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I ran across the following math problem where there is arithmetic involved:
$$9-4+1$$

Where supposedly the answer is 6? I entered into my computer and calculator and got the same result so I realized there was something strange in this math problem because it does not follow the PEMDAS pattern.

Can someone explain to me why the answer is equal to 6 and not 4?.

Your confusion lies in thinking that Addition supersedes Subtraction. The actual precedence set by "PEMDAS" is

• Parenthetical terms first


• Exponents


• Any multiplication OR division (equal precedence!)


• Any addition OR subtraction (equal precedence!)

This is why I suggested you think of it as PE(MD)(AS).

This is nothing more than a convenient agreement and not some mysterious law of nature. It allows us to write $3x-y$ without ambiguity, as it means $(3x)-y$ and not $3(x-y)$. If we mean the latter, we have to say it.

So, for example,
$$
5cdot 3 -2= 15-2=13
$$
and NOT $5 cdot 3-2=5$. If I wanted the latter I have to bracket it off to make it respect PEMDAS: $5cdot (3-2) = 5 cdot 1 =5$.

In general, when a "tie" in precedence comes, we operate left-to-right. For your case, $9-4+1=5+1=6$ since the subtraction is done first. You gave the second (addition) precedence, when really the subtraction gets done first, since they have equal precedence and the left-most one comes first.

To be honest, even this is sort of rigid. I can even do your addition first, as long as I respect the minus sign the right way. For instance
$$
9 - 4 + 1 = 9 + (-4+1) = 9 + (-3) = 9-3 =6.
$$

For some other examples,
$$
7+4-5+6= 11-5+6=6+6=12
$$
and
$$
2cdot 5 - 7 cdot 3 + 4 = 10 - 21 + 4 = -11 + 4 = -7.
$$

Probably getting too abstract... but subtraction doesn't really exist. What we are really doing when we subtract is adding a negative number.

$9 + (-4) + 1 = 6$

Once we have it in the this form, then we can do our addition in any order.

$(9 + (-4)) + 1 = 5+1 = 6\
(9 + ((-4) + 1) = 9+(-3) = 6$

That is addition is associative.

And we can even swap it around. Addition is commutative.

$9 + 1 + (-4) = 10+(-4) = 6$

In many ways "PEMDAS" creates more confusion and problems than it is worth.

"PEMDAS" does not mean addition comes before subtraction. In fact addition and subtraction are done from left to right (unless modified by brackets).

So for your problem, first do $9-4=5$, then $5+1=6$.

This is an excellent illustration of why people should not rely on memorisation without understanding!

Working from the left to the right,$$9-4+1= (9-4)+1=5+1=6$$