8 : 2 ( 2 + 2 ) = ?

If the 8 is a question number:

I would solve it like this:

2(2+2)

2(4)

2*4

8

ANSWER:8

If the 8 is part of a ratio/proportion thingy:

8/(2(2+2))

8/(2(4))

8/8

1

ANSWER:1

I can't access that site but

8:8

is

1:1

This was in the news yesterday-ish

(you have a typo in your question)

8/2(2+2) = ?      (edited...I had a typo in MY answer too!   I had posted :  8*2(2+2) = ? )

PEDMAS   says it's 1

BODMAS  says it's 16

See:

https://www.foxnews.com/tech/viral-math-problem-baffles-many-internet

Thanks EP for that link, but I'm pretty sure the information in that article is definitely incorrect.

The question is:

\(8\div2(2+2)\ =\ ?\)

And the ONLY correct answer is:

\(8\div2(2+2)\ =\ 8\div2(4)\ =\ \dfrac82(4)\ =\ 4(4)\ =\ 16\)

See here that WolframAlpha says the answer is 16:

https://www.wolframalpha.com/input/?i=8%C3%B72(2%2B2)

There is unfortunately much confusion about this question, but I do not think it is ambiguous.

PEMDAS and BODMAS mean the same thing, and they both say it's 16.

Parenthesses and Brackets are the same thing. Exponents and Orders are the same thing.

Multiplication and Division are on the same level and so can be written either as MD or DM.

We are also taught that when two operations are on the same level, we do them "left to right".

See this Khan Academy video:

https://www.khanacademy.org/math/pre-algebra/. . ./v/introduction-to-order-of-operations

Notice that he writes Mult/Div on one line and Add/Sub on one line.

And especially watch the example that starts around the time 4:10

The left-to-right rule does work for evaluating these expressions. However, in the case of division, instead of thinking that operations on the same level are done "left to right," I prefer to think that only the "item" immediately following the division symbol goes in the denominator, where an "item" is either a number or an expression enclosed in parenthesees.

The expression   \(1\div2+3+4\)   is equal to   \(\frac12+3+4\)   because we place

only the item immediately following the division symbol,  2 ,  in the denominator.

The expression   \(1\div(2+3)+4\)   is equal to   \(\frac{1}{(2+3)}+4\)   because we place

only the item immediately following the division symbol,  (2 + 3) ,  in the denominator.

Following this rule also means the expression   \(12/6/3\)   is equal to   \(\dfrac{\frac{12}{6}}{3}\)   which is  \(\dfrac23\)

The alternative rule would be that everything after the division symbol goes the denominator.

I personally do not like to say we must follow "left to right" because

for expressions like this:  1 + 2 + 3 + 4 + 5

we don't have to follow left to right. We could evaluate it like this:

1 + 2 + 3 + 4 + 5   =   1 + 2 + 3 + 9   =   1 + 5 + 9   =   10 + 5   =   15

It doesn't hurt to follow left-to-right to evaluate that expression; it just isn't necessary.

Another reason I prefer the rule "only use the item immediatley adjacent to the operator as the operand" is that we can use a similar rule for exponents. In the case of exponents, the rule would be "only the item immediately preceding the caret is the base."

4^2^3

Should  4^2^3  be  4^(2^3)  or  should it be  (4^2)^3   ?

If we just stick with the left-to-right rule, we would get

4^2^3  =  16^3  =  4096

If we use the rule "only the item immediately preceding the caret is the base," we would get

4^2^3  =  4^8  =  65536

Here, WolframAlpha says the answer is  65536:

https://www.wolframalpha.com/input/?i=4%5E2%5E3

The only operations that need such a rule are division, exponentiation, and subtraction.

It is unnecessary to follow left-to-right for multiplication and addition.

They are great teaching post Hectictar :)

This debate seems perpetual. The issue returns at least once every school year.

https://web2.0calc.com/questions/help-with-pemdas

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Another reason I prefer the rule "only use the item immediatley adjacent to the operator as the operand" is that we can use a similar rule for exponents. In the case of exponents, the rule would be "only the item immediately preceding the caret is the base."

4^2^3 

Should  4^2^3  be  4^(2^3)  or  should it be  (4^2)^3   ? 

If we just stick with the left-to-right rule, we would get

4^2^3  =  16^3  =  4096

Note that the web2.0calc calculator resolves this from the left to right or ascending order, where the resultant product becomes the BASE of the next exponent.  This operation is easy to see in the display above the calculator.  This is the only exception to the hierarchical order of operations. The reason for this is probably because the standard product of exponents to a base is well more common in physics and engineering than conventional stacked powers.   

If we use the rule "only the item immediately preceding the caret is the base," we would get

4^2^3  =  4^8  =  65536

Here, WolframAlpha says the answer is  65536:

Wolfram follows the (official) stacked power convention, where stacked powers are (exponentially) multiplied from the right to left (from the top down), and the resultant product becomes the EXPONENT to the base number. Generally, stacked power and power-towers are used in advanced, theoretical mathematics.

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According to the answer there,  a/bc  means  a/(bc)  which goes against my idea.

Hmm, I don't like that! But I guess it is the case. Though I still think  1/4(3)  =  3/4

For a/bc, the convention will equal a/(bc) 

Implicit multiplication of variables takes precedents over division – a noted exception, dating back to the late 1960s, to the normal convention of mathematical hierarchy. Herr Massow’s calculator is the only one I know of that that does this, and it’s probably because it allows the use of variables. (Comments from LancelotLink –2015)

Pasting a=20;b=2;c=5; a/(b)(c) in the calculator will result in (2).

Pasting 20/2*5 in the calculator will result in (50)

Note that pasting 20/(2)(5) will result in (2),  but pasting this 20/(2)*(5) will result in (50). The parenthesis without an operator between them triggers the variable precedents.

Note also, that the convention for precedents of variables multiplication applies only to the first two variables. 

Posts related to hierarchy-order of operations:

https://web2.0calc.com/questions/help-with-pemdas

https://web2.0calc.com/questions/4-2-x-2-2-x-56#r8

GA