48÷2(9+3) = ? is a math problem that, depending on the order of operations used, leads to two different answers: 2 and 288. 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) =” spiked on Google. Forum threads created that day include BodyBuilding.com, Physics Forums WallStreetOasis.com, SpartanTailgate, grasscity.com, Tennis Warehouse, Inside MD Sports, and The Escapist. On April 8th, it popped up on 6Theory, NIKETLK, Yahoo! Answers, DIYMA.com, and The Ill Community.
The Solution
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 and Google.
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 (such as 10÷2÷5 or 7÷2*9), those operations should be done from left to right.
Therefore, the problem “48÷2(9+3) =” would be solved like this:
48 ÷ 2 * (9+3)=
48 ÷ 2 * (12)=
48 ÷ 2 * 12=
24 * 12=
288
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:
48 ÷ 2 * (9+3)=
48 ÷ 2 * (12)=
48 ÷ 2 * 12=
48 ÷ 24=
2
However, solving the problem like this would be considered erroneous because multiplication and division hold equal precedence.
Some sources maintain that multiplication does not always comes before division:
- University of North Texas
- Northern Michigan University
- Deb Russell
- University of Minnesota Rochester
- Midland College
- Hofstra University
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.
Implied Multiplication
However, the answer 2 could be justified by the principle of implied multiplication. 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:
48 ÷ 2(9+3)=
48 ÷ 2(12)=
48 ÷ 24=
2
However, there is a lack of consensus on the value of implied multiplication.
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