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.

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Matteproblemet

π Løs

Øvelse (40)

Prøv hver enkelt på papir først. Vis svaret for å sjekke. Verifiserte kan åpnes i løseren for hvert steg.

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

   Arbeid gjennom det Tips

   Vis svaret

   \(0.75\)

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

   Arbeid gjennom det Tips

   Vis svaret

   \(16\)

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

   Arbeid gjennom det Tips

   Vis svaret

   \(12y+15\)

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

   Arbeid gjennom det Tips

   Vis svaret

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

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

   Arbeid gjennom det Tips

   Vis svaret

   4x + 8

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

   Arbeid gjennom det Tips

   Vis svaret

   6x + 42

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

   Arbeid gjennom det Tips

   Vis svaret

   Distribute.
   Multiply.

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

   Arbeid gjennom det Tips

   Vis svaret

   27y + 72

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

   Arbeid gjennom det Tips

   Vis svaret

   25w + 45

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Multiply.

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

    Arbeid gjennom det Tips

    Vis svaret

    7x − 42

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

    Arbeid gjennom det Tips

    Vis svaret

    8x − 40

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Simplify.

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

    Arbeid gjennom det Tips

    Vis svaret

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

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

    Arbeid gjennom det Tips

    Vis svaret

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

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Multiply.

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

    Arbeid gjennom det Tips

    Vis svaret

    5y + 3

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

    Arbeid gjennom det Tips

    Vis svaret

    4n + 9

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Multiply.

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

    Arbeid gjennom det Tips

    Vis svaret

    70 + 15p

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

    Arbeid gjennom det Tips

    Vis svaret

    4 + 35d

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

    Arbeid gjennom det Tips

    Vis svaret

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

    Arbeid gjennom det Tips

    Vis svaret

    rs − 2r

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

    Arbeid gjennom det Tips

    Vis svaret

    yz − 8y

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.

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

    Arbeid gjennom det Tips

    Vis svaret

    xp + 2p

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

    Arbeid gjennom det Tips

    Vis svaret

    yq + 4q

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Simplify.

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

    Arbeid gjennom det Tips

    Vis svaret

    −18m − 15

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

    Arbeid gjennom det Tips

    Vis svaret

    −48n − 66

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Multiply.
    Simplify.

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

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

    Arbeid gjennom det Tips

    Vis svaret

    −10 + 15a

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

    Arbeid gjennom det Tips

    Vis svaret

    −56 + 105y

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

    Arbeid gjennom det Tips

    Vis svaret

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

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

    Arbeid gjennom det Tips

    Vis svaret

    −z + 11

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

    Arbeid gjennom det Tips

    Vis svaret

    −x + 4

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Multiply.
    Combine like terms.

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

    Arbeid gjennom det Tips

    Vis svaret

    −3x + 3

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

    Arbeid gjennom det Tips

    Vis svaret

    2x − 20

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

    Arbeid gjennom det Tips

    Vis svaret

    Distribute.
    Combine like terms.

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Symboler som brukes her

Trykk på et symbol for den fulle definisjonen, et bilde og hva hver bokstav i det betyr.

\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.

Hvordan: Distributive Property

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

Spørsmål folk stiller

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.

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