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Order of Operations: An "Issue" (page 3 of 3)

I was recently berated by a very odd person who claimed that the order of operations is really a diabolical plot which was recently hatched by a cabal of math teachers in order to destroy students' ability to succeed in science. The "proof" of this alleged conspiracy was the fact that fractions, when written vertically, don't have parentheses bracketing their numerators (tops) and denominators (bottoms). No, I'm not kidding; I couldn't make this stuff up.

So, before someone of this ilk corners you somewhere, thumps a book at you, and tries to convert you to his belief system, let's handle this middle-school arithmetic "issue" right now.

  • Simplify the following:
    • -9 + (9 + 20)/[3(5)] - (-3)

Fractions mean something; specifically, "(all of this stuff on top) over (all of that stuff underneath)" means "(all of this) divided by (all of that)". With this in mind, recall the order of operations: You first have to handle any parentheses, simplifying inside them, before you can proceed to the other bits of a given expression. You cannot reach inside a parentheses, "understood" or otherwise, and rip out part of an addition or subtraction, and try to "cancel" that shredded portion with some other value. You can cancel only factors, and you have to simplify parentheses first. So:

    I should not try to cancel the 3 in the denominator with the 9 in the numerator; I'd have to reach "inside" the "understood" parentheses and rip the 9 off of the "+20", and that's wrong. In the same way, I should not try to cancel the 5 in the denominator with the 20 inside the sum in the numerator; this would not be mathematically legitimate.

    First, I must simplify the implicit (the unstated and unmarked, but still "understood") groupings. The 9 + 20 becomes a 29, and the 3(5) becomes a 15. Then I proceed as usual: I combine the integers, convert the result to a common-denominator fraction, and simplify to get a single value.

      -9 + [(9 + 20)/(3(5))] - (?3) = ... = - 61/15

To reiterate: By typesetting the fraction vertically, as shown in the original exercise above, we had created groupings; namely, the group of "everything above the line" and the group of "everything below the line". This grouping is implicit, so parentheses are not (generally) used, though the following would mean exactly the same thing:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved




    (9 + 20) / [3(5)]

When the vertical fraction above is reformatted horizontally (say, for typing it into an e-mail or a forum posting), you must convert the (vertically) implicit grouping into an (horizontally) explicit grouping, or this grouping could be "lost" or at least misunderstood. This conversion to explicit form might look like:

    (9 + 20) / [3(5)]

Whether you use square brackets, round parentheses, or curly braces, or use some or lots of spacing, is not the point. The point is that you are aware of the "understood" parentheses implicit in vertically-formatted fractions, and that you never violate the order of operations either by trying to reach inside those parentheses in your work or else by making it look like you're breaking the rules because of how you format your typing.

  • Simplify the following:
    • [3 + (15  (-3))] / 16

      (3 + [15  (-3)])/16 = ... = - 1/8

Remember to reduce fractions when you're done.

  • Simplify the following:
    • (4 + 8)/(2 + 1) - (3 - 1) + 2

    I need to remember that there are "understood" parentheses around the "4 + 8" in the numerator and the "2 + 1" in the denominator. I must not try to "cancel" the 2 with the 4 or the 8. That won't work!

    Instead, I work bit by bit, simplifying first inside all three sets of parentheses (two "understood", and one explicit).

      [4 + 8]/[2 + 1] - (3 - 1) + 2 = ... = 4

Remember: Be careful with fractions! Do not try to cancel the initial denominator's 2 with anything! You must first simplify the denominator, adding the 2 and the 1 to get 3 underneath, before you attempt anything with the 4 + 8 on top.

  • Simplify the following:
    • 6 + (16 - 4)/(2^2 + 2) - 2

    Can I start by cancelling the 2 in the denominator with the 4 in the numerator, or the 22 = 4 in the denominator with the 16 in the numerator? Of course not; that would be silly (and wrong). First, I must simplify inside the "understood" parentheses. Since 16 4 = 12, then:

      6 + [16 - 4]/[2 + 2] - 2 = ... = 6

Do not try to cancel the 4 or 2 into the 12; instead, first add the 4 and 2 to get 6. Only then can you cancel.

In general, take your time, respect the rules, work from the inside out, and be careful with your "minus" signs. If you do this, then you should be very successful with simplifying using the order of operations.

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Cite this article as:

Stapel, Elizabeth. "Order of Operations: An 'Issue'." Purplemath. Available from Accessed


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