If it's 2(1 + 2) that's considered one "term" and heavily implies that you should FOIL first before anything else. It isn't the same as 2 * (2 + 1). Of course you wouldn't likely get an equation like that without knowing "what" you're doing, which would answer any ambiguity.
2(1 + 2) does imply multiplication: 2 * (1 + 2). The reason it counts as one term, as I noted below, is because it is inside a two-dimensional fraction which has implicit parathenses in the numerator, denominator, and the fraction itself. The first equation is actually ((6) / (2(1 + 2))). When a fraction is written in two dimensions instead of a single string, the division between the numerator and the denominator is supposed to be done last.
The first equation is not 6 / 2(1 + 2). If it was, this means you get (6 / 2) * (1 + 2) as in the second equation, which means (1 + 2) is moved up to the numerator ((6(1+2)) / 2 = (6 / 2) * (1 + 2)), which means the two problems are not equal to each other. I believe this is the point of the "joke".
If I wrote 6 ÷ 2x, x=3 you wouldn't try to divide by just the 2. The 2 is "part of" the term (2x) which is how the majority of cases where you would actually see something like 2(some number) would work. PEMDAS BODMAS or whatever other mnemonic be damned
(I'm not arguing this passionately in any way I just like arguing <3)
x is still multiplied last. There's not a rule for implied multiplication shorthand preceding operations to the left. You still need to wrap 2x in parentheses if you want the operation to occur first.
https://www.wolframalpha.com/input?i=6%2F%282x%29
This isn't like a polynomial like ax^2 + bx + c as division is done between 6 and 2 before multiplication with x. Typically you wouldn't see such an equation (which is intended to trick you) as normally addition or subtraction would occur like in a polynomial or another variable equation (such as a linear graph), which would be done after the exponents, multiplication, and division with the variables are calculated. In the instance you wrote, it should be written as (6/2)x, or 3x, to avoid obscuring the equation. Though you intended for 6/(2x), or 3/x.
And no worries, comrade, I'm just meaning to help since I am good at math and like helping people (I don't mean this in an egotistical way). I'm not taking offense, and I am not meaning to offend anyone.
Assuming the first way is written correctly, the equation is actually 6 / (2 * (1 + 2)). The (1 + 2) is still inside the denominator. So it is solved as follows:
6 / (2 * (1 + 2))
6 / (2 * 3)
6 / 6
1
The second equation incorrectly takes out the (1 + 2) and places it as the numerator on the side. In order to take that piece out correctly, it would have to be: (6 / 2) * (1 / (1 + 2))
And to solve it, it would look like as follows:
(6 / 2) * (1 / (1 + 2))
3 * (1 / (1 + 2))
3 * (1 / 3)
3 / 3
1
Also, 3 * 3 = 9 in regards to second incorrect equation (incorrect meaning the second incorrectly refactored equation from the pic that you answered correctly up until the last operation).
I think The_sleepy_woke_dialectic forgot to put parentheses around the denominator, but I believe it was meant to be interpreted as the entire denominator as shown in the pic.
I think the image is discussing the two ways most people interpret the (deliberately slightly obtuse) equation 6 / 2(1+2)
Following BEDMAS BOMDAS PEMDAS or however you call it in your area as written, the correct interpretation is interpretation #2, which resolves to 9,
However many people also interpret the implitic multiplication in 2(1+2) to have higher priority, or makes the 2 and () into one unit, as if put into Parentheses, which leads to interpretation #1, which resolves to 1.
The real answer is to make the original question less obtuse, but any parsing algorithm correctly given the rules of mathematical notation would resolve it to 9
The rest of my response is just for extra clarification
If we assume the second equation is true, then the first would have to be represented as (6 * (1 + 2)) / 2, which is (6 * 3) / 2, which is 18 / 2, which is 9. This means the mistake was made by multiplying the fraction with (1 + 2) by incorrectly placing it in the denominator instead of the numerator. I think the image is supposed to be a tongue-in-cheek troll as the two equations are non-preserving transformations of each other. It's a common mistake that is made in arithmetic.
Unfortunately the step by step solutions are now locked behind a subscription.
Following BEDMAS BOMDAS PEMDAS or however you call it in your area as written, the correct interpretation is interpretation #2, which resolves to 9
Both problems are valid on their own. There's no correct interpretation. If we want to assume they were supposed to be equal to each other, this means one was incorrectly transformed from the other due to common mistakes that occur with two-dimensional fractions. The horizontal line between a fraction is not equal to / without implicit parentheses applied, otherwise the horizontal line would only apply to the first number, which is not the point of a two-dimensional fraction.
I'm not trying to start an argument over this, but I respectfully disagree.
6 / 2 * (1 + 2)
6 / 2 * 3
3 * 3
9 edit: accidentally said 6 here
Parentheses first, then division and multiplication granting priority to operations on the left.
If it's 2(1 + 2) that's considered one "term" and heavily implies that you should FOIL first before anything else. It isn't the same as 2 * (2 + 1). Of course you wouldn't likely get an equation like that without knowing "what" you're doing, which would answer any ambiguity.
2(1 + 2) does imply multiplication: 2 * (1 + 2). The reason it counts as one term, as I noted below, is because it is inside a two-dimensional fraction which has implicit parathenses in the numerator, denominator, and the fraction itself. The first equation is actually ((6) / (2(1 + 2))). When a fraction is written in two dimensions instead of a single string, the division between the numerator and the denominator is supposed to be done last.
The first equation is not 6 / 2(1 + 2). If it was, this means you get (6 / 2) * (1 + 2) as in the second equation, which means (1 + 2) is moved up to the numerator ((6(1+2)) / 2 = (6 / 2) * (1 + 2)), which means the two problems are not equal to each other. I believe this is the point of the "joke".
If I wrote
6 ÷ 2x, x=3you wouldn't try to divide by just the 2. The 2 is "part of" the term (2x) which is how the majority of cases where you would actually see something like 2(some number) would work. PEMDAS BODMAS or whatever other mnemonic be damned(I'm not arguing this passionately in any way I just like arguing <3)
2x still means 2 * x.
https://www.wolframalpha.com/input?i=6%2F2x
x is still multiplied last. There's not a rule for implied multiplication shorthand preceding operations to the left. You still need to wrap 2x in parentheses if you want the operation to occur first.
https://www.wolframalpha.com/input?i=6%2F%282x%29
This isn't like a polynomial like ax^2 + bx + c as division is done between 6 and 2 before multiplication with x. Typically you wouldn't see such an equation (which is intended to trick you) as normally addition or subtraction would occur like in a polynomial or another variable equation (such as a linear graph), which would be done after the exponents, multiplication, and division with the variables are calculated. In the instance you wrote, it should be written as (6/2)x, or 3x, to avoid obscuring the equation. Though you intended for 6/(2x), or 3/x.
And no worries, comrade, I'm just meaning to help since I am good at math and like helping people (I don't mean this in an egotistical way). I'm not taking offense, and I am not meaning to offend anyone.
Assuming the first way is written correctly, the equation is actually 6 / (2 * (1 + 2)). The (1 + 2) is still inside the denominator. So it is solved as follows:
6 / (2 * (1 + 2))
6 / (2 * 3)
6 / 6
1
The second equation incorrectly takes out the (1 + 2) and places it as the numerator on the side. In order to take that piece out correctly, it would have to be: (6 / 2) * (1 / (1 + 2))
And to solve it, it would look like as follows:
(6 / 2) * (1 / (1 + 2))
3 * (1 / (1 + 2))
3 * (1 / 3)
3 / 3
1
Also, 3 * 3 = 9 in regards to second incorrect equation (incorrect meaning the second incorrectly refactored equation from the pic that you answered correctly up until the last operation).
I think The_sleepy_woke_dialectic forgot to put parentheses around the denominator, but I believe it was meant to be interpreted as the entire denominator as shown in the pic.
My bad on typing 6 as the final number, typo.
I think the image is discussing the two ways most people interpret the (deliberately slightly obtuse) equation 6 / 2(1+2) Following BEDMAS BOMDAS PEMDAS or however you call it in your area as written, the correct interpretation is interpretation #2, which resolves to 9,
However many people also interpret the implitic multiplication in 2(1+2) to have higher priority, or makes the 2 and () into one unit, as if put into Parentheses, which leads to interpretation #1, which resolves to 1.
The real answer is to make the original question less obtuse, but any parsing algorithm correctly given the rules of mathematical notation would resolve it to 9
I just realized where the confusion is coming from:
A fraction written in the formation as shown in the image has implicit parentheses over the numerator and the denominator (as well as the entire fraction) that need to be explicitly written when converted to single-line form.
The rest of my response is just for extra clarification
If we assume the second equation is true, then the first would have to be represented as (6 * (1 + 2)) / 2, which is (6 * 3) / 2, which is 18 / 2, which is 9. This means the mistake was made by multiplying the fraction with (1 + 2) by incorrectly placing it in the denominator instead of the numerator. I think the image is supposed to be a tongue-in-cheek troll as the two equations are non-preserving transformations of each other. It's a common mistake that is made in arithmetic.
Here are the equations in WolframAlpha:
https://www.wolframalpha.com/input?i2d=true&i=Divide%5B6%2C2%5C%2840%291%2B2%5C%2841%29%5D
https://www.wolframalpha.com/input?i2d=true&i=Divide%5B6%2C2%5D%5C%2840%291%2B2%5C%2841%29
Unfortunately the step by step solutions are now locked behind a subscription.
Both problems are valid on their own. There's no correct interpretation. If we want to assume they were supposed to be equal to each other, this means one was incorrectly transformed from the other due to common mistakes that occur with two-dimensional fractions. The horizontal line between a fraction is not equal to / without implicit parentheses applied, otherwise the horizontal line would only apply to the first number, which is not the point of a two-dimensional fraction.