Distributive Property

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need \(\text{\$9.25};\) that is, \(9\) dollars and \(1\) quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

They need \(3\) times \(\text{\$9},\) so \(\text{\$27},\) and \(3\) times \(1\) quarter, so \(75\) cents. In total, they need \(\text{\$27.75}.\)

If you think about doing the math in this way, you are using the Distributive Property.

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

\[\begin{array}{lll} & 3(9.25) & \\ 3(9 & + & 0.25) \\ 3(9) & + & 3(0.25) \\ 27 & + & 0.75 \\ & 27.75 & \end{array}\]

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression \(3(x+4),\) the order of operations says to work in the parentheses first. But we cannot add \(x\) and \(4,\) since they are not like terms. So we use the Distributive Property,

Example

Try it.

Simplify: \(3(x+4).\)

Solution

	\(3(x+4)\)
Distribute.	\(3\cdot x+3\cdot 4\)
Multiply.	\(3x+12\)

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in would look like this:

Example

Try it.

Simplify: \(6(5y+1).\)

Solution

Distribute.
Multiply.

The distributive property can be used to simplify expressions that look slightly different from \(a(b+c).\) Here are two other forms.

Example

Try it.

Simplify: \(2(x-3).\)

Solution

Distribute.
Multiply.

Example

Try it.

Simplify: \(\frac{3}{4}(n+12).\)

Solution

Distribute.
Simplify.

Example

Try it.

Simplify: \(8(\frac{3}{8}x+\frac{1}{4}).\)

Solution

Distribute.
Multiply.

Example

Try it.

Simplify: \(100(0.3+0.25q).\)

Solution

Distribute.
Multiply.

Condensed: the full section is in OpenStax Prealgebra 2e.

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example

Try it.

When \(y=10\) evaluate: ⓐ \(\ 6(5y+1)\) ⓑ \(\ 6\cdot 5y+6\cdot 1.\)

Solution

ⓐ
	\(6(5y+1)\)
Simplify in the parentheses.	\(6(51)\)
Multiply.	\(306\)

ⓑ
Simplify.
Add.

Notice, the answers are the same. When \(y=10,\)

\[6(5y+1)=6\cdot 5y+6\cdot 1.\]

Try it yourself for a different value of \(y.\)

Example

Try it.

When \(y=3,\) evaluate ⓐ \(\ -2(4y+1)\)ⓑ \(\ -2\cdot 4y+(-2)\cdot 1.\)

Solution

ⓐ
	\(\ -2(4y+1)\)
Simplify in the parentheses.	\(-2(13)\)
Multiply.	\(-26\)

ⓑ
	\(\ -2\cdot 4y+(-2)\cdot 1\)
Multiply.	\(-24-2\)
Subtract.	\(-26\)
The answers are the same. When \(y=3,\)	\(-2(4y+1)=-8y-2\)

Example

Try it.

When \(y=35\) evaluate ⓐ \(\ \text{-}(y+5)\) and ⓑ \(\ \text{-}\text{y}-5\) to show that \(-(y+5)=\text{-}\text{y}-5.\)

Solution

ⓐ
	\(\ \text{-}(y+5)\)
Add in the parentheses.	\(-(40)\)
Simplify.	\(-40\)

ⓑ
	\(\ -y-5\)
Simplify.	\(-40\)
The answers are the same when \(y=35,\) demonstrating that	\(-(y+5)=\text{-}y-5\)

Distributive Property

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

Try it.

\(\frac{1}{2}(n+8)\)

Try it.

\(\frac{1}{3}(u+9)\)

Solution

\(\frac{1}{3}u+3\)

Try it.

\(\frac{1}{4}(3q+12)\)

Try it.

\(\frac{1}{5}(4m+20)\)

Solution

\(\frac{4}{5}m+4\)

Try it.

\(9(\frac{5}{9}y-\frac{1}{3})\)

Try it.

\(10(\frac{3}{10}x-\frac{2}{5})\)

Solution

3x − 4

Try it.

\(12(\frac{1}{4}+\frac{2}{3}r)\)

Try it.

\(12(\frac{1}{6}+\frac{3}{4}s)\)

Solution

2 + 9s

Try it.

\(-2(y+13)\)

Try it.

\(-3(a+11)\)

Solution

−3a − 33

Try it.

\(-7(4p+1)\)

Try it.

\(-9(9a+4)\)

Solution

−81a − 36

Try it.

\(-9(3a-7)\)

Try it.

\(-6(7x-8)\)

Solution

−42x + 48

Try it.

\(5+9(n-6)\)

Try it.

\(12+8(u-1)\)

Solution

8u + 4

Try it.

\(16-3(y+8)\)

Try it.

\(18-4(x+2)\)

Solution

−4x + 10

Try it.

\(4-11(3c-2)\)

Try it.

\(9-6(7n-5)\)

Solution

−42n + 39

Try it.

\(22-(a+3)\)

Try it.

\(-12-(u+10)\)

Try it.

\(-4-(c-10)\)

Solution

−c + 6

Try it.

\((5m-3)-(m+7)\)

Try it.

\((4y-1)-(y-2)\)

Solution

3y + 1

Try it.

\(5(2n+9)+12(n-3)\)

Try it.

\(9(5u+8)+2(u-6)\)

Solution

47u + 60

Try it.

\(9(8x-3)-(-2)\)

Try it.

\(4(6x-1)-(-8)\)

Solution

24x + 4

Try it.

\(14(c-1)-8(c-6)\)

Try it.

\(11(n-7)-5(n-1)\)

Solution

6n − 72

Try it.

\(6(7y+8)-(30y-15)\)

Try it.

\(7(3n+9)-(4n-13)\)

Solution

17n + 76

Evaluate Expressions Using the Distributive Property

In the following exercises, evaluate both expressions for the given value.

Try it.

If \(v=-2,\) evaluate

1. ⓐ \(\ 6(4v+7)\)
2. ⓑ \(\ 6\cdot 4v+6\cdot 7\)

Try it.

If \(u=-1,\) evaluate

1. ⓐ \(\ 8(5u+12)\)
2. ⓑ \(\ 8\cdot 5u+8\cdot 12\)

Solution

1. ⓐ 56
2. ⓑ 56

Try it.

If \(n=\frac{2}{3},\) evaluate

1. ⓐ \(\ 3(n+\frac{5}{6})\)
2. ⓑ \(\ 3\cdot n+3\cdot \frac{5}{6}\)

Try it.

If \(y=\frac{3}{4},\) evaluate

1. ⓐ \(\ 4(y+\frac{3}{8})\)
2. ⓑ \(\ 4\cdot y+4\cdot \frac{3}{8}\)

Solution

1. ⓐ \(\ \frac{9}{2}\)
2. ⓑ \(\ \frac{9}{2}\)

Try it.

If \(y=\frac{7}{12},\) evaluate

1. ⓐ \(\ -3(4y+15)\)
2. ⓑ \(\ -3\cdot 4y+(-3)\cdot 15\)

Try it.

If \(p=\frac{23}{30},\) evaluate

1. ⓐ \(\ -6(5p+11)\)
2. ⓑ \(\ -6\cdot 5p+(-6)\cdot 11\)

Solution

1. ⓐ −89
2. ⓑ −89

Try it.

If \(m=0.4,\) evaluate

1. ⓐ \(\ -10(3m-0.9)\)
2. ⓑ \(\ -10\cdot 3m-(-10)(0.9)\)

Try it.

If \(n=0.75,\) evaluate

1. ⓐ \(\ -100(5n+1.5)\)
2. ⓑ \(\ -100\cdot 5n+(-100)(1.5)\)

Solution

1. ⓐ −525
2. ⓑ −525

Try it.

If \(y=-25,\) evaluate

1. ⓐ \(\ -(y-25)\)
2. ⓑ \(\ -y+25\)

Try it.

If \(w=-80,\) evaluate

1. ⓐ \(\ -(w-80)\)
2. ⓑ \(\ -w+80\)

Solution

1. ⓐ 160
2. ⓑ 160

Try it.

If \(p=0.19,\) evaluate

1. ⓐ \(\ -(p+0.72)\)
2. ⓑ \(\ -p-0.72\)

Try it.

If \(q=0.55,\) evaluate

1. ⓐ \(\ -(q+0.48)\)
2. ⓑ \(\ -q-0.48\)

Solution

1. ⓐ −1.03
2. ⓑ −1.03

Try it.

Simplify \(8(x-\frac{1}{4})\) using the distributive property and explain each step.

Try it.

Explain how you can multiply \(4(\text{\$5.97})\) without paper or a calculator by thinking of \(\text{\$5.97}\) as \(6-0.03\) and then using the distributive property.

Solution

Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Condensed: the full section is in OpenStax Prealgebra 2e.

现在轮到你 没有计算器能解决这一个问题, 但其中的碎片是可以计算的。 请在下面尝试一个, 或者输入您自己的 。

⌨ 类型 ✎ 拔出它

你的数学问题

π 解决

实践 (40)

先在纸上试一个。 将答案显示为检查; 校验的答案可以在求解器中打开, 每一步都可用 。

1. Multiply: \(3(0.25).\)

   通过它工作 提示

   发送答案

   \(0.75\)

2. Simplify: \(10-(-2)(3).\)

   通过它工作 提示

   发送答案

   \(16\)

3. Combine like terms: \(9y+17+3y-2.\)

   通过它工作 提示

   发送答案

   \(12y+15\)

4. Simplify: \(3(x+4).\)

   通过它工作 提示

   发送答案

   	\(3(x+4)\)
   Distribute.	\(3\cdot x+3\cdot 4\)
   Multiply.	\(3x+12\)

5. Simplify: \(4(x+2).\)

   通过它工作 提示

   发送答案

   4x + 8

6. Simplify: \(6(x+7).\)

   通过它工作 提示

   发送答案

   6x + 42

7. Simplify: \(6(5y+1).\)

   通过它工作 提示

   发送答案

   Distribute.
   Multiply.

8. Simplify: \(9(3y+8).\)

   通过它工作 提示

   发送答案

   27y + 72

9. Simplify: \(5(5w+9).\)

   通过它工作 提示

   发送答案

   25w + 45

10. Simplify: \(2(x-3).\)

    通过它工作 提示

    发送答案

    Distribute.
    Multiply.

11. Simplify: \(7(x-6).\)

    通过它工作 提示

    发送答案

    7x − 42

12. Simplify: \(8(x-5).\)

    通过它工作 提示

    发送答案

    8x − 40

13. Simplify: \(\frac{3}{4}(n+12).\)

    通过它工作 提示

    发送答案

    Distribute.
    Simplify.

14. Simplify: \(\frac{2}{5}(p+10).\)

    通过它工作 提示

    发送答案

    \(\frac{2}{5}p+4\)

15. Simplify: \(\frac{3}{7}(u+21).\)

    通过它工作 提示

    发送答案

    \(\frac{3}{7}u+9\)

16. Simplify: \(8(\frac{3}{8}x+\frac{1}{4}).\)

    通过它工作 提示

    发送答案

    Distribute.
    Multiply.

17. Simplify: \(6(\frac{5}{6}y+\frac{1}{2}).\)

    通过它工作 提示

    发送答案

    5y + 3

18. Simplify: \(12(\frac{1}{3}n+\frac{3}{4}).\)

    通过它工作 提示

    发送答案

    4n + 9

19. Simplify: \(100(0.3+0.25q).\)

    通过它工作 提示

    发送答案

    Distribute.
    Multiply.

20. Simplify: \(100(0.7+0.15p).\)

    通过它工作 提示

    发送答案

    70 + 15p

21. Simplify: \(100(0.04+0.35d).\)

    通过它工作 提示

    发送答案

    4 + 35d

22. Simplify: \(m(n-4).\)

    通过它工作 提示

    发送答案

    Distribute.
    Multiply.

    Notice that we wrote \(m\cdot 4\ \text{as}\ 4m.\) We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

23. Simplify: \(r(s-2).\)

    通过它工作 提示

    发送答案

    rs − 2r

24. Simplify: \(y(z-8).\)

    通过它工作 提示

    发送答案

    yz − 8y

25. Simplify: \((x+8)p.\)

    通过它工作 提示

    发送答案

    Distribute.

26. Simplify: \((x+2)p.\)

    通过它工作 提示

    发送答案

    xp + 2p

27. Simplify: \((y+4)q.\)

    通过它工作 提示

    发送答案

    yq + 4q

28. Simplify: \(-2(4y+1).\)

    通过它工作 提示

    发送答案

    Distribute.
    Simplify.

29. Simplify: \(-3(6m+5).\)

    通过它工作 提示

    发送答案

    −18m − 15

30. Simplify: \(-6(8n+11).\)

    通过它工作 提示

    发送答案

    −48n − 66

31. Simplify: \(-11(4-3a).\)

    通过它工作 提示

    发送答案

    Distribute.
    Multiply.
    Simplify.

    You could also write the result as \(33a-44.\) Do you know why?

32. Simplify: \(-5(2-3a).\)

    通过它工作 提示

    发送答案

    −10 + 15a

33. Simplify: \(-7(8-15y).\)

    通过它工作 提示

    发送答案

    −56 + 105y

34. Simplify: \(-(y+5).\)

    通过它工作 提示

    发送答案

    Multiplying by −1 results in the opposite.
    Distribute.
    Simplify.
    Simplify.

35. Simplify: \(-(z-11).\)

    通过它工作 提示

    发送答案

    −z + 11

36. Simplify: \(-(x-4).\)

    通过它工作 提示

    发送答案

    −x + 4

37. Simplify: \(8-2(x+3).\)

    通过它工作 提示

    发送答案

    Distribute.
    Multiply.
    Combine like terms.

38. Simplify: \(9-3(x+2).\)

    通过它工作 提示

    发送答案

    −3x + 3

39. Simplify: \(7x-5(x+4).\)

    通过它工作 提示

    发送答案

    2x − 20

40. Simplify: \(4(x-8)-(x+3).\)

    通过它工作 提示

    发送答案

    Distribute.
    Combine like terms.

保持你自己的工作

免费账户在每一课上都添加注解, 记录你完成的, 在一个地方解决的, 以及您可以询问的关于此页的导师。 数学本身向每个人开放, 是否签名 。

签名上签名 登录

此处所用符号

输入完整定义、图片和其中每个字母的含义的符号。

\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\approxapproximately equal
Equal to the precision shown, not exactly.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.a \bmod nremainder
What is left after dividing a by n.\%per cent
Per hundred: 15% = 15/100.a : bratio, fraction
a for every b; a divided by b.

如何: Distributive Property

1. Simplify expressions using the distributive property
2. Evaluate expressions using the distributive property
3. If

人们对人们提出的问题

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

本页部分内容从 OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). 在这里进行浓缩和重新解释;错误属于我们。 由于其来源是非商业性用途许可证,这一课程是免费的,不出售任何物品,根据同一许可证分享。

更多 Arithmetic

Order of operationsFractionsPercentagesPowers and roots