This mathematics problem has been going around the internet recently and most of the answers I have seen for it published in comments, blogs and videos are actually incorrect. While the equation is stated in a deliberately ambiguous way, it can only have one correct answer and that is not the answer that most of the articles and videos provide.
Again, poor Albert Einstein is used to indicate genius, except he was not very good at school. The equation being asked is:
- 48 ÷ 8(14 – 8) =
There are two answers that are being given for this equation, but only one is correct and it is not the more common solution.
Wrong Answer
Here is the wrong answer, even though it correctly did what is inside the brackets/parentheses first:
- 48 ÷ 8(14 – 8) =
- 48 ÷ 8(6) =
- 48 ÷ 8 x 6 =
- 6 x 6 =
- 36
You might ask why this is wrong?
The reason is that 8(6) is NOT THE SAME as 8 x 6 even both are equal to 48.
If the challenge question asked was stated as 48 ÷ 8 x (14 – 8) = ?, I would happily agree that the answer is 36, but that is not the question as there is no multiplication symbol before the brackets/parentheses.
Correct Answer – Part 1
Let’s state the question again, but this time try to understand it better before solving. 48 is divided by the rest of the equation being 8(14 – 8).
- 48 ÷ 8(14 – 8) or 48 / 8(14 – 8) =
- 48
———— =
8(14 – 8) - 48
—— =
8(6) - 48
—- =
48 - 1
The correct answer is 1. This is because 8(14 – 8) is a single term and all of it becomes the denominator in the division.
Correct Answer – Part 2
Based on the Order of Operations mnemonic (BODMAS for UK or PEMDAS for US). When talking about solving the Brackets/Parentheses it means not just solving what is inside the brackets but also any factors directly attached to those brackets/parentheses. Without a multiplication symbol in the original question, the factor outside the brackets must be calculated as part of the Brackets/Parentheses stage and not as part of the later Multiplication/Division stage (using left to right). This “Implicit Multiplication” (aka Multiplication by Juxtaposition) has higher precedence than normal Multiplication.
- 48 ÷ 8(14 – 8) =
- 48 ÷ 8(6) =
- 48 ÷ 48 =
- 1
Again, the correct answer is 1. The 8 is directly attached to the (14 – 8) and must be calculated as part of that term in the equation.
Correct Answer – Part 3
Let’s try another approach. This time expanding 8(14 – 8) out using the Distributive Law. This simplifies the equation by moving the factor back inside the brackets/parentheses.
- 48 ÷ 8(14 – 8) =
- 48 ÷ (112 – 64) =
- 48 ÷ 48 =
- 1
Again, the correct answer is 1.
Note: A way to confirm if you have correctly expanded a term would be to factorise it again. Find the multiples of the numbers.
Factors of 112 = 1 2 4 7 8 16 28 56 112
Factors of 64 = 1 2 4 8 16 32 64
Finding the Highest Common Multiple (HCM) of 112 and 64 is actually 16. Therefore, we could factorise (112 – 64) to either 16(7 – 4) or 8(14 – 8).
Correct Answer – Part 4
How about using some algebra to solve the equation. Yes, it makes it a little more complicated, but it helps ensure the correct answer. Let’s assume that y = 8, so 48 = 6y.
- 48 ÷ 8(14 – 8) =
- 6y ÷ y(14 – 8) =
- 6y ÷ y(6) =
- 6y ÷ 6y =
- 1
Correct Answer – Part 5
Using the same y = 8 algebra, but this time expanding the factor with the distributive law first.
- 48 ÷ 8(14 – 8) =
- 6y ÷ y(14 – 8) =
- 6y ÷ (14y – 8y) =
- 6y ÷ 6y =
- 1
Correct Answer – Part 6
Another way to understand equations used in mathematics is to use language to understand what they are trying to solve.
For example: I have 48 apples, and I have 8 lines of people waiting. Each line is usually 14 people, but 8 people from each line were taken away to be given oranges instead. How many apples can each person have?
You guessed it, the answer is not 36 apples for each person unless you can perform miracles.
Conclusion
Mathematics is an exact science, equations always have the same solutions, but when the equation could be ambiguous, it should use brackets/parentheses or language to clarify. However, you cannot just add multiplication symbols or your own brackets/parentheses to an equation as that will change the solution.
For example:
- 48 ÷ 8(14 – 8) is not the same as 48 ÷ 8 x (14 – 8)
The first has a solution of 1 and the second has a solution of 36.
Stating the question as 48 ÷ (8(14 – 8)) would have been less ambiguous.
More Information
- BBC – Algebraic skills – Expanding Brackets – Distributive Law
- BBC – Algebraic skills – Adding Brackets – Factorising
- The PEMDAS Paradox
- Order of Operations: Implicit Multiplication?
Extra Conclusion
[EDIT] Watch the videos below to see why PEMDAS is wrong and that it should be PEJMDAS as Multiplication by Juxtaposition or Implicit Multiplication has precedence over explicit multiplication.
This also appears to be regional as North American teachers seem be ignoring Multiplication by Juxtaposition (watch the second video below for more information).
Not sure if primary school teachers should be overruling centuries of mathematics used by mathematicians, engineers and scientists just because it is easier to teach.
Videos
Interesting Videos on why PEMDAS is incomplete, and it should be PEJMDAS:
PEMDAS is wrong (direct link)
The Problem with PEMDAS: Why Calculators Disagree (Direct Link)
Also, have a look at this series of articles solving another maths question asked on the internet:
- Trending: Only for genius ?? 3 – 3 x 6 + 2 = ?? With Poll and Order of Operations
- Answer: Only for genius ?? 3 – 3 x 6 + 2 = ?? The answer with lots of explanations
- Conclusion: Only for genius ?? 3 – 3 x 6 + 2 = ?? Breaks down solution into steps
- The Last Word: Only for genius ?? 3 – 3 x 6 + 2 = ?? Explains ordering of DM and AS
- Viral: Only for genius ?? 3 – 3 x 6 + 2 = ?? Answer has gone viral and explanations
Enjoy
David
This article was originally posted on http://www.winthropdc.com/blog.