The distributive property
Sometimes, an expression will require us to perform both addition and subtraction within our parentheses. Not to worry! The distributive property can simplify these expressions too.
Let's see some examples:
- (X+2)Ć(X+3)=
X2+3X+2X+6=X2+5X+6
- (Xā4)Ć(Xā3)=
X2ā3Xā4X+12=X2ā7X+12
How would you go about using the distributive property in an equation with two sets of parentheses?
First, we multiply the first term of the first parenthesis by the first and second terms of the second parenthesis..
Next, we multiply the second term of the first parenthesis and multiply it by the first and second terms of the second parenthesis.
Remember to place the addition and subtraction signs in the correct places.
Another way to describe the distributive property:
(Z+T)Ć(X+Y)=ZX+ZY+TX+TY
(ZāT)Ć(XāY)=ZXāZYāTX+TY
The distributive property in elementary school
When first learning about the distributive property, we practice only with known whole numbers (without variables or fractions) in order to understand the idea of breaking down a larger number into smaller numbers. In this first stage, we use the distributive property mainly to simplify calculation, especially with large numbers.
For example:
3Ć102=3Ć(100+2)=300+6=306
7Ć96=7Ć(100ā4)=700ā28=672
By this point, most students have already mastered long addition and subtraction, but they might have less experience multiplying large numbers. The distributive property helps them to solve these problems by reducing into simpler multiplications equations.
Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
The distributive property in middle school
In middle school, the distributive property gets more interesting. Now, we will start to use not just whole numbers, but variables and exponents too!
For example:
- (X+5)Ć(X+6)=
X2+6X+5X+30=X2+11X+30
- (Xā7)Ć(Xā8)=
X2ā8Xā7X+56=X2ā15X+56
Other properties
As we mentioned earlier, there are other rules and properties out there that help us to simplify algebraic expressions. In this section we will briefly look at two of them: the associative property and the commutative property.
Do you know what the answer is?
The associative property
The associative property allows us to group several terms of an equation together without changing the final results, by moving the parentheses. However, we can only use this property to solve addition or multiplication exercises.
For example:
- (10+2)+8=
10+(2+8)=10+2+8=20
- 2Ć(3Ć6)=
(2Ć3)Ć6=2Ć3Ć6=36
To learn more about the associative property, you can read the following: The Associative Property.
The commutative property
The commutative property allows us to change the order of the terms in an equation without changing the outcome of the equation. Like the associative property, the commutative property can only be used for addition and multiplication.
Let us look at some examples:
- 2+6=6+2=8
- 3Ć4=4Ć3=12
To learn more about the commutative property, you can read the following: The Commutative Property.
Example exercises for seventh graders
Exercise 1
Task:
Using the distributive property, solve the following:
- 294:3=
- 105Ć4=
- 505:5=
- 207Ć5=
- 168:8=
Solutions:
- 294:3=(300ā6):3=300:3ā6:3=100ā2=98
- 105Ć4=(100+5)Ć4=100Ć4+5Ć4=400+20=420
- 505:5=(500+5):5=500:5+5:5=100+1=101
- 207Ć5=(200+7)Ć5=200Ć5+7Ć5=1000+35=1035
- 168:8=(160+8):8=160:8+8:8=20+1=21
Exercise 2
Task:
Three hundred fifty-one students from a high school were divided into nine equal groups.
How many students are in each group?
Solve the problem using the distributive property.
Solution:
We begin by expressing the problem numerically:
351:9=(360ā9):9=360:9ā9:9=40ā1=39
Answer:
There are 39 students in each group.
Do you think you will be able to solve it?
Exercise 3
Task:
Dani bought 15 boxes. In each box there were 9 pieces of candy.
How many pieces of candy did Dani buy in total?
Use the distributive property to solve the problem.
Solution:
We begin by expressing the problem numerically:
15Ć9=(10+5)Ć9=10Ć9+5Ć9=90+45=135
Answer:
Dani bought 135 pieces of candy.
Exercise 4
Task:
Isabel has packed 246 notebooks into 6 equal packages.
How many notebooks has Isabel put in each package?
Use the distributive property to solve the problem.
Solution:
We begin by expressing the problem numerically:
246:6=(240+6):6=240:6+6:6=40+1=41
Answer:
Isabel has put 41 notebooks in each packet.
Exercise 5
Task:
A mother had 894 euros. She divided the money equally among her three children.
How much money did each child receive?
Use the distributive property to solve the problem.
Solution:
We begin by expressing the problem numerically:
894:3=(900ā6):3=900:3ā6:3=300ā2=298
Answer:
Each child received 298 euros.
Exercise 6
Task:
Solve the following:
93:3=?
Solution:
We simplify the number 93 into 4 smaller numers to make it easier for us to divide it by 3.
For example 30:3=10
93:3=?
(30+30+30+3):3=
+30:3=10
+30:3=10
+30:3=10
+3:3=1
Then we will add the results and we will get:
93:3=31
Answer:
31
Do you know what the answer is?
Exercise 7
Task:
Solve the following:
=90:5
Solution:
We simplify the number 90 into 3 smaller numbers:
(50,20,20)
90=5ā+20+20
Then we divide each of them by 5 and add the three results.
+50:5=10
+20:5=4
+20:5=4
Which gives us:
90:5=10+4+4=18
Answer:
18
Exercise 8
Task:
Solve the following:
=72:18
Solution:
We simplify the number 72 into two smaller numbers (36+36) and then divide each of them by 18.
72:18=
(36+36):18=
+36:18=2
+36:18=2
72:18=4
4
Answer:
4
Exercise 9
Task:
Solve the following:
(40+70+35ā7)Ć9=
Solution:
(40+70+35ā7)Ć9=
40Ć9+70Ć9+35Ć9ā7Ć9=
=360+630+(30+5)9ā63
=360+630+270+45ā63=1242
Answer:
1242
Exercise 10
Task:
Solve the following:
(35+4)Ć(10+5)=
Solution:
(35+4)(10+5)=35Ć10+35Ć5+4Ć10+4Ć5
=350+175+40+20=585
Answer:
585
Do you think you will be able to solve it?
Exercise 11
Task:
Solve the following:
(7x+3)Ć(10+4)=
Solution:
(7X+3)(10+4)=7XĆ10+7XĆ4+3Ć10+3Ć4
=70X+28X+30+12=98X+42=238,ā42
98X=196,:98
X=98196ā=2
Answer:
2
More practice
- 187Ć(8ā5)=
- 32āĆ(12+0ā5)=
- 5Ć(221ā+161ā+43ā)=
- (10+5+18)Ć4=
- (5.5ā0.8)Ć5=
- 340:(12ā7)=
- (29ā4):5=
- 15ā:(6+1ā5)=
- 18:(5+7+4)=
- (71āā31ā):4=
- 97Ć12=
- 3Ć36=
- 120:97=
- 8:21ā=
- 151Ć23=
FAQs about the distributive property
What is the distributive property?
The distributive property is a method used to simplify expressions into smaller, more manageable pieces.
How is the distributive property used?
In an equation, we use the distributive property to break down a large number into two or more smaller numbers (using addition and subtraction), and then by distributing the multiplication.
Example
- 20Ć8Ć7=20+8Ć7=20Ć7+8Ć7=140+56=196
Can we use the distributive property in division?
Of course we can! In an expression with division, we break down the numerator into smaller numbers (using addition and subtraction), and then the division is distributed.
Example
- 150:6=120+30:6=120:6+30:6=20+5=25
If you found this article helpful, you may also be interested in the following:
For a wide range of mathematics articles, visit Tutorela website.
Do you know what the answer is?
Examples with solutions for The Distributive Property for 7th Grade
Exercise #1
Video Solution
Step-by-Step Solution
There are several ways to solve the following exercise,
We will present two of them.
In both ways, we begin by decomposing the number 84 into smaller units; 80 and 4.
44ā=1
Subsequently we are left with only the 80.
Continuing on with the first method, we will then further decompose 80 into smaller units; 10Ć8
We know that:48ā=2
And therefore, we are able to reduce the exercise as follows: 410āĆ8
Eventually we are left with2Ć10
which is equal to 20
In the second method, we decompose 80 into the following smaller units:40+40
We know that: 440ā=10
And therefore: 440+40ā=480ā=20=10+10
which is also equal to 20
Now, let's remember the 1 from the first step and add it in to our above answer:
20+1=21
Thus we are left with the following solution:484ā=21
Answer
Exercise #2
Solve the following exercise
?=24:12
Video Solution
Step-by-Step Solution
We will use the distributive property of division and split the number 24 into a sum of 12 and 12, which makes the division operation easier and allows us to solve the exercise without a calculator.
Note - it's best to choose to split the number based on knowledge of multiples. In this case of the number 12 because we need to divide by 12.
Reminder - The distributive property of division actually allows us to split the larger term in a division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator
We will use the formula of the distributive property
Ā (a+b):c=a:c+b:cĀ
24:12=(12+12):12
(12+12):12=12:12+12:12
12:12+12:12=1+1
1+1=2
Therefore the answer is section a - 2.
Answer
Exercise #3
Solve the following exercise
?=93:3
Video Solution
Step-by-Step Solution
We will use the distributive property of division and split the number 93 into a sum of 90 and 3, which makes the division operation easier and allows us to solve the exercise without a calculator.
Note - it's best to choose to split the number based on knowledge of multiples. In this case, we use 3 because we need to divide by 3. Additionally, in this case, splitting by tens and ones is suitable and makes the division operation easier.
Reminder - The distributive property of division essentially allows us to split the larger term in the division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator
We will use the formula of the distributive property
Ā (a+b):c=a:c+b:cĀ
93:3=(90+3):3
(90+3):3=90:3+3:3
90:3+3:3=30+1
30+1=31
Therefore, the answer is option B - 31.
Answer
Exercise #4
Video Solution
Step-by-Step Solution
In order to simplify the calculation , we first break down 94 and 72 into smaller and preferably round numbers.
We obtain the following exercise:
90+4+70+2=
Using the associative property, we then rearrange the exercise to be more functional.
90+70+4+2=
We solve the exercise in the following way, first the round numbers and then the small numbers.
90+70=160
4+2=6
Which results in the following exercise:
160+6=166
Answer
Exercise #5
Video Solution
Step-by-Step Solution
To solve the problem, first we will use the distributive property on the two numbers:
(60+3)-(30+6)
Now, we will use the substitution property to arrange the exercise in the way that is most convenient for us to solve:
60-30+3-6
It is important to pay attention that when we open the second parentheses, the minus sign moved to the two numbers inside.
30-3 =Ā
27
Answer