What is 60 5 7 5

I'm honestly about to freaking lose it. AAAAAAH I've been talking to people on this youtube video for a whole week now trying to convince people that their math is just wrong. I commented about two years ago stating the answer was 24, simply put at that, and that no complex explanation was necessary, and forgot about the video and moved on with my life for two years. Now all of a sudden the video has suddenly gotten spikes in views and I'm getting bombarded with people telling me the answer is 6. It always boils down to a few things such as... They turn a problem that was 60/5(7-5) into something like 60/((5(7-5)) without realizing.

They distribute the 5 to (7-5) and tell me "See it's 6 now!" but for some reason whenever I tell them 60/5 is the whole term, not just the 5 they ignore me and just spout some random hypothetical paragraph longer than this very post talking about parties and cake and tests and people getting sick or cake being lost or something. without even doing any math. Like why are people like this? I've been trying to tell people and even symbolab states it's 24.

so, since you guys clearly know a lot of math, what do you guys think of this problem? Is 60/5(7-5) really *ambiguous*? Because personally I don't think so. I think if you just follow the flow of math it's pretty obvious. Two years ago I heard people trying to spin this problem to something like, "both 6 and 24 are right because it's ambiguous, so everybody can be right!" which is just silly, especially for math. I just don't get so many people are like this. Is math really that esoteric that even simple problems like these involve somewhat esoteric knowledge such as what terms are and how to accurately multiply and distribute? If I'm wrong though, so be it but I am confident it is 24

2 Answers By Expert Tutors

Kenneth S. answered • 08/26/16

Expert Help in Algebra/Trig/(Pre)calculus to Guarantee Success in 2018

First, 60/5 = 12.  THEN multiply by two; answer = 24.

Neal D. answered • 08/27/16

Education Made Understandable

60 ÷ 5 ( 7 - 5 )

60 ÷ 5 (2)

12 (2)

24

Robert H.

If you use pemdas correctly the equation would look like this 

60÷5(7-5)

60÷5(2)

60÷10

6

Neal D.

I always thought that it was do All Multiplication and/or Division starting

on the left side and go to the right.

Believe me, I looked at this problem several times both ways, knew that 

the answer would be different depending on the way I did it.

Multiplication and/or Division, left to right

You may have seen some of my videos on this topic, such as:

What is 6÷2(1+2) = ? The Correct Answer Explained (over 9 million views)

9 – 3 ÷ 1/3 + 1 = ? The Correct Answer (Viral Problem In Japan) (over 7 million views)

There’s another problem that’s going viral right now, so it’s time for the order of operations to save the day!

What is the correct answer to the following expression?

60÷5(7 – 5) =

Watch the video where I explain the correct answer. (I do two main new things in this video: I illustrate the answer with binary expression trees and I relate that to how Google evaluates the expression.)

What is 60÷5(7 – 5) = ? The Correct Answer Explained

Or keep reading.
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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

(I copy/pasted most of the text from my previous order of operations post, so hopefully I updated all the numbers! But if not, please let me know if there are any typos/errors and I will correct them, thanks).

The correct answer is 24 according to the modern interpretation of the order of operations.

The order of operations

The expression can be simplified by the order of operations, often remembered by the acronyms PEMDAS/BODMAS.

First evaluate Parentheses/Brackets, then evaluate Exponents/Orders, then evaluate Multiplication-Division, and finally evaluate Addition-Subtraction.

Everyone is in agreement about the first step: simplify the addition inside the parentheses.

60÷5(7 – 5)
= 60÷5(2)

This is where the debate starts.

The answer is 24

If you type 60÷5(2) into a calculator, the input has to be parsed and then computed. Most calculators will convert the parentheses into an implied multiplication, so we get

60÷5(2)
= 60÷5×2

According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right. First take 60 and divide it by 5, and then multiply by 2.

60÷5×2
= 12×2
= 24

This gets to the correct answer of 24.

This is without argument the correct answer of how to evaluate this expression according to current usage.

Some people have a different interpretation. And while it’s not the correct answer today, it would have been regarded as the correct answer 100 years ago. Some people may have learned this other interpretation more recently too, but this is not the way calculators would evaluate the expression today.

The other result of 6

Suppose it was 1917 and you saw 60÷5(2) in a textbook. What would you think the author was trying to write?

Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below).

Under that interpretation:

60÷5(2)
= 60÷(5(2))
(Important: this is outdated usage!)

From this stage, the rest of the calculation works by the order of operations. First we evaluate the multiplication inside the parentheses. So we multiply 5 by 2 to get 10. And then we divide 60 by 10.

60÷(5(2))
= 60÷10
= 6

This gives the result of 6. This is not the correct answer that calculators will evaluate; rather it is what someone might have interpreted the expression according to older usage.

Binary expression trees

Since some people think the answer is 24, and others think it is 6, many people argue this problem is ambiguous: it is a poorly written expression with no single correct answer.

But here’s my counter-point: a calculator is not going to say “it’s an ambiguous expression.” Just as courts rule about ambiguous legal sentences, calculators evaluate seemingly ambiguous numerical expressions. So if we take the expression as written, what would a calculator evaluate it as?

There are two possible binary expression trees.

I suggested the binary expression tree on the left is consistent with PEMDAS/BODMAS. But what does a calculator actually do?

If you try Google (see it evaluate 60÷5(7-5)) you’ll get an answer of 24. Furthermore, the Google output even inserts parentheses to indicate it is using the binary tree on the left of (60/5)*(7 – 5).

The Android calculator app also gives an answer of 24.

So I would argue 24 is correct by the order of operations, and it is what calculators also evaluate the expression (and notice that scientific calculators are programmed to evaluate according to the order of operations).

(I wouldn’t be surprised, however, if a calculator gave an answer of 6. There might be some that would evaluate like the binary tree on the right hand side. But from what I’ve seen, many calculators give an answer of 24.)

The symbol ÷ historical use

Textbooks often used ÷ to denote the divisor was the whole expression to the right of the symbol. For example, a textbook would have written:

9a2÷3a
= 3a

This indicates that the divisor is the entire product on the right of the symbol. In other words, the problem is evaluated:

9a2÷3a
= 9a2÷(3a)
(Important: this is outdated usage!)

I suspect the custom was out of practical considerations. The in-line expression would have been easier to typeset, and it takes up less space compared to writing a fraction as a numerator over a denominator:

The in-line expression also omits the parentheses of the divisor. This is like how trigonometry books commonly write sin 2θ to mean sin (2θ) because the argument of the function is understood, and writing parentheses every time would be cumbersome.

However, that practice of the division symbol was confusing, and it went against the order of operations. It was something of a well-accepted exception to the rule.

Today this practice is discouraged, and I have never seen a mathematician write an ambiguous expression using the division symbol. Textbooks always have proper parentheses, or they explain what is to be divided. Because mathematical typesetting is much easier today, we almost never see ÷ as a symbol, and instead fractions are written with the numerator vertically above the denominator.

*Note: I get many, many emails arguing with me about these order of operations problems, and most of the time people have misunderstood my point, not read the post fully, or not read the sources. If you send an email on this problem, I may not have time to reply.

Sources

1. Some examples of the debate

Twitter BrookeOnAir
https://twitter.com/BrookeOnAir/status/1037059377591549952

Quora
https://www.quora.com/For-the-equation-60%C3%B75-7-5-I-got-6-and-12-Which-is-the-more-acceptable-answer-and-why

2. Web archive of Matthew Compher’s Arguing Semantics: the obelus, or division symbol: ÷

3. In 2013, Slate explained this problem and provided a bit about the history of the division symbol.

http://www.slate.com/articles/health_and_science/science/2013/03/facebook_math_problem_why_pemdas_doesn_t_always_give_a_clear_answer.html

4. The historical usage of ÷ is documented the following journal article from 1917. Notice the author points out this was an “exception” to the order of operations which did cause confusion. With modern typesetting we can avoid confusing expressions altogether.

Lennes, N. J. “Discussions: Relating to the Order of Operations in Algebra.” The American Mathematical Monthly 24.2 (1917): 93-95. Web. http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents

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