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.

今度はあなたの番 計算機では解けませんが、計算可能です。下の一つを試してみましょう。もしくは自分で入力してください。

⌨ 入力 ✎ 描いて

数学の問題

π 解く

練習 (40)

まず紙で解いてみてください。答えを表示して確認できます。検証済みの問題は、ソルバーで全ステップを開けます。

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

   ステップごとに解く ヒント

   答えを表示

   \(0.75\)

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

   ステップごとに解く ヒント

   答えを表示

   \(16\)

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

   ステップごとに解く ヒント

   答えを表示

   \(12y+15\)

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

   ステップごとに解く ヒント

   答えを表示

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

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

   ステップごとに解く ヒント

   答えを表示

   4x + 8

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

   ステップごとに解く ヒント

   答えを表示

   6x + 42

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

   ステップごとに解く ヒント

   答えを表示

   Distribute.
   Multiply.

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

   ステップごとに解く ヒント

   答えを表示

   27y + 72

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

   ステップごとに解く ヒント

   答えを表示

   25w + 45

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Multiply.

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

    ステップごとに解く ヒント

    答えを表示

    7x − 42

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

    ステップごとに解く ヒント

    答えを表示

    8x − 40

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Simplify.

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

    ステップごとに解く ヒント

    答えを表示

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

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

    ステップごとに解く ヒント

    答えを表示

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

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Multiply.

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

    ステップごとに解く ヒント

    答えを表示

    5y + 3

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

    ステップごとに解く ヒント

    答えを表示

    4n + 9

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Multiply.

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

    ステップごとに解く ヒント

    答えを表示

    70 + 15p

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

    ステップごとに解く ヒント

    答えを表示

    4 + 35d

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

    ステップごとに解く ヒント

    答えを表示

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

    ステップごとに解く ヒント

    答えを表示

    rs − 2r

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

    ステップごとに解く ヒント

    答えを表示

    yz − 8y

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.

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

    ステップごとに解く ヒント

    答えを表示

    xp + 2p

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

    ステップごとに解く ヒント

    答えを表示

    yq + 4q

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Simplify.

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

    ステップごとに解く ヒント

    答えを表示

    −18m − 15

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

    ステップごとに解く ヒント

    答えを表示

    −48n − 66

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Multiply.
    Simplify.

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

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

    ステップごとに解く ヒント

    答えを表示

    −10 + 15a

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

    ステップごとに解く ヒント

    答えを表示

    −56 + 105y

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

    ステップごとに解く ヒント

    答えを表示

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

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

    ステップごとに解く ヒント

    答えを表示

    −z + 11

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

    ステップごとに解く ヒント

    答えを表示

    −x + 4

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Multiply.
    Combine like terms.

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

    ステップごとに解く ヒント

    答えを表示

    −3x + 3

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

    ステップごとに解く ヒント

    答えを表示

    2x − 20

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

    ステップごとに解く ヒント

    答えを表示

    Distribute.
    Combine like terms.

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ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

\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.

どうやって: Distributive Property

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

よくある質問

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.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). ここでは要約して説明し直しています。誤りは私たちの責任です。 ソースは非商用利用のためにライセンスされているため、このレッスンは無料で、売ることはなく、同じライセンスの下で共有されている。

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