Expert: Josh - 11/29/2004

Question
Howdy!
Back when I was in school, we were taught to perform mathematic operations in the MDAS order...multiply, divide, add, subtract.  We used that old mnemonic device "My Dear Aunt Sally" to help remember this.  (Understanding, of course, that a set of parentheses or brackets takes precedence even over the MDAS rule.)
So I always thought that you'd go through an equation and multiply first, then go back and divide, then go back and add, then go back and subtract.   But I've been told lately that the MDAS rule means that multiplication and division are given "equal" priority, and that you go through first and do all multiplication AND division...THEN you go through and do all addition AND subtraction.  So I'm all discombobulated.

Bottom line--You're given the following equation:

12 divided by 3 times 2 = X

What's the correct value for X?  Is the answer 8 or 2?

Answer
Hi David,

Thanks for the question.
Yes, indeed. Multiplication and division are given the same precedance over addition and subtraction.

Priority
1        () before [] before {}
2         *, /
3         +, -

In fact, there is no need for having division and subtraction at all and I will explain why.

Consider the expression Y divided Z.
We can write Y/Z as Y x (1/Z).
In general dividing Y by Z is the same as multiplying Y by Z inverse, viz., "1/Z".

e.g., 10/2 = 10 x (1/2)
Dividing ten by two is equivalent to multiplying it by a half.

Similarly, subtraction may be viewed as the inverse operation (undoing) of addition. All we have to do is to flip the sign. Consider the expression Y subtracted by Z. We can write Y-Z as Y+(-Z).

e.g., 10-2 = 10+(-2)

For these reasons, we can always do without division and subtraction and use multiplication and addition instead.

In summary, you always work from left to right. You first compute expressions enclosed inside parentheses. In this context, you always evaluate either multiplication or division first from left to right, then, evaluate addition or subtraction from left to right.

e.g.,X = 12/3*2
      = (12/3)*2
      = 4*2

consider another example
we will translate the following step by step (remember, working from left to right)

Y = 1 - 2 / 3 * 4 + 5 * 6 / 7 - 8 * 9
 = 1 -(2/3) * 4 + 5 * 6 / 7 - 8 * 9
 = 1 -((2/3)*4) + 5 * 6 / 7 - 8 * 9
 = 1 -((2/3)*4) + (5*6)/7 - 8 * 9
 = 1 -((2/3)*4) + ((5*6)/7) - 8 * 9
 = 1 -((2/3)*4) + ((5*6)/7) - (8*9)
 = 1 - (8/3) + (30/7) - 72
 etc.

Don't hesitate to write back if you have further questions.

Cheers.