48÷2(9+3) = ?

About

48÷2(9+3) = ? is a math problem that, depending on the order of operations used, leads to two different answers: 2 and 288.^[1] An alternative version 6÷2(1+2)= ? with the answers 1 or 9, has popped up online as well. It can be a hot topic for debate, and is sometimes used to troll other users because of the argument that can result afterward.

Spread

This internet phenomenon exploded on April 7th, 2011, around the same time when searches for “48÷2(9+3) =”^[2] spiked on Google. The thread that first sparked interest in this problem was on Hot Pursuit^[3], a small local forum based in Texas. Shortly afterwards a member of the site posted the query on BodyBuilding.com and was spread onwards.^[4] Other forum posts from that day include Physics Forums^[5], Wall Street Oasis^[6], SpartanTailgate^[7], GrassCity^[7], Tennis Warehouse^[8], Inside MD Sports^[9], and \The Escapist. On April 8th, it popped up on 6Theory^[11], NIKETLK^[12], Yahoo! Answers^[13], DIYMA.com^[14], and The Ill Community^[15].

The Solution

Inputting the problem on different calculators can lead to different results depending on if the calculator is non-scientific or scientific, and how the calculator interprets order of operations.

There are considerable arguments for both answers, but the general consensus is that writing ambiguous fractions like “2/6x” makes solving such problems confusing, and it is considered bad form to write ambiguously written fractions in the first place.

Standard Order of Operations

If one strictly uses the standard order of operations to solve mathematical expressions, the answer to the problem would be 288, which is also the same solution provided by WolframAlpha^[17] and Google.^[18]

By convention, the order of precedence in a mathematical expression is as follows:

• Terms inside of Brackets or Parentheses.
• Exponents and Roots.
• Multiplication and Division.
• Addition and Subtraction.

If there are two or more operations with equal precedence – that are not exponents or roots (such as 10÷2÷5 or 7÷2*9) – those operations should be done from left to right. However, exponents and roots are generally evaluated from right to left (e.g. a^b^c = a^(b^c)).

Therefore, the problem “48÷2(9+3) =” would be solved like this:

PEMDAS

Solving for the answer 2 is sometimes a result of doing multiplication before division. Much of the confusion can be blamed on PEMDAS (sometimes known as, “Please Excuse My Dear Aunt Sally”) and other similar mnemonics used to teach order of operations in schools.

As an example, PEMDAS stands for:

• Parentheses
• Exponentiation
• Multiplication
• Division
• Addition
• Subtraction

Whereas BEDMAS stands for:

• Brackets
• Exponentiation
• Division
• Multiplication
• Addition
• Subtraction

The former can lead to the implication that addition always comes before subtraction, and that multiplication always comes before division. The latter can lead to the implication that addition always comes before subtraction, and that division always comes before multiplication.

If one uses multiplication before division (PEMDAS being especially popular in the United States), the problem would be solved like this:

However, solving the problem using this line of reasoning would be considered erroneous because multiplication and division hold equal precedence.^[19]

It is helpful to remember that division and multiplication are inverse operations, and thus represent the same operation written in a different way. Division is the same as multiplication of the reciprocal, and multiplication is the same as division of the reciprocal. This is similar to how addition is the same as subtraction of the negative, and how raising to the nth power is the same as taking the 1/nth root.

Distributive Property

One can justify the answer 2 by asserting that the distributive property shows that the answer is 2. The idea is that 2(9+3), by the distributive property, is equal to 2(9) + 2(3). So the evaluation would go like this:

Unfortunately, the answer is still ambiguous because it rests on the assumption that 2(9+3) should be evaluated before 48 ÷ 2. If we evaluate 48 ÷ 2 first, then we can use the same distributive property to get a different answer:

Part of the confusion may stem from the fact that the distributive property is usually presented as x(y+z) = xy+xz. It is true that wx(y+z) = w(xy+xz). However, it is not the case that (w/x)(y+z) = w/(xy+xz), and so one has to decide whether to evaluate (48/2) first or 2(9+3) first in the expression 48 ÷ 2(9+3), in order to get an unambiguous answer. Thus, the distributive property does not decide the correct answer.

Implied Multiplication Precedence

There does exist the argument that implied multiplication should be considered to have higher precedence than normal multiplication or division. For example, consider the problem "2/5x."

If one strictly follows the standard order of operations, the correct interpretation would be “(2/5)*(x).”

But many calculators and textbooks state that a higher value of precedence should be placed on implied multiplication than on explicit multiplication:

Because “5x” is implied to be "5*x," it gets higher priority than "2/5." In this case, "2/5x" would be interpreted as "(2)/(5*x)."

Returning to the original problem, if one utilizes the principles of implied multiplication, then “2(9+3)” gets higher precedence than the explicit “48/2,” and would be solved like this:

Proponents of higher precedence on implied multiplication would agree that 48 ÷ 2 * (9+3) = 288, but argue that 48 ÷ 2(9+3) = 2, which is how the problem was presented originally. However, there is a lack of consensus on the precedence of implied multiplication.

The Obelus (÷)

The obelus, the name for the symbol denoting “÷”, is argued by some^[20][21] to represent the division of all terms preceding it by all terms after it. The obelus supposedly separates the two components of the fraction, with the top dot representing the numerator and the bottom dot representing the denominator. There is some support that the obelus did mean this,^[22] but whether people still mean this today, or use it interchangeably with “/” today is unclear.

ISO 80000-2-9.6^[23] states that the obelus should not be used.

Using this interpretation of the obelus, we get:

6÷2(1+2)= ?

On April 27th, 2011, a Redditor posted a slightly different version to the “wtf” subreddit. As of April 29th, 2011, it has received 1281 comments, and has a karma score of 776. A Redditor posted the following comment to the thread which was the highest rated comment at 574 upvotes:

I’m a math professor, and my view is that although the standard convention, if applied precisely and rigorously, does give an unambiguous procedure to follow, nobody, and that includes professional mathematicians, would ever write a formula like this. This is mostly because, after about 3rd grade, none of us ever use the division symbol ever again.^[16]

This version of the problem quickly spread to facebook. ^[17]

External References

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