2.5: The Distributive Property
At the end of the school year, an elementary school teacher makes a little gift bag for each of his students. Each bag contains one class photograph, two party favors, and five pieces of candy. The teacher will distribute the bags among his 28 students. How many of each item does the teacher need?
You could begin this problem by deciding your variables.
Let \begin{align*}p=photograph, \ f=favors,\end{align*}
Next you can write an expression to represent the situation: \begin{align*}p + 2f + 5c.\end{align*}
There are 28 students in class, so the teacher needs to repeat the bag 28 times. An easier way to write this is \begin{align*}28 \cdot (p + 2f + 5c).\end{align*}
We can omit the multiplication symbol and write \begin{align*}28(p + 2f + 5c)\end{align*}
Therefore, the teacher needs \begin{align*}28p + 28(2f) + 28(5c)\end{align*}
The teacher needs 28 photographs, 56 favors, and 140 pieces of candy to complete the endofyear gift bags.
When you multiply an algebraic expression by another expression, you apply the Distributive Property.
The Distributive Property: For any real expressions \begin{align*}M, \ N,\end{align*}
\begin{align*}& M(N+K)= MN+MK\\
& M(NK)= MNMK\end{align*}
Example 1: Determine the value of \begin{align*}11(2 + 6)\end{align*}
Solution: Using the Order of Operations: \begin{align*}11(2 + 6) = 11(8)= 88.\end{align*}
Using the Distributive Property: \begin{align*}11(2 + 6) = 11(2) + 11(6)= 22 + 66 = 88.\end{align*}
Regardless of the method, the answer is the same.
Example 2: Simplify \begin{align*}7(3x  5).\end{align*}
Solution 1: Think of this expression as seven groups of \begin{align*}(3x 5)\end{align*}
Solution 2: Apply the Distributive Property. \begin{align*}7(3x5)= 7(3x)+7(5)= 21x35\end{align*}
Example 3: Simplify \begin{align*}\frac{2}{7} (3y^2  11).\end{align*}
Solution: Apply the Distributive Property.
\begin{align*}&\frac{2}{7} (3y^2 + 11)= \frac{2}{7} (3y^2) + \frac{2}{7}(11)\\
&\frac{6y^2}{7}\frac{22}{7}\end{align*}
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 Lesson 1.2, we saw how the fraction bar acts as a grouping symbol. The following example involves using the Distributive Property with fractions.
Example 4: Simplify \begin{align*}\frac{2x+4}{8}.\end{align*}
Solution: Think of the denominator as: \begin{align*}\frac{2x+4}{8}= \frac{1}{8} (2x+4).\end{align*}
Now apply the Distributive Property: \begin{align*}\frac{1}{8} (2x)+ \frac{1}{8}(4) = \frac{2x}{8} + \frac{4}{8}.\end{align*}
Simplify: \begin{align*}\frac{x}{4} + \frac{1}{2}.\end{align*}
Solve RealWorld 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 5: An octagonal gazebo is to be built as shown below. Building code requires fivefootlong steel supports to be added along the base and fourfootlong steel supports to be added to the roofline 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*}
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*}
A total of 72 feet of steel is needed for this project.
Practice Set
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. CK12 Basic Algebra: Distributive Property (5:39)
Use the Distributive Property to simplify the following expressions.

\begin{align*}(x + 4)  2(x + 5)\end{align*}
(x+4)−2(x+5) 
\begin{align*}\frac{1}{2}(4z + 6)\end{align*}
12(4z+6) 
\begin{align*}(4 + 5)(5 + 2)\end{align*}
(4+5)−(5+2) 
\begin{align*}(x + 2 + 7)\end{align*}
(x+2+7) 
\begin{align*}0.25 (6q + 32)\end{align*}
0.25(6q+32) 
\begin{align*}y(x + 7)\end{align*}
y(x+7) 
\begin{align*}4.2(h  11)\end{align*}
−4.2(h−11) 
\begin{align*}13x(3y + z)\end{align*}
13x(3y+z) 
\begin{align*}\frac{1}{2}(x  y)  4\end{align*}
12(x−y)−4 
\begin{align*}0.6(0.2x + 0.7)\end{align*}
0.6(0.2x+0.7) 
\begin{align*}(2  j)(6)\end{align*}
(2−j)(−6) 
\begin{align*}(r + 3)(5)\end{align*}
(r+3)(−5) 
\begin{align*}6 + (x  5) + 7\end{align*}
6+(x−5)+7 
\begin{align*}6  (x  5) + 7\end{align*}
6−(x−5)+7 
\begin{align*}4(m + 7) 6(4  m)\end{align*}
4(m+7)−6(4−m) 
\begin{align*}5(y  11) + 2y\end{align*}
−5(y−11)+2y
Use the Distributive Property to simplify the following fractions.
 \begin{align*}\frac{8x + 12}{4}\end{align*}
 \begin{align*}\frac{9x + 12}{3}\end{align*}
 \begin{align*}\frac{11x + 12}{2}\end{align*}
 \begin{align*}\frac{3y + 2}{6}\end{align*}
 \begin{align*} \frac{6z  2}{3}\end{align*}
 \begin{align*}\frac{7  6p}{3}\end{align*}
In 23 – 25, write an expression for each phrase.
 \begin{align*}\frac{2}{3}\end{align*} times the quantity of \begin{align*}n\end{align*} plus 16
 Twice the quantity of \begin{align*}m\end{align*} minus 3
 \begin{align*}4x\end{align*} times the quantity of \begin{align*}x\end{align*} plus 2
 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?
 Use the Distributive Property to show how to simplify 6(19.99) in your head.
 A student rewrote \begin{align*}4(9x + 10)\end{align*} as \begin{align*}36x + 10\end{align*}. Explain the student’s error.
 Use the Distributive Property to simplify 9(5998) in your head.
 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*} and 60 macadamia nuts, calculate the following.
 How many (full) cookies can he make?
 How many macadamia nuts can he put on each cookie, if each is supposed to be identical?
Mixed Review
 Translate into an inequality: Jacob wants to go to Chicago for his class trip. He needs at least $244 for the bus, hotel stay, and spending money. He already has $104. How much more does he need to pay for his trip?
 Underline the math verb(s) in this sentence: The product of 6 and a number is 4 less than 16.
 Draw a picture to represent \begin{align*}3 \frac{3}{4}\end{align*}.
 Determine the change in \begin{align*}y\end{align*} of the equation \begin{align*}y = \frac{1}{6} x4\end{align*} between \begin{align*}x=3\end{align*} and \begin{align*}x=9\end{align*}.
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