisest of us.
You have spoken wisely, David Lee.
Especially if she is wrong!
She's owed an apology then.
@@ferengiprofiteer9145 In the immortal words of Will Rogers, "There are two theories to arguing with a woman. Neither works."
@Joe Doe I hope you are joking man! And yes I just assumed your gender, but you have got to lighten up dude!
50 years old and still waiting to encounter a problem like this in real life.
Right.
I'm wondering what units would be used for an equation like this?
I'm 27 and I use this equation all of the time.
Whenever I get stopped for drunk driving, the officer always asks, "Miss, how many beers have you had?"
I always say, "60÷5(7-5)"
While he stands there tryna figure it out, I slip out of the cuffs and steal his patrol car.
Voila!
Watch more youtube and you will get more of these. ;)
I guess you never saw the the man with binoculars
It's an equation for time travel. You'll be able to go back in time and get those software programmers to fix this problem first. Then we don't have to waste our time on problems like these.
Pemdas doesn't actually work because division and subtraction don't actually work correctly left to right. Kind of. To make it work you need to change everything to multiplacation:
60* (0.2)(7-5)
This works whether or not you distribute into the parenthesis.
60*0.2*2=24
Or
60*(.4)=24
I think
You are correct.
I’m 53 years old and calculated 24 as the answer due to the way I was taught mathematics at school.
You calculated correctly.
For some reason these kids are wanting to do the multiplication on the right before the division on the left, madmen all of them.
It’s easy to see that if you take
60 / 5 (7-5) you start with the parenthesis
60 / 5 (2)
So you have 60 / 5 x 2
If you do math incorrectly and do the multiplication on the right first, you get a sum of 6, but anyone who passed 5th grade math knows you go from left to right
12 x 2 is the final product before solution
Same here Im 56.
Same here ans is 24. I'm 42 btw
I’m 58. I get 24.
I’m 55... I got 24
I was taught that the 5 before the parentheses would multiply what was inside so 7-5 is 2 times 5 equals 10 divided by 60 equals 6
That's what most people remember but what you forget is the TERM outside the parentheses is multiplied by the value of the parentheses not just the factor next to it.
TERMS are seperated by addition and subtraction not multiplication or division. 60÷5 is one TERM attached to and multiplied with the value of the parentheses 2... The correct answer is 24
60÷5(7-5)= 24
60÷(5(7-5))=6
60+5(7-5)= 60+5×2= 60+10=70
60-5(7-5)= 60-5×2= 60-10= 50
True. 6
@@stephenkinyanjui477 WRONG. The correct answer is 24 not 6
@@JJJJ-hp9oz there is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to group and give priority to operations INSIDE the parentheses and nothing more.
5(2) is not a parenthetical priority and is exactly the same as 5×2
You then demonstrate the Distributive Property incorrectly. The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses.
60÷5(7-5)=
60÷5*7-60÷5*5 parentheses eliminated
12*7-12*5=
84-60=
24
60÷(5(7-5))=
60÷(5*7-5*5) inner parentheses removed
60÷(35-25)=
60÷10=
6
60÷5(7-5) does NOT equal 60÷(35-25)
@@JJJJ-hp9oz LMAO... The Order of Operations were formally established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's... New Math is an excuse for people who fail to understand the basic rules of math... The correct answer is and always has been 24 not 6
You FAIL to understand what constitutes a TERM and you FAIL to understand that when written in an inline format only the number to the right of the obelus is in the denominator unless WITHIN a grouping symbol...
The way I was trained 40 years ago, is if the order was to divide into 60 first, then the 5(7-5) would have to be expressed 5 x (7-5), in order to require calculating from left to right. When you have 5(7-5), then you're supposed to multiply 5 x 2 first, with (7-5) being akin to an algebraic variable of multiplication no different than 5x or 5y, for example. It seems strange that they've changed a number things, that to me make no sense in the last 40 years.
You were trained wrong.
@@MrGreensweightHist ...Well, then I guess the entire Midwest got trained wrong, and thousands of engineers and mathematicians in university did too. They changed the process for this in the late 90's and after 2000, from what I remember.
@@Condor1970 "Well, then I guess the entire Midwest"
Egotistical of you to think everyone in the region was taught the same as you were.
"and thousands of engineers and mathematicians in university did too."
Many do, but I think you are overestimating how many people agree with you.
Not that it matters, a million wrong people are still wrong.
"hey changed the process for this in..."
No they did not.
The rules go back over 500 years, but some people are always teaching it wrong.
IT happens.
It sucks.
It's life.
Sorry you are one of the people that were taught wrong.
@@Condor1970Don't worry, you're not wrong.
That interpretation is still widely used today.
Academically, multiplication by juxtaposition implies grouping.
Literally/programming-wise, multiplication by juxtaposition implies only multiplication.
The person replying to you is wrong though.
They believe ridiculous things like parentheses are explicit multiplication operators (*, × and •, for example, are three explicit multiplication).
Easy to show too.
If you have two multiplication or division operators side by side, you get a syntax error.
Like 4××3 or 8÷÷2 or 8÷×2.
These are invalid expressions.
4×(3) and 4÷(3) or (4)×3 or (4)÷3 or even (4)×(3), are all valid syntax. So, how could brackets possibly be explicit multiplication operators then?
Their whole argument is based on that false belief, so they have no real argument at all. They are just biased and why they think millions of people are wrong (as well as the American Mathematical Society, the American Physical Society, Harvard, Berkeley, calculator manufacturers like CASIO, Sharp, HP and TI, and the following litany of professors and mathematicians:
Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc.).
If that guy was a competent maths teacher they would be teaching proper notation writing, like what modern international standards like ISO-80000-1 mentions on this case, and not promoting poor, ambiguous notation that only causes issues, and has zero benefits.
The solution is use one of these if forced to write on one line:
(a/b)(c+d)
or
a/(b(c+d))
and two line fractions in all other cases.
No arguments then when proper notation is used.