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The Order of Operations: Examples (page 2 of 2)

  • Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.

      • 16 ÷ 2[8 – 3(4 – 2)] + 1
          = 16 ÷ 2[8 – 3(2)] + 1
          = 16 ÷ 2[8 – 6] + 1
          = 16 ÷ 2[2] + 1         (**)
          = 16 ÷ 4 + 1
          = 4 + 1
          =       5

The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy:

    calculator screen-shot: 16 / 2(8) = 1, but 16 / 2 * 8 = 64

Note that different software will process things differently; even different models of Texas Instruments graphing calculators will process things differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!

(And please do not send me an e-mail either asking for or else proffering a definitive verdict on this issue. As far as I know, there isn't any such verdict, and I have no interest in being drawn into the debate.)

  • Simplify 14x + 5[6 – (2x + 3)].

    • If you have trouble taking a subtraction through a parentheses, try turning it into multiplying a negative 1 through the parentheses (note the highlighted red "1" below):

      14x + 5[6 – (2x + 3)]
          = 14x + 5[6 –       1(2x + 3)]
          = 14x + 5[6 – 2x – 3]
          = 14x + 5[3 – 2x]
          = 14x + 15 – 10x
          =       4x + 15

  • Simplify the following:

      • -9 + (9 + 20)/3(5) - (-3)

    Be careful when simplifying fractions. You cannot cancel a term from part of a sum, only a stand-alone factor in a product. Do not try to cancel the 3 with the 9, or the 5 with the 20; this is not legitimate.

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

  • Simplify the following:

      • [3 + (15 ÷ (-3))] / 16

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

Remember to reduce fractions when you're done.   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

  • Simplify the following:

      • (4 + 8)/(2 + 1) - (3 - 1) + 2

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

Be careful with fractions! Do not try to cancel the 2 with anything! You have to add in the 1 first, to get 3 underneath.

  • Simplify the following:

      • 6 + (16 - 4)/(2^2 + 2) - 2

      6 + [16 - 4]/[2² + 2] - 2 = ... = 6

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

  • Simplify –{2x – [3 – (4 – 3x)] + 6x}.

    • If you have trouble with subtracting parentheses, convert the subtraction to multiplying a negative 1 through the parentheses. Remember to simplify at each step, combining like terms where possible:

      –{2x – [3 – (4 – 3x)] + 6x}
          = –      1{2x – 1[3 – 1(4 – 3x)] + 6x}
          = –      1{2x – 1[3 – 4 + 3x] + 6x}
          = –      1{2x – 1[– 1 + 3x] + 6x}
          = –      1{2x + 1 – 3x + 6x}
          = –      1{2x + 6x – 3x + 1}
          = –      1{5x + 1}
          =       –5x – 1

The most important things to remember with Order of Operations is to work from the inside out, and to do each step completely and separately. If you try to do two steps at once, or try to do stuff in your head, you will be much more likely to make mistakes. Take the time to get in the habit of being careful. The skill will stand you in good stead, especially on tests where you can't afford mistakes.

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Cite this article as:

Stapel, Elizabeth. "The Order of Operations: Examples." Purplemath. Available from
    http://www.purplemath.com/modules/orderops2.htm. Accessed
 

 

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