Inverse Operations and the Commutative Property

The Big Four operations — addition, subtraction, multiplication, and division — are actually two pairs of inverse operations, which means that the operations can undo each other:

  • Addition and subtraction: Subtraction undoes addition. For example, if you start with 3 and add 4, you get 7. Then, when you subtract 4, you undo the original addition and arrive back at 3:

    3 + 4 = 7 → 7 – 4 = 3

    This idea of inverse operations makes a lot of sense when you look at the number line. On a number line, 3 + 4 means start at 3, up 4. And 7 – 4 means start at 7, down 4. So when you add 4 and then subtract 4, you end up back where you started.

  • Multiplication and division: Division undoes multiplication. For example, if you start with 6 and multiply by 2, you get 12. Then, when you divide by 2, you undo the original multiplication and arrive back at 6:

    6 x 2 = 12 → 12 / 2 = 6

The commutative property of addition tells you that you can change the order of the numbers in an addition problem without changing the result, and the commutative property of multiplication says you can change the order of the numbers in a multiplication problem without changing the result. For example,

2 + 5 = 7 → 5 + 2 = 7
3 x 4 = 12 → 4 x 3 = 12

Through the commutative property and inverse operations, every equation has four alternative forms that contain the same information expressed in slightly different ways.

For example, 2 + 3 = 5 and 3 + 2 = 5 are alternative forms of the same equation but tweaked using the commutative property. And 5 – 3 = 2 is the inverse of 2 + 3 = 5. Finally, 5 – 2 = 3 is the inverse of 3 + 2 = 5.

You can use alternative forms of equations to solve fill-in-the-blank problems. As long as you know two numbers in an equation, you can always find the remaining number. Just figure out a way to get the blank to the other side of the equal sign:

  • When the first number is missing in any problem, use the inverse to turn the problem around:

    _______________ + 6 = 10 → 10 – 6 = _______________

  • When the second number is missing in an addition or multiplication problem, use the commutative property and then the inverse:

    9 + _______________ = 17 → _______________ + 9 = 17 → 17 – 9 = _______________

  • When the second number is missing in a subtraction or multiplication problem, just switch around the two values that are next to the equal sign (that is, the blank and the equal sign):

    15 – _______________ = 8 → 15 – 8 = _______________

Sample questions

  1. What’s the inverse equation to 16 – 9 = 7?

    7 + 9 = 16. In the equation 16 – 9 = 7, you start at 16 and subtract 9, which brings you to 7. The inverse equation undoes this process, so you start at 7 and add 9, which brings you back to 16:

    16 – 9 = 7 → 7 + 9 = 16

  2. Use inverse operations and the commutative property to find three alternative forms of the equation 7 – 2 = 5.

    5 + 2 = 7, 2 + 5 = 7, and 7 – 5 = 2. First, use inverse operations to change subtraction to addition:

    7 – 2 = 5 → 5 + 2 = 7

    Now use the commutative property to change the order of this addition:

    5 + 2 = 7 → 2 + 5 = 7

    Finally, use inverse operations to change addition to subtraction:

    2 + 5 = 7 → 7 – 5 = 2

Practice questions

  1. Using inverse operations, write down an alternative form of each equation:

    a. 8 + 9 = 17
    b. 23 − 13 = 10
    c. 15 x 5 = 75
    d. 132 / 11 = 12
  2. Use the commutative property to write down an alternative form of each equation:

    a. 19 + 35 = 54
    b. 175 + 88 = 263
    c. 22 x 8 = 176
    d. 101 x 99 = 9,999
  3. Use inverse operations and the commutative property to find all three alternative forms for each equation:

    a. 7 + 3 = 10
    b. 12 − 4 = 8
    c. 6 x 5 = 30
    d. 18 / 2 = 9
  4. Fill in the blank in each equation:

    a. _______________ – 74 = 36
    b. _______________ x 7 = 105
    c. 45 + _______________ = 132
    d. 273 – _______________ = 70
    e. 8 x _______________ = 648
    f. 180 / _______________ = 9

Following are the answers to the practice questions:

1.
a. 8 + 9 = 17: 17 − 9 = 8
b. 23 − 13 = 10: 10 + 13 = 23
c. 15 x 5 = 75: 75 / 5 = 15
d. 132 / 11 = 12: 12 x 11 = 132
2.
a. 19 + 35 = 54: 35 + 19 = 54
b. 175 + 88 = 263: 88 + 175 = 263
c. 22 x 8 = 176: 8 x 22 = 176
d. 101 x 99 = 9,999: 99 x 101 = 9,999
3.
a. 7 + 3 = 10: 10 − 3 = 7, 3 + 7 = 10, and 10 − 7 = 3
b. 12 − 4 = 8: 8 + 4 = 12, 4 + 8 = 12, and 12 – 8 = 4
c. 6 x 5 = 30: 30 / 5 = 6, 5 x 6 = 30, and 30 / 6 = 5
d. 18 / 2 = 9: 9 x 2 = 18, 2 x 9 = 18, 18 / 9 = 2
4.
a. 110. Rewrite _______________ – 74 = 36 as its inverse:
36 + 74 = _______________
Therefore, 36 + 74 = 110.
b. 15. Rewrite _______________ x 7 = 105 as its inverse:
105 / 7 = _______________
So, 105 / 7 = 15.
c. 87. Rewrite 45 + _______________ = 132 using the commutative property:
_______________ + 45 = 132
Now rewrite this equation as its inverse:
132 – 45 = _______________
Therefore, 132 − 45 = 87.
d. 203. Rewrite 273 – _______________ = 70 by switching around the two numbers next to the equal sign:
273 – 70 = _______________
So, 273 − 70 = 203.
e. 81. Rewrite 8 x _______________ = 648 using the commutative property:
_______________ x 8 = 648
Now rewrite this equation as its inverse:
648 / 8 = _______________
So, 648 / 8 = 81.
f. 20. Rewrite 180 / _______________ = 9 by switching around the two numbers next to the equal sign:
180 / 9 = _______________
So, 180 / 9 = 20.
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