What is the correct answer to 50÷5×2 + 10 = ? – Mind Your Decisions
It seems like every year some equation divides the internet with people arguing about the correct answer.
What is 6÷2(1+2) = ? The Correct Answer Explained (22 million views)
9 – 3 ÷ (1/3) + 1 = ? The Correct Answer (Viral Problem In Japan) (12 million views)
What is 8÷2(2 + 2) = ? The Correct Answer Explained (3 million views)
Since there is another problem that’s going viral right now, it’s time for the order of operations to save the day!
What is the correct answer to the following expression?
50÷5×2 + 10 = ?
As usual, watch the video for a solution.
Or keep reading.
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Answer To 50÷5×2 + 10 = ?
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
The correct answer is 30 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. If two operations of the same precedence appear, evaluate from left to right.
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 50 and divide it by 5, and then multiply by 2. Finally add 10
50÷5×2 + 10
= (50÷5)×2 + 10
= 10×2 + 10
= 20 + 10
= 30
This is without a doubt the correct answer according to the modern interpretation of the order of operations.
But some people may have learned it a different way.
The other result of 15
Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below). In other words you need to do the multiplications first and then the division.
Under that interpretation:
50÷5×2 + 10
= 50÷(5×2) + 10
(Important: this is outdated usage!)
= 50÷10 + 10
= 5 + 10
= 15
This gives the result of 15. This is not the correct answer that calculators will evaluate; rather it is what someone might have interpreted the expression according to older usage.
Here’s a textbook from 1969 sent to me by Pete P.
1969 Textbook Mathematics for Management and Finance, with Basic and Modern Algebra 2nd Ed., Stephen P. Shao, Ph. D. Professor of Management and Statistics, School of Business Administration, Old Dominion University.
So if some people learned it one way, and other people learned it another way, is there really any saying what the correct answer is?
Binary expression trees
For this expression, there are two possible binary expression trees.
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 50÷5×2 + 10) you’ll get an answer of 30. Furthermore, the Google output even inserts parentheses to indicate it is using the binary tree on the left of (50÷5)×2.
Most popular calculators evaluate the expression the same way, and I would argue that is NOT a coincidence, but rather a reflection that calculators are programmed to the same PEMDAS/BODMAS rules we learn in school.
In computer programming, it would be written as 50/5*2. I found the answer to this expression was 16 in Python, C#, Java, Javascript, and Excel. Even if you think the answer is 1, you should know that computers–which basically run our lives–will evaluate this as 16. If you want to be a programmer and use computers you should at least understand how they will evaluate the expression.
Isn’t the answer ambiguous?
Some mathematicians believe the expression is incorrectly written, and therefore can have multiple interpretations. I strongly disagree with this point. The main cause of confusion is the order of operations!
For example, consider the problem 9 – 3 ÷ (1/3) + 1 (over 9 million views). This is an unambiguous expression and has only a single answer. But the problem went viral in Japan after a study found 60 percent of 20 somethings could get the correct answer, down from a rate of 90 percent in the 1980s. It is clear the problem is students do not learn the order of operations.
Mathematicians who say “the answer is ambiguous” overlook that students get unambiguous expressions wrong at an alarming rate. It is our duty as mathematicians to emphasize the order of operations in its modern form so that we can write proper expressions and interpret them correctly. Not a single person who disagrees with me has considered why students get the wrong answer to 9 – 3 ÷ (1/3) + 1.
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
DailyMail
https://www.dailymail.co.uk/news/article-15137377/how-solve-math-problem-pemdas-divides-internet.html
1. Google evaluation
https://www.google.com/search?q=50%C3%B75%C3%972+%2B+10
2. WolframAlpha evaluation
https://www.wolframalpha.com/input?i=50%C3%B75%C3%972+%2B+10
3. Web archive of Matthew Compher’s Arguing Semantics: the obelus, or division symbol: ÷
4. 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
5. 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
6. 1969 Textbook Mathematics for Management and Finance, with Basic and Modern Algebra 2nd Ed., Stephen P. Shao, Ph. D. Professor of Management and Statistics, School of Business Administration, Old Dominion University.
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