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When to Use the Distributive Property

Write the numerator in brackets and the denominator as a fraction in front

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When to Use the Distributive Property

David and Denise are having an argument. David says that you can't use the Distributive Property to simplify the expression \begin{align*}\frac{4x + 5}{8}\end{align*}4x+58, while Denise says that you can. Who do you think is right? After completing this Concept, you'll know when to use the Distributive Property to simplify expressions so that you can settle arguments such as these.


Identifying Expressions Involving the Distributive Property

The Distributive Property often appears in expressions, and many times it does not involve parentheses as grouping symbols. In a previous Concept, we saw how the fraction bar acts as a grouping symbol. The following example involves using the Distributive Property with fractions.

Example A

Simplify \begin{align*} \frac{9-6y}{3}.\end{align*}96y3.


The denominator needs to be distributed to each part of the expression in the numerator.

We can rewrite the expression so that we can see how the Distributive Property should be used:

\begin{align*} &\frac{9-6y}{3}=\frac{1}{3}(9-6y)=\\ &\frac{1}{3}(9)-\frac{1}{3}(6y)=\frac{9}{3}-\frac{6y}{3}=\\ & 3-2y.\end{align*}96y3=13(96y)=13(9)13(6y)=936y3=32y.

Example B

Simplify \begin{align*}\frac{2x+4}{8}.\end{align*}2x+48.


Think of the denominator as \begin{align*} \frac{1}{8}\end{align*}18: \begin{align*}\frac{2x+4}{8}= \frac{1}{8} (2x+4).\end{align*}2x+48=18(2x+4).

Now apply the Distributive Property: \begin{align*}\frac{1}{8} (2x)+ \frac{1}{8}(4) = \frac{2x}{8} + \frac{4}{8}.\end{align*}18(2x)+18(4)=2x8+48.

Simplified: \begin{align*}\frac{x}{4} + \frac{1}{2}.\end{align*}x4+12.

Solve Real-World Problems Using the Distributive Property

The Distributive Property is one of the most common mathematical properties seen in everyday life. It crops up in business and in geometry. Anytime we have two or more groups of objects, the Distributive Property can help us solve for an unknown.

Example C

An octagonal gazebo is to be built as shown below. Building code requires five-foot-long steel supports to be added along the base and four-foot-long steel supports to be added to the roof-line of the gazebo. What length of steel will be required to complete the project?

Solution: Each side will require two lengths, one of five and one of four feet respectively. There are eight sides, so here is our equation.

Steel required \begin{align*}= 8(4 + 5)\end{align*}=8(4+5) feet.

We can use the Distributive Property to find the total amount of steel.

Steel required \begin{align*}= 8 \times 4 + 8 \times 5 = 32 + 40\end{align*}=8×4+8×5=32+40 feet.

A total of 72 feet of steel is needed for this project.

Video Review


Guided Practice

Simplify \begin{align*}\frac{10x+8y-1}{2}.\end{align*}10x+8y12.


First we rewrite the expression so we can see how to distribute the denominator:

\begin{align*}& \frac{10x+8y-1}{2}=\frac{1}{2}(10x+8y-1)= \\ &\frac{1}{2}(10x)+\frac{1}{2}(8y)-\frac{1}{2}(1)= 5x+4y-\frac{1}{2}\end{align*}10x+8y12=12(10x+8y1)=12(10x)+12(8y)12(1)=5x+4y12

Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Distributive Property (5:39)

Use the Distributive Property to simplify the following expressions.

  1. \begin{align*}(2 - j)(-6)\end{align*}(2j)(6)
  2. \begin{align*}(r + 3)(-5)\end{align*}(r+3)(5)
  3. \begin{align*}6 + (x - 5) + 7\end{align*}6+(x5)+7

Use the Distributive Property to simplify the following fractions.

  1. \begin{align*}\frac{8x + 12}{4}\end{align*}8x+124
  2. \begin{align*}\frac{9x + 12}{3}\end{align*}9x+123
  3. \begin{align*}\frac{11x + 12}{2}\end{align*}11x+122
  4. \begin{align*}\frac{3y + 2}{6}\end{align*}3y+26
  5. \begin{align*}- \frac{6z - 2}{3}\end{align*}6z23
  6. \begin{align*}\frac{7 - 6p}{3}\end{align*}76p3

In 10 – 17, write an expression for each phrase.

  1. \begin{align*}\frac{2}{3}\end{align*}23 times the quantity of \begin{align*}n\end{align*}n plus 16
  2. Twice the quantity of \begin{align*}m\end{align*}m minus 3
  3. \begin{align*}-4x\end{align*}4x times the quantity of \begin{align*}x\end{align*}x plus 2
  4. A bookshelf has five shelves, and each shelf contains seven poetry books and eleven novels. How many of each type of book does the bookcase contain?
  5. Use the Distributive Property to show how to simplify 6(19.99) in your head.
  6. A student rewrote \begin{align*}4(9x + 10)\end{align*}4(9x+10) as \begin{align*}36x + 10\end{align*}36x+10. Explain the student’s error.
  7. Use the Distributive Property to simplify 9(5998) in your head.
  8. Amar is making giant holiday cookies for his friends at school. He makes each cookie with 6 oz of cookie dough and decorates each one with macadamia nuts. If Amar has 5 lbs of cookie dough \begin{align*}(1 \ lb = 16 \ oz)\end{align*}(1 lb=16 oz)and 60 macadamia nuts, calculate the following.
    1. How many (full) cookies can he make?
    2. How many macadamia nuts can he put on each cookie if each is supposed to be identical?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.9. 

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Distributive Property

For any real numbers M, \ N, and K: &M(N+K)= MN+MK\\
&M(N-K)= MN-MK

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