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Parentheses and the Associative Property

Parentheses group operations together, telling you to do any operations inside a set of parentheses before you do operations outside of it. Parentheses can make a big difference in the result you get when solving a problem, especially in a problem with mixed operations. In two important cases, however, moving parentheses doesn’t change the answer to a problem.

  • The associative property of addition says that when every operation is addition, you can group numbers however you like and choose which pair of numbers to add first; you can move parentheses without changing the answer.

  • The associative property of multiplication says you can choose which pair of numbers to multiply first, so when every operation is multiplication, you can move parentheses without changing the answer.

Taken together, the associative property and the commutative property allow you to completely rearrange all the numbers in any problem that’s either all addition or all multiplication.

Sample questions

  1. What’s (21 – 6) / 3? What’s 21 – (6 / 3)?

    5 and 19. To calculate (21 – 6) / 3, first do the operation inside the parentheses — that is, 21 − 6 = 15:

    (21 – 6) / 3 = 15 / 3

    Now finish the problem by dividing: 15 / 3 = 5.

    To solve 21 – (6 / 3), first do the operation inside the parentheses — that is, 6 / 3 = 2:

    21 – (6 / 3) = 21 – 2

    Finish up by subtracting 21 − 2 = 19. Notice that the placement of the parentheses changes the answer.

  2. Solve 1 + (9 + 2) and (1 + 9) + 2.

    12 and 12. To solve 1 + (9 + 2), first do the operation inside the parentheses — that is, 9 + 2 = 11:

    1 + (9 + 2) = 1 + 11

    Finish up by adding 1 + 11 = 12.

    To solve (1 + 9) + 2, first do the operation inside the parentheses — that is, 1 + 9 = 10:

    (1 + 9) + 2 = 10 + 2

    Finish up by adding 10 + 2 = 12. Notice that the only difference between the two problems is the placement of the parentheses, but because both operations are addition, moving the parentheses doesn’t change the answer.

  3. Solve 2 x (4 x 3) and (2 x 4) x 3.

    24 and 24. To solve 2 x (4 x 3), first do the operation inside the parentheses — that is, 4 x 3 = 12:

    2 x (4 x 3) = 2 x 12

    Finish by multiplying 2 x 12 = 24.

    To solve (2 x 4) x 3, first do the operation inside the parentheses — that is, 2 x 4 = 8:

    (2 x 4) x 3 = 8 x 3

    Finish by multiplying 8 x 3 = 24. No matter how you group the multiplication, the answer is the same.

  4. Solve 41 x 5 x 2.

    410. The last two numbers are small, so place parentheses around these numbers:

    41 x 5 x 2 = 41 x (5 x 2)

    First, do the multiplication inside the parentheses:

    41 x (5 x 2) = 41 x 10

    Now you can easily multiply 41 x 10 = 410.

Practice questions

  1. Find the value of (8 x 6) + 10.

  2. Find the value of 123 / (145 – 144).

  3. Solve the following two problems:

    a. (40 / 2) + 6 = ?
    b. 40 / (2 + 6) = ?

    Do the parentheses make a difference in the answers?

  4. Solve the following two problems:

    a. (16 + 24) + 19
    b. 16 + (24 + 19)

    Do the parentheses make a difference in the answers?

  5. Solve the following two problems:

    a. (18 x 25) x 4
    b. 18 x (25 x 4)

    Do the parentheses make a difference in the answers?

  6. Find the value of 93,769 x 2 x 5. (Hint: Use the associative property for multiplication to make the problem easier.)

Following are the answers to the practice questions:

  1. 58.

    First, do the multiplication inside the parentheses:

    (8 x 6) + 10 = 48 + 10

    Now add: 48 + 10 = 58.

  2. 123.

    First, do the subtraction inside the parentheses:

    123 / (145 – 144) = 123 / 1

    Now simply divide 123 / 1 = 123.

  3. Solve the following two problems:

    a. (40 / 2) + 6 = 20 + 6 = 26
    b. 40 / (2 + 6) = 40 / 8 = 5

    Yes, the placement of parentheses changes the result.

  4. Solve the following two problems:

    a. (16 + 24) + 19 = 40 + 19 = 59
    b. 16 + (24 + 19) = 16 + 43 = 59

    No, because of the associative property of addition, the placement of parentheses doesn’t change the result.

  5. Solve the following two problems:

    a. (18 x 25) x 4 = 450 x 4 = 1,800
    b. 18 x (25 x 4) = 18 x 100 = 1,800

    No, because of the associative property of multiplication, the placement of parentheses doesn’t change the result.

  6. 93,769 x 2 x 5 = 937,690.

    The problem is easiest to solve by placing parentheses around 2 x 5:

    93,769 x (2 x 5) = 93,769 x 10 = 937,690

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