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Distributive Property

Simplify expressions using the 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, as shown in .

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.

$4(x+8)$

Try it.

$3(a+9)$

Solution

3a + 27

Try it.

$8(4y+9)$

Try it.

$9(3w+7)$

Solution

27w + 63

Try it.

$6(c-13)$

Try it.

$7(y-13)$

Solution

7y − 91

Try it.

$7(3p-8)$

Try it.

$5(7u-4)$

Solution

35u − 20

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.

$r(s-18)$

Try it.

$u(v-10)$

Solution

uv − 10u

Try it.

$(y+4)p$

Try it.

$(a+7)x$

Solution

ax + 7x

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.

$-3(x-6)$

Try it.

$-4(q-7)$

Solution

−4q + 28

Try it.

$-9(3a-7)$

Try it.

$-6(7x-8)$

Solution

−42x + 48

Try it.

$-(r+7)$

Try it.

$-(q+11)$

Solution

−q − 11

Try it.

$-(3x-7)$

Try it.

$-(5p-4)$

Solution

−5p + 4

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.

$8-(r-7)$

Solution

−r + 15

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

ⓐ $\ 6(4v+7)$
ⓑ $\ 6\cdot 4v+6\cdot 7$

Try it.

If $u=-1,$ evaluate

ⓐ $\ 8(5u+12)$
ⓑ $\ 8\cdot 5u+8\cdot 12$

Solution

ⓐ 56
ⓑ 56

Try it.

If $n=\frac{2}{3},$ evaluate

ⓐ $\ 3(n+\frac{5}{6})$
ⓑ $\ 3\cdot n+3\cdot \frac{5}{6}$

Try it.

If $y=\frac{3}{4},$ evaluate

ⓐ $\ 4(y+\frac{3}{8})$
ⓑ $\ 4\cdot y+4\cdot \frac{3}{8}$

Solution

ⓐ $\ \frac{9}{2}$
ⓑ $\ \frac{9}{2}$

Try it.

If $y=\frac{7}{12},$ evaluate

ⓐ $\ -3(4y+15)$
ⓑ $\ -3\cdot 4y+(-3)\cdot 15$

Try it.

If $p=\frac{23}{30},$ evaluate

ⓐ $\ -6(5p+11)$
ⓑ $\ -6\cdot 5p+(-6)\cdot 11$

Solution

ⓐ −89
ⓑ −89

Try it.

If $m=0.4,$ evaluate

ⓐ $\ -10(3m-0.9)$
ⓑ $\ -10\cdot 3m-(-10)(0.9)$

Try it.

If $n=0.75,$ evaluate

ⓐ $\ -100(5n+1.5)$
ⓑ $\ -100\cdot 5n+(-100)(1.5)$

Solution

ⓐ −525
ⓑ −525

Try it.

If $y=-25,$ evaluate

ⓐ $\ -(y-25)$
ⓑ $\ -y+25$

Try it.

If $w=-80,$ evaluate

ⓐ $\ -(w-80)$
ⓑ $\ -w+80$

Solution

ⓐ 160
ⓑ 160

Try it.

If $p=0.19,$ evaluate

ⓐ $\ -(p+0.72)$
ⓑ $\ -p-0.72$

Try it.

If $q=0.55,$ evaluate

ⓐ $\ -(q+0.48)$
ⓑ $\ -q-0.48$

Solution

ⓐ −1.03
ⓑ −1.03

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

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Multiply: $3(0.25).$
If you missed this problem, review

Otkrij odgovor

$0.75$
Simplify: $10-(-2)(3).$
If you missed this problem, review

Otkrij odgovor

$16$
Combine like terms: $9y+17+3y-2.$
If you missed this problem, review .

Otkrij odgovor

$12y+15$
Simplify: $3(x+4).$

Otkrij odgovor

$3(x+4)$
Distribute. $3\cdot x+3\cdot 4$
Multiply. $3x+12$
Simplify: $4(x+2).$

Otkrij odgovor

4x + 8
Simplify: $6(x+7).$

Otkrij odgovor

6x + 42
Simplify: $6(5y+1).$

Otkrij odgovor

Distribute.
Multiply.
Simplify: $9(3y+8).$

Otkrij odgovor

27y + 72
Simplify: $5(5w+9).$

Otkrij odgovor

25w + 45
Simplify: $2(x-3).$

Otkrij odgovor

Distribute.
Multiply.
Simplify: $7(x-6).$

Otkrij odgovor

7x − 42
Simplify: $8(x-5).$

Otkrij odgovor

8x − 40
Simplify: $\frac{3}{4}(n+12).$

Otkrij odgovor

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

Otkrij odgovor

$\frac{2}{5}p+4$
Simplify: $\frac{3}{7}(u+21).$

Otkrij odgovor

$\frac{3}{7}u+9$
Simplify: $8(\frac{3}{8}x+\frac{1}{4}).$

Otkrij odgovor

Distribute.
Multiply.
Simplify: $6(\frac{5}{6}y+\frac{1}{2}).$

Otkrij odgovor

5y + 3
Simplify: $12(\frac{1}{3}n+\frac{3}{4}).$

Otkrij odgovor

4n + 9
Simplify: $100(0.3+0.25q).$

Otkrij odgovor

Distribute.
Multiply.
Simplify: $100(0.7+0.15p).$

Otkrij odgovor

70 + 15p
Simplify: $100(0.04+0.35d).$

Otkrij odgovor

4 + 35d
Simplify: $m(n-4).$

Otkrij odgovor

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.
Simplify: $r(s-2).$

Otkrij odgovor

rs − 2r
Simplify: $y(z-8).$

Otkrij odgovor

yz − 8y
Simplify: $(x+8)p.$

Otkrij odgovor

Distribute.
Simplify: $(x+2)p.$

Otkrij odgovor

xp + 2p
Simplify: $(y+4)q.$

Otkrij odgovor

yq + 4q
Simplify: $-2(4y+1).$

Otkrij odgovor

Distribute.
Simplify.
Simplify: $-3(6m+5).$

Otkrij odgovor

−18m − 15
Simplify: $-6(8n+11).$

Otkrij odgovor

−48n − 66
Simplify: $-11(4-3a).$

Otkrij odgovor

Distribute.
Multiply.
Simplify.
You could also write the result as $33a-44.$ Do you know why?
Simplify: $-5(2-3a).$

Otkrij odgovor

−10 + 15a
Simplify: $-7(8-15y).$

Otkrij odgovor

−56 + 105y
Simplify: $-(y+5).$

Otkrij odgovor

Multiplying by −1 results in the opposite.
Distribute.
Simplify.
Simplify.
Simplify: $-(z-11).$

Otkrij odgovor

−z + 11
Simplify: $-(x-4).$

Otkrij odgovor

−x + 4
Simplify: $8-2(x+3).$

Otkrij odgovor

Distribute.
Multiply.
Combine like terms.
Simplify: $9-3(x+2).$

Otkrij odgovor

−3x + 3
Simplify: $7x-5(x+4).$

Otkrij odgovor

2x − 20
Simplify: $4(x-8)-(x+3).$

Otkrij odgovor

Distribute.
Combine like terms.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Distributive Property

Simplify expressions using the distributive property
Evaluate expressions using the distributive property
If

Questions people ask

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.

Pokušaj sam.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Više u Arithmetic

Order of operations Fractions Percentages Powers and roots

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

ⓑ
	\(\ -2\cdot 4y+(-2)\cdot 1\)

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

ⓑ
	\(\ -y-5\)

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

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

ⓐ
	\(6(5y+1)\)

Simplify in the parentheses.	\(6(51)\)
Multiply.	\(306\)

ⓐ
	\(\ -2(4y+1)\)

Simplify in the parentheses.	\(-2(13)\)
Multiply.	\(-26\)

ⓐ
	\(\ \text{-}(y+5)\)

Add in the parentheses.	\(-(40)\)
Simplify.	\(-40\)


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