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?
- Wikipedia: Order of Operations: Mixed Division and Multiplication
- Wikipedia: Multiplication: Notation
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.
As you said, the answer can only be 1. Even though I finished my maths degree nearly 40 years ago, BODMAS/BOMDAS is still the correct calculation method for any of these equations.
I have added some videos that explain that BODMAS and PEMDAS does not take into account Multiplication by Juxtaposition.
But BODMAS/PEMDAS both apply to 48+8(14-8) or 48-8(14-8) where there is multiplication by juxtaposition.
My take is that the 8 is a divisor and should be treated as such.
48(14/8-8/8)=
48(1.75-1)=
48(.75)=36 or just
48(6/8)=36
Sadly, you expanded the denominator incorrectly, to put the 8 inside the brackets means you need to multiply not divide the values inside the brackets.
48 ÷ 8(14 – 8) = 48 ÷ (14×8 – 8×8) = 48 ÷ (112 – 64) = 48 ÷ (48) = 48 ÷ 48 = 1
This is because the denominator is 8(14 – 8) = 8(6) = 48 or 8(14 – 8) = (112 – 64) = 48. The answer is always the same.
For the apple problem, the expression should be
48➗[8(14-8)].
48 students show up for a test. The students are divided into 8 equal groups and sent to 8 different classrooms with a teacher in each classroom. There are 14 problems presented. The students can eliminate 8 problems of their choice and do the remaining problems. How many total problems must each teacher grade?
How many proofs do you want?
1. Order of Operations
48÷8(14-8)
48÷8(6)
48÷8×6
6×6
36
2. The multiplicative by reciprocal rule
48÷8(14-8)
48x⅛(14-8)
48x⅛(6)
48x⅛x6
6×6
36
3. Commutative property
48÷8(14-8)
(14-8)48÷8
(6)48÷8
288÷8
36
4. Distributive property
48÷8(14-8)
48÷8×14-48÷8×8
6×14-6×8
84-48
36
the posted expression is
48
—(14-8)= 48÷8(14-8) = 36
8
the posted expression is NOT
48
—— = 48÷ [8(14-8)] = 1
8(14-8)
By adding a vinculum you’ve added a second set of grouping symbols and changed the expression completely
In your proofs you have treated multiplication by juxtaposition as just normal multiplication, this is wrong by the mathematics taught everywhere in the world except USA. The issue is that 8(6) is not the same as 8×6. 8(6) must be calculated at the brackets/parenthesis step of order of operations.
Why don’t you expand 8(14-8) out by multiplying all the terms inside the brackets by 8, so 8(14-8) = (112-64) = 48.
So
Proof 1: 48÷8(6) is not the same as 48÷8×6, it should be 48÷(8×6)
Proof 2: 48x⅛(6) is not the same as 48x⅛x6, it should be 48x(1/(8×6))
Proof 3: 48÷8(14-8) is not the same as (14-8)48÷8 as you have broken up the single term 8(14-8)
Proof 4: 48÷8(14-8) is not the same as 48÷8×14-48÷8×8, it should be 48÷(8×14-8×8)
Yes, 48÷8(14-8) IS THE SAME as 48÷[8(14-8)], that is the whole idea of multiplication by juxtaposition or implied multiplication. Which is sadly not taught correctly in the USA.
Try 48÷8(14-8), let’s make x = 14-8, so 48÷8(x) = 48÷8x, x = 6 so 8x = 48. There for equation is 48÷48 = 1
You don’t need to specify extra brackets around 8x, we all know that 8x is the same as 8(x) or (8 * x), you cannot just separate the 8 from the x.
If you keep making fundamental mistakes you can easily prove that 0 = 1, or black = white and get run over at the next Zebra/Pedestrian crossing.
PS: If the question was 48÷8x(14-8) then the answer would be 36. But 48÷8(14-8) is not the same as 48÷8x(14-8).
Sorry no. Multiplication by juxtaposition is an algebraic concept and does not apply to elementary arithmetic. 48÷8(6) is mathematically identical to 48÷8×6. The presence or lack of an explicit multiplication sign does not effect the solution in any way.
You are confusing and conflating two different types of Implicit multiplication …. One without a delimiter and one with a delimiter..
Type 1… Implicit Multiplication between a coefficient and variable… A special relationship given to coefficients and variables that are directly prefixed i.e. juxtaposed WITHOUT a delimiter and forms a composite quantity by Algebraic Convention… Example 2y or BC This type of Implicit Multiplication is given priority over Division and most other operations but not all other operations… This can be seen in most Algebra text books or Physics book. Physics uses this type of Implicit Multiplication quite heavily..
Type 2… Implicit Multiplication between a TERM and a Parenthetical value that have been juxtaposed without an explicit operator but WITH a delimiter…The parentheses serve to delimit the two sub-expressions.. Parenthetical implicit multiplication. The act of placing a constant, variable or TERM next to parentheses without a physical operator. The multiplication SYMBOL is implicit, implied though not plainly expressed, meaning you multiply the constant, variable or TERM with the value of the parentheses or across each TERM within the parenthetical sub-expression. Parentheses group and give priority to operations WITHIN the symbol of INCLUSION not outside the symbol.
Watch the videos on the blog post. It explains the history. US teachers changed the rules. US websites/videos show the answer that US teachers decided to teach because they simplified maths. Everywhere else in the world does not differentiate by between simple maths and algebraic maths. Multiplication by juxtaposition existed before the PEMDAS order of operations mnemonic.
see the second video at 9:00 https://youtu.be/4x-BcYCiKCk?si=M7edvG3D6m5J9ImB&t=539
You will never persuade me to change a mathematical fundamental principle just because the US teachers simplified it.
These internet questions are designed to cause confusion with equations that they know can be misinterpreted. If you were to describe the problem with words first, I would hope you will get the correct answer regardless.
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?
Solve this:
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?
Answer should be that each of the remaining 6 people in each of the 8 lines will get 1 apple.
If you give each person 36 apples from the 48 apples you started with, then you have performed a miracle. Congratulations, you have now given out 1728 apples meaning you created 1680 apples from thin air.
48÷8(14-8) ≠ 48÷[8(14-8)]
Try this. Go to the Symbolab Solver (https://www.symbolab.com/solver) and click on the “Enter a problem” section and type in the characters:
48/8(14-8)
Then click go. It will show the answer as 1.
Symbolab says it’s 36
You typed it differently. but again it is incorrectly treating Multiplication by Juxtaposition as the same as standard Multiplication.
There is no difference between simple maths and algebraic maths. Implied Multiplication is higher priority.
I am sorry that the US teachers simplified the rules for you and that US based internet resources follow that same simplification.
Please can you stop creating videos and posting them on my comments now.
From Wikipedia: https://en.wikipedia.org/wiki/Order_of_operations
Multiplication denoted by juxtaposition (also known as implied multiplication) creates a visual unit and is often given higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n
….
Mathematics education researcher Hung-Hsi Wu points out that “one never gets a computation of this type in real life”, and calls such contrived examples “a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules”.
I entered it exactly as written. But if you like I can use the slash vs the obelus. The solution is the same. .
You clicked with the mouse after entering the 8 to change the position and move the position of the final part of the equation, this breaking the juxtaposition of the final 8(14-6) term.
I have 48 bags of apples to divide between 8 people
48÷8
Inside each bag is 14 apples. Upon examination 8 of the apples are rotten and are discarded leaving 6 apples in each bag.
(14-8)=6
How many apples does each person get?
6 bags multiplied by 6 apples in each bag = 36 apples per person.
How can you give 36 apples to each person when you only have 48 apples!!!!
I just realised you changed the question. You have again split the juxtaposition of 8(14-8). You cannot do that without changing the equation.
8(14-8) is not the same as 8*(14-8).
Except juxtaposition describes the relationship between a variable and a coefficient. Since neither exist in this expression the concept does not apply. Correlation does not imply causation. Just because they look similar doesn’t mean they are the same.
I can post a dozen videos from as many different calculators showing that 8(6) and 8×6 are IDENTICAL. Can you post even one that proves otherwise? I can post a dozen videos from as many calculators that prove 48÷8(6) and 48÷[8(6)] are DIFFERENT. Can you post even one that proves they are the same?
8*6 and 8(6) by themselves are the same. Both equal 48.
Multiplication by Juxtaposition is whenever terms are placed next to each other without a mathematical operation symbol. It is not limited to algebra or when variables are used.
Multiplication by juxtaposition takes priority over division. Which is why it must be calculated before the division in this equation. Just expand our the bracketed term which includes the juxtaposed term 8(14-8) = (112-64) = 48.
Just watch the second video on my blog to see the history and how teachers in the USA simplified how they wanted to teach maths.
Please stop posting as you obviously are refusing to acknowledge that multiplication by juxtaposition exists and has priority over division.
The only place you’ll find that concept is on Youtube and Facebook. It does not exist in any textbook printed except when talking about variables and coefficients. I’d sooner acknowledge the Earth is flat that your juxtaposition theory.
It won’t be in America textbooks. It was definitely in my English and Australian textbooks.
You’re also using distributive property incorrectly. Distribution removes the parentheses in the same step. a(b+c)=ab+ac not (ab+ac). You method gives you 48÷112-64 which equals -63.57 not 1. You are altering the posted expression by adding grouping symbols that do not exist in the Original post. Would that strategy work when you submit your taxes?
The denominator was not distributed incorrectly. The obelus is not a grouping symbol. It applies only to the number directly to its right. The 8 is the sole divisor.
People always use a(b+c), but that really doesn’t apply. It should be d➗a(b+c).
Dividing by a number is the same as multiplying by the reciprocal, so 48✖️.125(14-8). That still equals 36.