The     Distributive  Property

Your teacher asks you to distribute rulers to the class.  What does she mean?  You are to give a ruler to each student, right? You can't skip anyone.  Remember this when you use the distributive property in math.


 
 
The Distributive Property

a ( b + c )  =  ab  +  ac

You distribute to simplify an expression or equation.  Just like when you give out rulers to each student, you must "give out" the leading number to each term inside the parentheses.

Since parentheses signal multiplication, the leading number will be multiplied by the terms inside the parentheses.  Keep the sign that is between the terms.
 
 

7 ( x + 4)
 <   read "7 times the quantity of x + 4"

Notice in the example that everything inside the parentheses is read as "the quantity of."  Seven is multiplied by whatever is inside the parentheses, be it x+4, x+3y-2z, or 12.

To simplify this example, distribute the 7 to each term.  (Terms are separated by + and - signs.)
 
 

7 ( x + 4)   =   7x + 7*4

7x + 28

Don't be tempted to combine this any further.  7x and 28 are not like terms.  See combining like terms for more information.
 

You may find it helpful to draw arcs from the 7 to each term as you multiply, and to talk yourself through it step-by-step.
 

Some more examples:

3(4p - 2)   =   3*4p - 3*2

12p - 6
12(9 + 3w)    =    12*9 + 12*3w

108 + 36w
10(6m+7n) = 10*6m + 10*7n

60m + 70n
5(3x+2y-8z) = 5*3x + 5*2y - 5*8z

15x + 10y - 40z

Practice with some problems!
 
 
 

Of course, if you can "give out" to each term, you must be able to "take from" each term.  This "undistributing" is called factoring.

Use the "linkers" (+ and - signs) to help separate the terms.
 
 

 6(78) - 6(8)
 

6(78) - 6(8)
 

6 (78 - 8)
 
 
 

6 ( 70)

420

 <There is a quicker way than to multiply this out!

<Look at the terms (think parts).  Is there anything in common?  YES!  6!

<Take out the common # (6), and rewrite what wasn't used (78-8).  Your first thought is to distribute...but you'll undo what you just did.

<Simplify inside the parentheses.

<Multiply.

 

Some more examples:
 

11(14) + 11(6)
11(14 + 6)
11(20)
220
4(39) - 4(9)
4(39 - 9)
4(30)
120
8(17) - 8(3) + 8(6)
8(17 - 3 + 6)
8(20)
160

Practice with some problems!
 
 
 

There is yet another way that the distributive property can help simplify problems and aid in mental math computations.  It involves rewriting a number using addition or subtraction.
 
 
 

7(108)

7(100 + 8)

7*100 + 7*8

700 + 56

756

 < 108 = 100 + 8

< Distribute.
 
 

 

There is more than one way to solve these problems.  Look at this example.  On the left, addition is used, and on the right, subtraction is used.  Both are correct and acceptable.  It is suggested that you use the one with the fewest steps and the least chance for error (in this case, subtraction).
 
 

4(293)
4(200 + 90 + 3)
4*200 + 4*90 + 4*3
800 + 360 + 12
1172
4(293)
4(300 - 7)
4*300 - 4*7
1200 - 28
1172

Practice with some problems!
 
 

Practice Problems

Use the distributive property to simplify.  Check your answers.
 

    1.  8(x+2)

    2.  4(y+1)

    3.  2(3a+7)

    4.  5(4-x)

    5.  6(3p+2)

    6.  1/2(14w+12)

    7.  1/3(27-9r)

    8.  3(x+y)

    9.  7(a-b)

    10.  9(2x + 3y)

    11.  5(4c - 8d)

    12.  1/4(4g - 16m)

    13.  10( x + 3y - 14z)

    14.  5(48) + 5(12)

    15.  9(38) + 9(2)

    16.  6(23) - 6(3)

    17.  8(178) - 8(8)  + 8(30)

    18.  6(36)

    19.  7(598)

    20.  4(210)

    21.  7(1023)

    22.  3(279)
     

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ANSWERS
1.  8x + 16        2.  4y + 4        3.  6a + 14        4.  20 - 5x        5.  18p + 12        6.  7w + 6        7.  9 - 3r
8.  3x + 3y        9.  7a - 7b        10.  18x + 27y        11.  20c - 40d        12.  g - 4m        13.  10x + 30y - 140z
14.  5(48+12)=5(60)=300        15.  9(38+2)=9(40)=360        16.  6(23-3)=6(20)=120
17.  8(178-8+30)=8(200)=1600        18.  6(30+6)=180+36=216        19.  7(600-2)=4200-14=4186
20.  4(200+10)=800+40=840        21.  7(1000+20+3)=7000+140+21=7161        22.  3(300-20-1)=900-60-3=837
 
 


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