According to the calculator they have different answers, the first being 32 and the latter being 17. I got the answer 22 when i worked it out.
In: 0
Calculators have gotten good since I’ve been out of school lol.
PEMDAS: parentheses, exponents, multiplication/division, addition/subtraction. Those must be done in that order.
The first one becomes (11)*3-1, then 33-1, =32
The second becomes 5+6*(2), then 5+12, =17
To get 22, you were doing addition/subtraction before multiplication, which is incorrect order of operations. Effectively, you had done (5+6)*(3-1)
BEDMAS
Algabraic convention sets the order of operations as follows:
1. Brackets;
2. Exponents;
3. Division;
4. Multiplication;
5. Addition; and
6. Subtraction.
Calculators and computer programming languages follow this convention.
So following this convention yields different results for each equation.
This topic has been the bane of many students’ math careers before they ever get a chance to learn the love of math and how powerful it can be. The order in which you perform the operations in math is defined by a convention. Some operations have higher priorities than others. As far as I’m aware, no matter where you studied, there’s a 95+% chance you learned the same thing that I did:
Parentheses/Brackets always get the highest priority. Anything wrapped up will always get evaluated first before anything else. Parentheses (actually all operation orders) can be recursive ((5+2)*(3+6)+4)*7. So you work from outside in. If there is an expression inside a large parentheses block like above, just apply the order of operations to what’s inside first, before anything else.
Exponents/Orders/Powers are always next. There are no exponents in this equation, so it’s not relevant.
Depending on what you were taught, this next part can change. I’ll explain the rare, but still valid order of operations difference that I’ve heard of at the end of the answer. For now, most everybody I’ve ever heard of has been taught that:
all Multiplication and Division are 3rd most important. They both get exactly the same priority. That means, if there are both * and / (or ÷) symbols, you should apply them from left to right to sort out which is which.
There is often confusion here, some people think because the acronym lists M before D that multiplication should take priority over division. This is NEVER the case. They get the exact same priority. Use left to right. Additionally, some equations are written vertically as fractions with a horizontal bar between the top and bottom half to denote division. There is no real structured convention for this, but it can eliminate some confusion. In this case, apply the shortest horizontal line first. Divide those things which are immediately above and below that line, then move to the next longer division line. (If lines are the same length, then someone is deliberately trying to make it ambiguous to get internet strangers to argue, guaranteed).
Addition and Subtraction are always last. And just like M/D, they get the exact same priority. Addition does not come before subtraction. If both are involved, apply left to right to solve.
This is something that should have been taught day 1 as soon as you learned to multiply and divide. Then again as soon as you learned exponents. And probably brought up once per month in your first algebra class. It’s so fundamental to everything that is mathematics, and so often doesn’t get the attention it deserves. With those rules, the answers of your two problems go like this:
parentheses first. (5+6) = 11, meaning (5+6)*3-1 becomes 11*3-1
skip exponents, there are none. Multiplication and division next. 11*3 = 33, meaning 11*3-1 becomes 33-1
addition and subtraction last. 33-1 = 32
—
For the bottom equation, parentheses first. (3-1) = 2, meaning 5+6*(3-1) becomes 5+6*2
skip exponents, there are none. Multiplication and division next. 6*2 = 12, meaning 5+6*2 becomes 5+12
addition and subtraction last. 5+12 = 17
now, as promised, there is another order of operations that can be used. I think its dumb. I hate it. I hate whoever came up with it. I don’t hate whoever uses it, but I wish they’d stop. Apparently it’s popular in physics, which is news to me because I’m a physics major and I’ve never heard of it. It’s annoying, leads to added confusion, and is unnecessary… but… that being said, it does exist and is valid. If two terms are butted up against each other with no symbol in between it is called “implied multiplication” example would be 5(2) = 5*2. If there is a symbol (either * or x or •) then it is called “explicit multiplication.”
There is a convention out there where implicit multiplication actually has a higher priority than division and explicit multiplication. The latter 2 still get exactly the same priority and use left to right, but there are problems where this can make a difference.
Example: 8/2(3+1) the way I think is better, ALL multiplication and division get the same priority and should be applied left to right. That means 8/2 comes first (after parentheses and exponents) and the problem becomes 4(4) = 16.
The alternate way to interpret this is that implied multiplication comes first (after parentheses and exponents) meaning 2(4) comes first. So the problem becomes 8/8 = 1. And since 1≠16 there is a chance that this problem could be interpreted 2 different ways. Fortunately for you, implicit multiplication is not used in either of your problems, so you don’t have to worry about it. Also, just like the horizontal fraction bars, anyone with enough training in math to get past high school is likely to write the problems in a way where this ambiguity is erased by adding parentheses or rearranging completely. So you likely won’t have to worry about it until the next viral math problem comes around.
Learning math means you have to learn the language, grammar and punctuation of math. Each of the symbols in those equations is an instruction about what to do and what order to do it in. u/cavalier78 has a great example of how grammar and punctuation can totally change the meaning of a sentence. The same is true for those equations. The parentheses are in different places (the punctuation is different), so they have different meanings.
The other answers mentioned the order of operations or PEMDAS. This is the main grammar rule for these types of equations. Several other comments explained this well.
One thing that can confusing when this is taught is that the multiplication and division (MD) have equal precedence. When you get to the multiplication and division step you start at the left hand side of the equation and work towards the right, doing all of the multiplication and division in the order that they are written. The same is true for addition and subtraction (AS).
This language of math is something that mathematicians came to an agreement on. As long as everyone is speaking the same mathematical language and following the same math grammar rules, we can communicate with each other about math. We could have come up with different rules, but it would be really confusing to communicate with people who don’t use those same rules.
(5+6)x3 – 1
is the same as
(5+6) + (5+6) + (5+6) -1
=======~=======~=======
5 + 6x(3-1)
is the same as
5 + (3-1) + (3-1) + (3-1) + (3-1) + (3-1) + (3-1)
=======~=======~=======
Hopefully you can see how these differ
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