Solving algebraic equations is made easier by understanding some basic rules and properties. A few examples of properties that we will learn to use in the seventh grade are: the distributive, associative and commutative properties. These properties get learned and relearned throughout our time in school, each time adding new layers to or understanding. Today we will focus on the distributive property. We will go into depth on what it is and how to use it, and we will briefly get to know the associative and commutative properties as well.

What is the distributive property?

The distributive property is a method to simplify multiplication and division exercises. Essentially, it breaks down expressions into smaller, easier to manage terms.

Let's see some examples:

  • 6×26=6×(20+6)=120+36=1566 \times 26 = 6 \times (20 + 6) = 120 + 36 = 156
  • 7×32=7×(30+2)=210+14=2247 \times 32 = 7 \times (30 + 2) = 210 + 14 = 224
  • 104:4=(100+4):4=100:4+4:4=25+1=26104:4 = (100+4):4 = 100:4 + 4:4 = 25+1 = 26

If we look at the following examples, we can see that we have broken down the larger number into several smaller numbers that are more manageable. The value is the same as before, but now we can distribute a complex operation into several easy operations.

The distributive property can be described as:

Z×(X+Y)=ZX+ZYZ \times (X + Y) = ZX + ZY

Z×(XY)=ZXZYZ \times (X - Y) = ZX - ZY

A - The Distributive Property for Seventh Graders

Suggested Topics to Practice in Advance

  1. The commutative property
  2. The Commutative Property of Addition
  3. The Commutative Property of Multiplication

Practice The Distributive Property for 7th Grade

Examples with solutions for The Distributive Property for 7th Grade

Exercise #1

94+72= 94+72=

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= 90+4+70+2=

Using the associative property, we then rearrange the exercise to be more functional.

90+70+4+2= 90+70+4+2=

We solve the exercise in the following way, first the round numbers and then the small numbers.

90+70=160 90+70=160

4+2=6 4+2=6

Which results in the following exercise:

160+6=166 160+6=166

Answer

166

Exercise #2

6336= 63-36=

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 = 

27

Answer

27

Exercise #3

133+30= 133+30=

Video Solution

Step-by-Step Solution

In order to solve the question, we first use the distributive property for 133:

(100+33)+30= (100+33)+30=

We then use the distributive property for 33:

100+30+3+30= 100+30+3+30=

We rearrange the exercise into a more practical form:

100+30+30+3= 100+30+30+3=

We solve the middle exercise:

30+30=60 30+30=60

Which results in the final exercise as seen below:

100+60+3=163 100+60+3=163

Answer

163

Exercise #4

14070= 140-70=

Video Solution

Step-by-Step Solution

In order to simplify the resolution process, we begin by using the distributive property for 140:

100+4070= 100+40-70=

We then rearrange the exercise using the substitution property into a more practical form:

10070+40= 100-70+40=

Lastly we solve the exercise from left to right:

10070=30 100-70=30

30+40=70 30+40=70

Answer

70

Exercise #5

Solve the exercise:

84:4=

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.

44=1 \frac{4}{4}=1

Subsequently we are left with only the 80.

 

Continuing on with the first method, we will then further decompose 80 into smaller units; 10×8 10\times8

We know that:84=2 \frac{8}{4}=2

And therefore, we are able to reduce the exercise as follows: 104×8 \frac{10}{4}\times8

Eventually we are left with2×10 2\times10

which is equal to 20

In the second method, we decompose 80 into the following smaller units:40+40 40+40

We know that: 404=10 \frac{40}{4}=10

And therefore: 40+404=804=20=10+10 \frac{40+40}{4}=\frac{80}{4}=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 20+1=21

Thus we are left with the following solution:844=21 \frac{84}{4}=21

Answer

21

Exercise #6

Which equation is the same as the following?

13×29 13\times29

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

(10+3)×(301)=13×29 (10+3)\times(30-1)=13\times29

b.

10×3×30×1=30×30×1=900 10\times3\times30\times1=30\times30\times1=900

c.

(10×3)×30=13×30 (10\times3)\times30=13\times30

d.

10×3+29=30+29=59 10\times3+29=30+29=59

Therefore, the answer is option A.

Answer

(10+3)×(301) (10+3)\times(30-1)

Exercise #7

Which equation is the same as the following?

36×4 36\times4

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

36+4=40 36+4=40

b.

4×(30+6)=4×36 4\times(30+6)=4\times36

c.

404+4=36+4=40 40-4+4=36+4=40

d.

4×30+6=120+6=126 4\times30+6=120+6=126

Therefore, the answer is option B.

Answer

4×(30+6) 4\times(30+6)

Exercise #8

Which equation is the same as the following?

3×83 3\times83

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

3×8×3=24×3=72 3\times8\times3=24\times3=72

b.

(2+1)×(80+3)=3×83 (2+1)\times(80+3)=3\times83

c.

3+(80+3)=3+83 3+(80+3)=3+83

d.

3+83=86 3+83=86

Therefore, the answer is option B.

Answer

(2+1)×(80+3) (2+1)\times(80+3)

Exercise #9

Which equation is the same as the following?

14×42 14\times42

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

10+4+40+2=14+42=36 10+4+40+2=14+42=36

b.

(10×4)+(40×2)=40+80 (10\times4)+(40\times2)=40+80

c.

(10+4)×(40+2)=14×42 (10+4)\times(40+2)=14\times42

d.

10×4×40×2=40×40×2=160×2=360 10\times4\times40\times2=40\times40\times2=160\times2=360

Therefore, the answer is option C.

Answer

(10+4)×(40+2) (10+4)\times(40+2)

Exercise #10

Which equation is the same as the following?

160×6 160\times6

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation inside parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

160+6=166 160+6=166

b.

(100×60)×6=6,000×6 (100\times60)\times6=6,000\times6

c.

(100+60)+(3+3)=160+6 (100+60)+(3+3)=160+6

d.

(100+60)×(3+3)=160×6 (100+60)\times(3+3)=160\times6

Therefore, the answer is option D.

Answer

(100+60)×(3+3) (100+60)\times(3+3)

Exercise #11

Which equation is the same as the following?

34×11 34\times11

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

(30+4)+11=34+11 (30+4)+11=34+11

b.

30×4×11=120×11=1,320 30\times4\times11=120\times11=1,320

c.

(30+4)+10+1=34+11 (30+4)+10+1=34+11

d.

(30+4)×(10+1)=34×11 (30+4)\times(10+1)=34\times11

Therefore, the answer is option D.

Answer

(30+4)×(10+1) (30+4)\times(10+1)

Exercise #12

Which equation is the same as the following?

39×19 39\times19

Video Solution

Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

(30×9)+(10×9)=270+90 (30×9)+(10×9)= 270+90

b.

(30+9)×(10×9)=39×90 (30+9)×(10×9)= 39×90

c.

(401)×(201)=39×19 (40-1)×(20-1)= 39×19

d.

(401)+(20×1)=39+20 (40-1)+(20×1)= 39+20

Therefore, the answer is option C.

Answer

(401)×(201) (40-1)\times(20-1)

Exercise #13

4×53= 4\times53=

Video Solution

Step-by-Step Solution

To make solving easier, we break down 53 into more comfortable numbers, preferably round ones.

We obtain:

4×(50+3)= 4\times(50+3)=

We multiply 2 by each of the terms inside the parentheses:

(4×50)+(4×3)= (4\times50)+(4\times3)=

We solve the exercises inside the parentheses and obtain:

200+12=212 200+12=212

Answer

212

Exercise #14

11×34= 11\times34=

Video Solution

Step-by-Step Solution

To make solving easier, we break down 11 into more comfortable numbers, preferably round ones.

We obtain:

(10+1)×34= (10+1)\times34=

We multiply 34 by each of the terms in parentheses:

(34×10)+(34×1)= (34\times10)+(34\times1)=

We solve the exercises in parentheses and obtain:

340+34=374 340+34=374

Answer

374

Exercise #15

6×29= 6\times29=

Video Solution

Step-by-Step Solution

To make solving easier, we break down 29 into more comfortable numbers, preferably round ones.

We obtain:

6×(301)= 6\times(30-1)=

We multiply 6 by each of the terms in parentheses:

(6×30)(6×1)= (6\times30)-(6\times1)=

We solve the exercises in parentheses and obtain:

1806=174 180-6=174

Answer

174

Topics learned in later sections

  1. The Distributive Property
  2. The Distributive Property of Division
  3. The Distributive Property in the Case of Multiplication
  4. The commutative properties of addition and multiplication, and the distributive property
  5. The Associative Property
  6. The Associative Property of Addition
  7. The Associative Property of Multiplication
  8. Advanced Arithmetic Operations
  9. Subtracting Whole Numbers with Addition in Parentheses
  10. Division of Whole Numbers Within Parentheses Involving Division
  11. Subtracting Whole Numbers with Subtraction in Parentheses
  12. Division of Whole Numbers with Multiplication in Parentheses