In math, distributive property says that the sum of two or more addends multiplied by a number gives you the same answer as distributing the multiplier, multiplying each addend separately, and adding the products together.
PEMDAS Order of Operations | Using Distributive Property |
---|---|
$(5+7+3)\times 4$ | $(5+7+3)\times 4$ |
$=15\times 4$ | $=(5\times 4)+(7\times 4)+(3\times 4)$ |
$=60$ | $=20+78+12$ |
$=60$ |
Distributive property is one of the most used properties in mathematics. It is used to simplify and solve multiplication equations by distributing the multiplier to each number in the parentheses and then adding those products together to get your answer.
Step | |
$7\times \left(7+5+8\right)=?$ | Given equation |
$7\left(7\right)+7\left(5\right)+7\left(8\right)=?$ | Distribute the multiplier |
$49+35+56=?$ | Multiply |
$\mathbf{=}\mathbf{140}$ | Add |
Distributive property connects three basic mathematic operations in two pairings: multiplication and addition; and multiplication and subtraction.
The Distributive Property states that, for real numbers $a$, $b$, and $c$, two conditions are always true:
You can use distributive property to turn one complex multiplication equation into two simpler multiplication problems, then add or subtract the two answers as required.
The distributive property is the same as the distributive property of multiplication, and it can be used over addition or subtraction.
Here are examples of the distributive property of multiplication at work:
Distributive Property Over Addition | Distributive Property Over Subtraction |
---|---|
$6\times (10+5)$ | $6\times (10-5)$ |
$=(6\times 10)+(6\times 5)$ | $=(6\times 10)-(6\times 5)$ |
$=60+30$ | $=60-30$ |
$=90$ | $=30$ |
The distributive property does not apply to division in the same since as it does with multiplication, but the idea of distributing or “breaking apart” can be used in division.
The distributive law of division can be used to simplify division problems by breaking apart or distributing the numerator into smaller amounts to make the division problems easier to solve.
Instead of trying to solve $\frac{125}{5}$, you can use the distributive law of division to simplify the numerator and turn this one problem into three smaller, easier division problems that you can solve much easier.
$\frac{50}{5}+\frac{50}{5}+\frac{25}{5}$
$10+10+5$
$\mathbf{25}$
These example problems that may help you to understand the power of the Distributive Property:
Too easy? Let’s try a real-life word problem using money amounts:
You buy nine boxed lunches for the members of Math Club at $\$7.90$ each. Using mental math, how much should you be reimbursed for the lunches? You notice that $\$7.90$ is only $\$0.10$ away from $\$8$, so you use the Distributive Property:
The Math Club treasurer should reimburse you $\$71.10$ for the lunches.
In basic operations, the Distributive Property applies to multiplication of the multiplicand to all terms inside parentheses. This is true whether you add or subtract terms:
$2(3+4+5)-6(7+8)=?$
The Distributive Property allows you to distribute the multiplicands or factors outside the parentheses (in this case, $2$ and $-6$), to each term inside the parentheses:
$2\left(3\right)+2\left(4\right)+2\left(5\right)-6\left(7\right)-6\left(8\right)=?$
$6+8+10-42-48=?$
$24-90=-66$
You can use the characteristics of the Distributive Property to “break apart” something that is too hard to do as mental math, too:
$9\times \mathrm{1,847}=?$
Expand the multiplier and distribute the multiplicand to each place value:
$9\left(\mathrm{1,000}\right)+9\left(800\right)+9\left(40\right)+9\left(7\right)=?$
$\mathrm{9,000}+\mathrm{7,200}+360+63=?$
Associate (group) addends for easier mental addition:
$(\mathrm{9,000}+\mathrm{7,200})+(360+63)=?$
$\mathrm{16,200}+423=\mathrm{16,623}$
In algebra, the Distributive Property is used to help you simplify algebraic expressions, combine like terms, and find the value of variables. This works with monomials and when multiplying two binomials:
$3(5a+12)-(a+8)=?$
Distribute the $3$ and the $-1$:
$3\left(5a\right)+3\left(12\right)+(-1)(a+8)=?$
$15a+36-a-8=?$
Combine like terms:
$14a+28=?$
Subtract $28$ from both sides:
$14a=-28$
Divide both sides by $14$:
$a=-2$
Here is another more example of how to use the distributive property to simplify an algebraic expression:
$(3x+4)(x-7)$
$(3x+4)(x+-7)$
Use the Distributive Property:
$\left(3x\right)\left(x\right)+\left(3x\right)(-7)+\left(4\right)\left(x\right)+\left(4\right)(-7)$
Simplify:
$3{x}^{2}-21x+4x-28$
Combine like terms:
$3{x}^{2}-17x-28$
You may be familiar with the steps to solving binomials as the FOIL method:
Distributive Property works with all real numbers, which includes positive and negative integers. In algebra especially, you need to pay careful attention to negative signs in expressions.
Review the steps above we used in this problem, above:
$3(5a+12)-(a+8)=?$
We added a $+$ sign after the first term and distributed a $-1$ to a and $8$, like this:
$3(5a+12)+(-1)(a+8)=?$
We then distributed the negative sign to both terms within the second parentheses:
$15a+36-a-8=?$
We can show this distribution of the negative sign with two generic formulas, one for addition and one for subtraction:
We can apply the Distributive Property to geometry when working with problems involving area of rectangles. Though algebra may seem unrelated to geometry, the two fields are strongly connected.
Suppose we are presented with a drawing that lacks numbers but does show a relationship.
We have no idea what the width and length are, but we are told that the rectangle has an area of $65$ square meters. How do we calculate width and length?
We know that area is width times length ($w\times l$), which in this case is $x$ for the width and $x+8$ for the length, or $\left(x\right)(x+8)$.
Write down what we know:
$x(x+8)=65{m}^{2}$
Distribute $x$:
${x}^{2}+8x=65{m}^{2}$
Convert it to a quadratic equation (subtract to make one side equal to $0$):
${x}^{2}+8x-65=0$
Factor the quadratic equation:
$(x-5)(x+13)=0$
$x-5=0andx+13=0$
$x=5andx=-13$
We cannot have a negative number for a width or length, so the correct answer is that $x$, the width, is equal to $5$. This means the length, $x+8$, is equal to $13$. We can check our work:
$\left(5m\right)\left(8m\right)=65{m}^{2}$
After working your way through this lesson and video, you've learned:
The Distributive Property states that, for real numbers a, b, and c, two conditions — a(b + c) = ab + ac and a(b - c) = ab - ac — are always true. The Distributive property applies to all real numbers with multiplication and addition, and multiplication and subtraction. It is especially useful in algebra when using the FOIL method to multiply two binomials. It can also speed up mental math, help solve geometry problems involving area, and improve your understanding of factoring.
One area of caution is to observe the negative and positive signs, especially in the second term of an algebraic expression, because the negative sign is distributed following the generic formulas -(a + b) = -a - b and -(a - b) = -a + b.
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