Mathematical Equation that Confused the Social Media

8÷2(2+2) = ?

Many are confused in evaluating this equation. There were different results even the calculators also had different answers.

Some of those who evluated this equation got an answer of 1 and some got 16.

How did they come to that answer?

According to the PEMDAS rules or the rule for evaluating an eqution, the content of the Parenthesis will be evaluated first, then the Exponents will follow, followed by the Multiplication or Division. According to this rule, Multiplication and Division are equal in position in the equation. So if you see consecutive Division and Multiplication evaluate it from left to right whichever of them comes first. And followed by Addition and Subraction. Addition and Subraction also have the same precedence, so whichever comes first from left to right is the first to evaluate.

So, how did the answer turn out to be 16? Here:

Following PEMDAS rule.

• 8÷2(2+2) = ?
• 8÷2x4 = ?
• 4x4 = 16

How did the answer become 1? Here:

Following PEMDAS rule.

• 8÷2(2+2) = ?
• 8÷2(4) = ?
• 8÷8 = 1

Both methods follow the PEMDAS rule. The question is which is the correct answer? Let's first examine another method of calculation. Since we are in the computer age, let's try in Microsoft Excel.

If we will going to input the equation into the MS Excel formula, that will goes like this formula:

• =8/2(2+2)

MS Excel will prompt you that Microsoft Excel found an error in the formula you entered. Do you want to accept the correction proposed below?

• =8/2*(2+2)

When we accept, Ms Excel will change the formula you entered to = 8/2*(2+2) and the result will be 16.

So, is your answer 16 correct?

Take note, MS Excel did not accept your formula instead it changed it like this:

• 8÷2*(2+2)

So, we are still not sure.

Let's refer to the Electronic Calculator.

We found something on the internet through Google Search. He used a TI-84 Plus C Silver Edition by Texas Instruments and the answer came out is 16.

As for Casio's fx-115MS S-V.P.A.M the answer came out is 1.

Generally speaking, even electronic calculators do not match the answer.

An equation like this first appears in this equation:

• 8÷2(1+2) = ?

Their evaluation results are also inconsistent. Some got 1 and some 9.

In fact we have seen in Google search that the same brand of Electronic Calculator with different models also have different answers.

On the Casio fx-570MS SVPAM, the answer that came out was 1, while 9 on the Casio fx50FH Super-FX Plus.

Let us examine the equation by means of Algebra.

In Algebra, we have something called Polynomials. When we say Polynomials, it is an expression or equation that consists of variables, constants and exponents combined through mathematical operations such as addition, subtraction, multiplication and division. A variable can have a constant number but cannot be divided by a variable. Usually it is Multiplication, like the example below:

• 2a = 2 x a

In writing mathematical expressions, parenthesis is also used as a multiplication in an expression. It is often used in groupings. For example:

• 2(a+b) = 2a + 2b

The operation used in the above example is called distributive property. Even though there is no multiplication sign between 2 and a+b we multiply it by each entity inside the parenthesis. It also means that the entity outside the parenthesis is part of the entity inside so they cannot be separated.

Note the following examples:

1. 2a + 2a = 4a
2. 2a + 2 = 2a + 2
3. 2a + 2(a+b) = 2a + 2a + 2b = 4a + 2b
4. 2 + 2(a+b) = 2 + 2a + 2b or 2a + 2b + 2

Note:

In the fourth example, 2 is not added to 2 instead it is distributed to a + b, so it becomes 2a + 2b.

Now, let's go back to the confusing equation:

• 8÷2(2+2) = ?

In this case we have 2(2+2). similar to 2(a+b). Basically, we will use distributive property. Will we still follow the PEMDAS rules? Of course yes. But this time, let's first distribute what needs to be distributed.

• 8÷2(2+2) = ?
• 8÷(4+4) = ?
• 8÷8 = 1

It is also stated in the evaluation of polynomials that identical variables can be combined and of course those constants without variables. For example:

• 2a+2a+2+b+3 = 4a+b+5

Let's go back to the equation we are analyzing:

• 8÷2(2+2) = ?

Notice that all entities are constant or have no variables. That means we can put them together now.

• 8÷2(2+2) = 8÷2(4)

We prioritize 2+2 within the parenthesis because according to PEMDAS we first evaluate what is inside the parenthesis. So we evaluate 2+2 = 4.

Next,

• 8÷2(4)

4 has a numerical coefficient of 2. It is similar to 2a (2 is the numerical coeficient of a). We need to distribute it. 2(4) = 2*4 = 8, so,

• 8÷8=1

Why do we put 2*4 first and not 8÷2?

Because 2 in this equation is still part of 4. We cannot evaluate immediately from left to right because we still have one entity that needs to distribute its coefficient.

The expression 8÷2*4 is different from 8÷2(4), 8÷2*b is not equal to 8÷2b.

Based on that equation, let's put it in a fraction:

Equation 1:

• 8÷2*b

8	b
2

Equation 2:

• 8÷2b

8

2b

now, solve for b. let b=4.

Equation 1:

8	4
2

• 4 x 4 = 16

Equation 2:

8

2*4

8	= 1
8

Placing 2 outside the parenthesis is extremely important. So it should not be considered just a simple multiplication. 2 is associated with each entity inside the parenthesis.

If the purpose of the parenthesis is for grouping only, it should be written like this:

• 8÷2*(2+2) = ?

Conclusion:

The correct answer to 8÷2(2+2) is 1. Some are just confused in interpreting 2(2+2) because they consider it to be just a multiplication that has nothing to do with the entities inside the parenthesis. This number is called a numerial coefficient. They forget that every entity or even variable in a polynomial can have a numerical coefficient. And in order to eliminate the parenthesis it needs to be evaluated by distributive property.