## Sunday, January 25, 2009

### BODMAS or PEMDAS: It is elementary!

Growing up in India I learnt the mnemonic BODMAS (pronounced: bodmas!) to help me with remembering the order of operations when solving equations. It stands for:

 B Brackets O Order D Division M Multiplication A Addition S Subtraction

My wife, who grew up here in the U.S., uses the phrase “Please excuse my dear aunt Sally” to remember the order of operations.

 P Parenthesis E Exponents M Multiplication D Division A Addition S Subtraction

But, hold on, there is a difference between BODMAS and PEMDAS, in India we seem to be performing the division before the multiplication, and here we seem to be multiplying before division! Yikes! what is going on here, who is correct?

They both are, the one thing the mnemonic does not show is that some operations have the same precedence as follows:

(Multiplication and Division) and

Thus M&D and D&M are exactly the same and so is A&S and S&A.

Why is this? Simple, because division can be replaced by a multiplication operation and subtraction can be replaced by an addition, as follows:

x / y = x * (1/y)

and x – y = (x + (-y))

http://mathforum.org/dr.math/faq/faq.order.operations.html

Alex said...

What people seem to forget is the DM and AS of the BODMAS and PEMDAS situations are grouped, which means that both mean the same thing.

SEoF said...

Actually, just to throw a spanner in your works, M and D aren't the same, so aren't interchangeable. These examples show them as being the same, but actually, using something like

8÷2(2+0)

PEMDAS requires you to rewrite the equation, otherwise, by following PEMDAS strictly, you get the following:
⇒8÷2(2)
⇒8÷4
⇒2

BODMAS teaches us correctly that the division must be done first, and you get the following:
⇒8÷2(2)
⇒4(2)
⇒8

Two VERY different results.

3G said...

just passing thru and realise u made a mistake. i'm pretty sure the 'B' in BODMAS stands for brackets if i remember properly. therefore u do brackets first and u'd get 2 as the answer

Russ said...

SEof has used the brackets correctly, and if you reverse the calculation to.

8x2÷(2+0)

Both PEMDAS and BODMAS give the correct answer of 8

8x2÷2
BODMAS: 8x1 = 8
PEMDAS: 16÷2 = 8

So it 'appears' that PEMDAS is a flawed rule.

GLally said...

I know the confusion as well. Kids in US are using PEMDAS and in India we used BODMAS. So using the same logic, what is the answer for:

48-16x3+3
48-48+3
0+3 = 3

or

48-48+3
48-51 = -3

I am confused. So when you get to Multiply and divide; or plus and minus only in the equation; do you go from left to right solving the equation. Or still follow PEMDAS even when the equation only has DM or AS. Please help!! Thanks.

Raj M. Rao said...

Glally

Ur 2nd calculation is wrong:
48 - 48 + 3
51 - 48 = 3

Ur mistake was using -51 which will never happen.

ellehcir1006 said...

I'm sorry but I think PEMDAS and BODMAS will not give you the same results.
For example,
6/2(1+2)
will give you
1 from Pemdas
9 from BODMAS

Raj Rao said...

ellehcir1006, here is why you are not getting the correct answer when you applied PEMDAS - you are multiplying (1+2) with 2, but 2 is part of the denominator, whereas (1+2) is part of the numerator.

A good way to relook at your problem is:
6/2(1+2)
= 6/1 * 1/2 * (1+2)/1
= 6/1 * 1/2 * (3)/1
= 6 * 0.5 * 3
= 9

hope that makes sense!

GLally said...

I agree with Roy. Here is the answer:

6/2(1+2)
= 6/2*3 - At this point in PEMDAS or BODMAS, you will solve the M or D by starting the first equation from the LEFT. In this case it is D = Divide
= 3*3
= 9

So, here the issue was that you can't litterally go word to word of PEMDAS or BODMAS. When it comes to Multiplication and Division, go with what ever comes first from LEFT. In case of addition and substraction go with the first from LEFT again. So Multiplication & Division is one group, the other group is Addition & Substraction. When solving these groups go with what comes first from the LEFT. Hope this helps.

GLALLY

John MacKenzie said...

Just to confuse things further, growing up in Ireland, my secondary school principle and math teacher taught us "BOMDAS"

:)

Well I guess people did not get the most important memo:))..When two operations of same precedence in this case division and multiplication are to be done convert them into one operation first!
6÷2(1+2)
= 6÷2(3)
= 6÷2X3
= 6 X ½ X 3 (KEY STEP)
= 9
PE(MD)(AS) or BO(DM)(AS) are the same rules. M&D and A&S are of same precedence Anonymous said...

Futures Trader, I certainly agree with your logic when you say dividing by 2 is the same as multiplying by 1/2. However, what we don't seem to agree with is the fast that 2(3) is actially one term, and so, when converting to multiplication, we get 6 X 1/2(3) = 6/6 = 1.

Tj said...

Parentheses means that you do any work inside the parentheses, not that you solve any term containing them.

GLally said...

I Agree TJ that parentheses or bracket solves first. Now if there is equation within the parentheses you solve it first using the same BODMAS or PEMDAS method. Hope this clarifies my point. Thanks.

GLALLY Anonymous said...

Hello,
2 things give 2 different answers
PEMDAS
6÷3X2
=>6÷6
=>1

BODMAS
6÷3X2
=>2X2
=>4

Mausam Das said...

6/2(1+2)=1

To solve 6/2(1+2) ,we should not
confuse what method to be
used-BODMAS or PEMDAS.
We have to know the difference between
multiplication symbols X and (nil).So,

(i) 6/2(1+2)=1 and
(ii) 6/2 X (1+2)=9

For (i),take 1+2=a.So the expression will be 6/2a
which will result into 1,not 6a/2 which gives 9.
Remember, 2a and 2Xa are different.

Kumbirai Manungo said...

Just rather stick to BODMAS as PEMDAS can lead you to a wrong answer if not used properly.

Esn said...

I'm from Canada and I can definitely say that the correct answer for "6÷2(1+2)" would be "1" according to the system that was taught to us in public school.

Mausam Das's comment explains it well.

For the answer to be "9", it would have to be written like this: "(6÷2)(1+2)"

I am astonished that something this basic to math is apparently different in different countries.

Rudra Ganguly said...

They are same. Esn, 6÷2(1+2) will give you 9 because when having a series of M&D, it is required to start from LEFT. So first do D and then do M.

Rudra Ganguly said...

For a series of M and D, please start from LEFT. So 6÷2(1+2) is 9 as per PEDMAS.

coolkid said...

6 ÷ 3 x 2

try and prove that one gives the same answers

Agnieszka MT said...

The answer is only one, both in BODMAS amd PEMDAS. Although PEMDAS states multiplication first, it asks you to do the calculations starting from LEFT, so if division comes first on the left, you do division firs. This way you come to same results.

Dan Andrew said...

OK, Let's clear this all up!

I was taught BODMAS at school, but I've been teaching PEMDAS to my students, so I fully understand the differences and key to getting this right. Here goes...

The key is that when teaching PEMDAS, it is IMPERATIVE that you teach that the 'MD' is a group, and within that group, YOU MUST GO FROM LEFT TO RIGHT. Using this rule, the results are the same for both PEMDAS and BODMAS.

Notice that all of the exceptions above that people have correctly pointed out begin with division, and end with subtraction. If you use the LEFT TO RIGHT rule, then PEMDAS becomes PEDMAS anyway.

Work through the example 8 / 4 * 2 yourself, using both methods, but FORGETTING the left to right rule for PEMDAS, and you'll see why this works.

In conslusion, this make PEMDAS more complex to use, but easier to remember (if using the mnemonic 'Please Excuse My Dear Aunt Sally'.

However, I don't think it's particularly difficult to remember BODMAS as a sort of word, without a mnemonic, so I'd suggest that you teach this method, as it is simpler for a young student to understand, and doesn't require the student to remember to be careful with Division and Multiplication.

Phew - That was an essay!
Cheers,
Dan

Harish said...

Thanks, I was wondering about it for quite a time too!

abhishek saxena said...

Hi Dan Andrew
if you use the grouping in PEMDAS for 'AS' as well unlike in BODMAS we do get diff result like in this : 6-1X0+2/2 = 5(BODMAS) and 7(PEMDAS)

Unknown said...

Parenthesis always first, then exponent... Then decision/multiplication or multiplication/devision which ever comes first.. Then addition/subtract or subtract/addition which ever comes first..

Emmanuel said...

You didn't finish with the bracket before moving on.

Unknown said...

The way we are taught pemdas is, if they're are both m and d in an equation without parenthesis, you move left to right and do whichever of the two is first. Then go back to the beginning and repeat with a and s.

Benjamin Hicks said...

I see a lot of confusion here. Both are certainly correct because multiplication and division have the same precedence, as do addition and subtraction. When dealing with operators of the same precedence, one works from left to right, thus one will always arrive at the same answer when using either acronym. The acronym assumes that one understands the equal precedence of */ and +-

jayjaymcboogie said...

It is the same with pemdas. The parentheses is separating the equation not defining what must come first. 8÷2(2+0) is the same as (8÷2)(2+0). Once both sides are figured then they can be multiplied together. That's why you are tought to go from left to write with pemdas. It is the same.

Gastronomist said...

Try solving 64÷8x4.

By BODMAS, it will solve to 32.

By PEMDAS, it will solve to 2 (incorrect application of PEMDAS).

As you see both yield different results.

The PEMDAS rule states that in the absence of parenthesis, if the sign is from the same family (÷,x) or (+,-), one must solve from left to right.

Going back to the example, PEMDAS will yield 32, too.

I think BOMDAS is simpler as one doesn't have to know these additional trivial rules.

Enjoy.

Kohinoor Basu said...

BODMAS and PEMDAS will give totally different resultfor the following problem

2 14 2 5
( - div -- times 28) div (- div - )
7 3 3 6 Anonymous said...

No, you don't get the same answer. Do it again but do the division before multiplication. You get 8.

Nkululeko Fuyane said...

In yo first, the rule is convert into one operation by saying 6*1/3 for 6/3. So yo new working will be 6*1/3*2=4

I've always thought that BO DM AS applies first, then you solve the problem by going from left to right...no?

Then for Order (power, power of powers etc), you have to solve the problem starting from the top down. then left to right.

BO | DM (or MD)| AS. Top down for Order, left to right after you presolve/expand the BODMAS/BOMDAS or PEDMAS/PEMDAS priorities. You will get the same result every damn time. Class dismissed.

Ivan Nel said...

You removed the divide. Why did you divide again?

6 x 1/2 (3)
6 × 0.5 x 3
= 9

Unknown said...

First of all we should solve brackets, so your example is totally wring

Bertin Paul Xavier said...

12/2(6-7+4)*2

ANS IS 4 OR 1

WHICH IS CORRECT

Bertin Paul Xavier said...

12/2(6-7+4)*2

can some one help me with this pls

Raj Rao said...

36!

6*3*2

LaToya Hall said...

You have to finish the B (or P in this case) then move to the division. That makes the answer still 2

Phyllis M'ya said...

BODMAS is the correct order. I learnt that in Kenya and passed with flying colors.

Unknown said...

But this creates a problem say with the equation 6÷2(1+2) as social media is finding.

Raj Rao said...

6÷2×(2+1)
Has just 1 answer : 9
The rules are in your phone's calculator app, so try it there.

The order is:

6÷2×(2+1)
3×(2+1). Do the division first
3×(3). Next comes multiplication, but you have to do the work in parenthesis first
9

Shaan Mishra said...

I think the issue here isn't which method to use but instead how the problems are the problems displayed on the internet. Without proper structuring it's easier to get confused.

As someone brought up on Bodmas system, when I see 6/2(2 + 1) it instantly becomes: (6)(1/2)(1/(1+2)) which gives us (6) x (1/2) x (1/3) which gives us (6) x (1/6) and thus the answer: 1.

However the people solving it as 9 are doing the following equation: (6) x (1/2) x (2+1), which gives 9. The google character shows it 9 because of you enter the equation.

Now why 6/(2*(2+1)) and not (6/2) x (2+1) is because of how it's written. It's 2(2+1), therefore the addition equation is also included in the division.

The problems is at the source :)

Simple Mathematics said...

Its a nice way to look at the BODMAS/PEDMAS questions.

thanks
Martin

Unknown said...

In pemdas you are supposed to multiply and divide left to right so for your equation 6÷3×2=4 6÷3=2 2×2=4. Do you see where you get it they both give the same answer but the more everyone trys to complicate it the more confusion we creat. By just simply doing it in two steps its easier. It does not matter if you have division fist or multiply first in the equation as long as you go from left to right.
If you just follow pemdas with out remembering that when you reach md and as you do that from left to right if you have add before subtract you add first if you have divide before multiply you divide first. Everyone is trying to make ot so hard yet its so easy

Abelardo León González said...

Hi,
PEMDAS = Parenthesis, Exponent, Multiplication, Division, Addition and Substraction;

are not the unique expressions. They incite us to think that a priority exists between multiplications and divisions (same for additions and substractions). They confuses, they are not the unique names:

PEMDAS = PEMDSA = PEDMAS = PEDMSA
BOMDAS = BOMDSA = BODMAS = BODMSA

A unique name should not be used to indicate the priority of the operations.

Best regards.

Unknown said...

Maam u have to add all the positive numbers together and then the negative numbers together and then substract.u cannot add -48 and +3 and even if u do that u donot get +51!!u get-45!!

Arunita Sarkar Mookherjee said...

Maam u have to add all the positive numbers together and then the negative numbers together and then substract.u cannot add -48 and +3 and even if u do that u donot get +51!!u get-45!!

FIRSAN said...

6÷2(3) is not equal to 6÷2x3
In first case the simplification is 6÷6=1 and in the next case it is 3x3=9

Unknown said...

Everyone is missing the whole point of brackets. Division and multiplication are equal. And addition and subtraction are equal. When writing an equation you write it from left to right and you insert brackets if anything needs to be done in a different order than left to right. Keeping in mind that div/multi comes before add/Sub.

6÷2x(2+1) would be written like this:

6÷2(2+1)
6÷2(3)
6÷2x3
3x3
9

The wrong way people do it is by forgetting multi and division are equals and you do whichever of them is farther left first unless there's a bracket.

6÷(2×(2+1)) is the equation you're solving when you're doing 6÷2(2+1) the wrong way. 6÷2(3) the 3 is in brackets does not mean you multiply it first . The brackets at that point are moot and it's just a multiplication. 6÷2×3 . Now left to right because there are no more brackets and all the equations are even. You could convert it all from a mix of division to only containing multiplication (to 6 × 1/2 × 3 ) and you'll still get 9. But that's for advanced mathematics.

6÷(2x3)
6÷6
1

Or

(1/6)(2×3)
(1/6)(6)
1

This is actually correct 100% guaranteed

rhp said...

A late comment ... PEMDAS and BODMAS are just mnemonics help children learn precedence of operations. They have the same effect!

Division and multiplication are equivalent operations, since you can replace them in an equation, as is often done when solving problems:

6÷2 is the same as 6x(1/2)

Addition and subtraction have a similar relationship:

6 - 2 is the same as 6 + (-2)

So when dealing with multiplication and division (or addition and subtraction) neither takes precedence over the other, and you perform the calculation from left to right.

Part of the problem (confusion?) is from the typesetting of the content

6÷2x(2+1) can be written (as on a piece of paper) as:

6
--- x (2+1) = 9
2

6÷2x(2+1) is NOT

6
--------
2(2+1)

The issue stems from how we visualise the problem. Have a look at https://en.wikipedia.org/wiki/Order_of_operations

Luvie Doobiedoo said...

Actually, we can have a different answer in the simple math problem of 8÷2×2.

Using PEMDAS will give you an answer of 2, but using BODMAS will give you 8.

Unknown said...

1+2/1 or 1/1+2 . 😅

tblount said...

PEMDAS and BODMAS ..... reverse the order of multiplication vs division ..... but in REALITY they are EQUAL ...not actually prioritized because the order is opposite for dm and md. So the ONLY way to get the same answer is to solve the equation left to right and treat multiplication and division as equal. The correct way to write the acronym cannot be linear.

TeRysa Patterson said...

You did pemdas wrong. Multiply & divide left to right you will also get 8.

Ovidio Dulay said...

M and D have the same rank as well as A and S are on the same rank.
Whoever is at the left side will be the commanding officer. So, as the officer in command will get the priority to lead.

Unknown said...

Using PEMDAS for multiplication and division requires a rule::You have to start solving from left to right...

On the other hand using BODMAS does not require any rule.. Just do the division before you do the multiplication.. Whether left to right or right to left or middle to anywhere as long as division is done before multiplication.. Eg:

6/2*3 =9

6*2/3 using PEMDAS: (6*2)/3=4
6*2/3 using BODMAS: 6*(2/3)=4

Just make sure to start from left to right while using PEMDAS... on the other hand, while using BODMAS, do not worry, you will get the correct solution.

Unknown said...

Why not! It can be the part of denominator. Bodmas or bedmas is the correct way.. According to you it is actually (6/2)(1+2).. so actually you are not following pemdas..you are just adding parenthesis and seperating them..

Dale Williams said...

PEMDAS and BODMAS are not rules of math. They are both just acronyms used to try to apply the SAME order of operations rules. They CANNOT and DO NOT provide different results. They also mean exactly the same thing.
Parentheses or Brackets
Exponents or Orders
Multiplication and Division or Division and Multiplication

MD means exactly the same thing as DM

Arik Iam said...

The rule is simple - when you have operations of the same level do them starting from the left. So it doesn't matter division or multiplications - they are of the same level of importance so just do one on the left first, then the next which becomes left. Example: 6/2/1*2/2*4*5/2 => 6/2 => 3/1*2/2*4*5/2 => 3/1 => 3*2/2*4*5/2 => 3*2 => 6/2*4*5/2 => 6/2 => 3*4*5/2 => 12 => 12*5/2 => 12*5 => 60/2 = 30. This is how it works and either BODMAS(BEDMAS) and PEMDAS will work fine.

Unknown said...

I don't use a specific procedure, the way I get it right according to a calculator is that you do brackets first, and then rewrite the equation with the answer of the brackets in place for example 3+8÷2(2+0) would be 3+8÷2x2 as I answered the brackets. Then you would answer all multiplication problems from left to right which would make the above equation equate to 3+8. Then you proceed to complete any addition or subtraction from left to right again and you will come to your answer of 11, check it on a calculator.

mortalsolace said...

As others have stated, you have worked through the problem incorrectly using PEMDAS. Yes, the parentheses will be solved first (2), however then you complete the equation left to right if there are no exponents, doing either division or multiplication (whichever comes first, left to right). Once all multiplication and division is completed left to right, you complete any remaining addition/subtraction left to right.

You do PEMDAS and BODMAS the same - in the direction the equation is written. They'll both give you the correct answer. Anonymous said...

Lol u didn't go left to right, that's ur problem. Funny thing is you were so matter of fact about your answer and didn't even bother to follow the steps. PEMDAS and BODMAS are literally the same but I like pemdas cause I think it sounds better

edwin boakye said...

Rao
48 - 48+3
48- 51

The - is just a sign to be used after addition in Bodmas.

GLally said...

Still a lot of people literally follow PEMDAS or BODMAS. When it comes to group of Multiplication and Divide in an equation, you should solve left to right. Same goes for Addition and Subtraction. Therefore:

4-6/3+5*3
= 4-2+15
= 2+15
= 17 Correct

https://www.mathsisfun.com/operation-order-bodmas.html

GLally said...

100% agree with this statement:

As others have stated, you have worked through the problem incorrectly using PEMDAS. Yes, the parentheses will be solved first (2), however then you complete the equation left to right if there are no exponents, doing either division or multiplication (whichever comes first, left to right). Once all multiplication and division is completed left to right, you complete any remaining addition/subtraction left to right.

You do PEMDAS and BODMAS the same - in the direction the equation is written. They'll both give you the correct answer.

Like I said before try Excel and you will get the correct answer.

Unknown said...

Result = p*q/s+r*t/y*t; //where p=9, q=8, r=7, s=6, t=5, y=5

= 9*8/6+7*5/5*5
= 72/6+35/5*5
= 12+7*5
= 12+35 = 47
Using BODMAS, I get 47

= 9*8/6+7*5/5*5
= 72/6+35/25
= 12+1.4 = 13.4
Using PEMDAS, I get 13.4

L said...

Both BODMAS and PEMDAS are used to help children learn mathematics. As you grow, you can ignore these acronyms.

I will try to show how to use these acronyms using the problems given in this forum.

8÷2(2+0)

BODMAS:
=8÷2(2)
=4(2)
=8

PEMDAS:
=8÷2(2)
=16÷2 (note that 8÷2 can be seen as a fraction, 8÷2 = 8/2. When we multiply a fraction with a number, we just multiply the numerator so 8/2 x 2 = 16/2)
=8

Rule 3:
=8(1+0)
=8(1)
=8

Rule 4:
=(16+0)÷2
=16÷2
=8

Rule 5:
=4(2+0)
=4(2)
=8

Rule 6:
=4(2+0)
=(8+0)
=8

How many rules do you need to learn?
If you are still bounded to these acronym, then you are not supposed to go to secondary school yet. Master your basic on primary school first.