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# One Math Problem But Two Different Answers

But what really is the right answer to this equation? It’s time to settle this once and for all. Here’s a real math problem

All of us have been taught on how to add, subtract, multiply, and divide in grade school. These are the core of problem solving, which starts from simple arithmetic until it gets complicated with exponentiation and grouping included.

While to perform these mathematical operations seems easy, there has been confusion going around the Internet when it comes to the order of operations. And it’s just right to settle this once and for all.

To put simply, the order of operations states which operations take precedence, or are taken care of, before which other operations. In other words, this mathematical rule should tell us how to answer an equation with the plus, minus, times, and divide signs put altogether. Because it’s not automatically done from left to right like in reading.

### PEMDAS

The most common technique in remembering the order of operations is through the acronym “PEMDAS” which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

What this essentially tells us is the ranks of how the operations are performed: expressions within the parentheses come first; the exponents follow; multiplication and division are next, whichever of the two comes first; and addition and subtraction are last, whichever of the two comes first.

### For a much clearer precedence:

• Parentheses (simplify inside them)
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Take for example this problem that I found in Quora, which asks for the solution to the equation 6/2(2+1).

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Two Casio calculators – the left is an fx-570MS and the right is an fx-570ES – have the same input equation but different answers. But there is only one real answer. How come?

By following the order of operations or PEMDAS, it is clear that the expression inside the parenthesis, which is the (2+1), is done first. For that, we are left with 6/2*3. The next to perform is exponents but since there is none in this case, this should be skipped.

With 6/2*3 left and either of the multiplication or division to do next, we should get 3*3 and eventually 9. So the answer is 9.

Remember that multiplication and division are in the same rank so whichever operation is in the leftmost, that should be performed first. This is the most common source of confusion for some who answered 1, thinking that the 2*3 part should be done first, hence 6/6 or 1.

Moreover, by simply looking at the original equation, one would answer 1 because he or she could see that the expression 2(2+1) is more tightly bound than 6/2. But that’s not should be the case when the order of operations, a rule of mathematics with origin as far back as the 1500s, is observed.

Another solution is that the division part of the 6/2*3 can be converted into the multiplication of its reciprocal. That gives us 6*(1/2)*3, which is still equal to 9.

So the answer is 9. I am sure about that. Even Microsoft Excel, which follows the order of operations, is sure with that value.

### But how come one calculator of the same brand said that the answer is 1?

The difference has something to do with the mode or the program of the calculator. For one, not all calculators follow the order-of-operations hierarchy, which leads to a different interpretation of the equation.

In this case, the true form of the equation is 6/2(2+1), which yields 9. However, a calculator could read it as 6/[2(2+1)], which interprets everything after the division sign as a group. That is how one Casio calculator got the answer to be 1.

Sources: Purple Math | HP Museum | Quora

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1. Brent Burns says:

The answer is 1. 100%. Your logic fails as the 2 of the 2(2+1) part of the equation is implicitly implied to be part of the brackets. 6/2*(1+2) is a completely different equation than 6/2(1+2). Typing 6/2(1+2) gets rejected when you type it into Excel, and it forces you to change the equation to 6/2*(1+2) which gives the answer 9.
Example. You’d agree with me that 6/(4+2) is equal to 1. Factoring a 2 out of the brackets gives 6/2(2+1) which still has to equal 1, as we haven’t changed the equation. And it still does equal 1, as you can either factor the 2 back in to solve, or solve 6/2(3), with the 2 still being implicitly attached to the brackets. Make sense?

1. Dimitri says:

It does not make sense since 6÷(2+4) = 6÷(2(1+2)) and not 6÷2(1+2)

1. Dakota says:

All 3 of those equations result in the exact same answer. 6÷2(1+2) = 6÷(2+4) because the 2 factors into what’s inside the parenthesis so it would be 6÷(2*1 +2*2)=1. You can also consider 6÷2(1+2) to be equal to 6÷2(3) because the parenthesis don’t just disappear after you add what is inside of them, and since 2(3) would take precedence bc of the parenthesis, you’d get 1.

1. Lucian says:

On which year did you study this level of math at school? There is no precedence between multiplication and division. Period. There are both equally important operations, and since 6÷2(2+1) is an abbreviation of 6÷2×(2+1) there is a logic (mathematical thinking, not opinionated opinion!) that the right order to calculate the above mentioned equation is parenthesis, after which MUST be done in the reading order (left to right). There is no such a thing as 2×(3), since 3 is the total of the parenthesis.

1. Cameron Hidalgo says:

Operations occur between terms. 2(3) is a singular term. As such the division must apply to the entire term not just the 2.

2. Damian loris says:

The answer is one. It is a Juxtaposition. That is solved first in true order of operation. This can be proven simple. Take everything to the left of the devided by and put it on top everything to the right on bottom. Now that we simply changed it to a fraction we should not get a different answer.

3. Ron says:

The answer is definitely 9. Those of you who espouse 1 do not quite understand the Order of Operations…the first division sign is NOT all inclusive…parentheses contents must be evaluated first so you have 6 /2 x 3…now you MUST do division prior to multiplication as neither has priority over the other, so now you evaluate what 6/2 is and that’s 3…finally you evaluate 3 x 3 and you get the final answer of 9…end of story….you can “but”, “but” all you want, BUT you’d be wrong

1. Lucian says:

Excellent answer, Sir! You can, but, but… but you’d be wrong!!!! Hahaha, genial idea!!!! Thumbs up!

4. sam says:

“For that, we are left with 6/2*3.”

But you’re not left with 6/2*3. You are left with 6/2(3). 6/2*3 is different than 6/2(3). Or expressed like this 6/(2*3)

## One Math Problem But Two Different Answers

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