The Distributive Property of Division

πŸ†Practice distributive property for seventh grade

The distributive property of division allows us to break down the first term of a division expression into a smaller number. This simplifies the division operation and allows us to solve the exercise without a calculator.

When using the distributive property of division, we begin by breaking down the number being divided by another, the dividend.

For example:

54:3=(60βˆ’6):3=60:3βˆ’6:3=20βˆ’2=1854:3= (60-6):3= 60:3-6:3= 20-2=18

We break down 54 54 into 60βˆ’6 60-6 .
The value remains the same since 60βˆ’6=54 60-6=54
Both 60 60 and 6 6 are divisible by 3 3 and, therefore, the calculation is much easier.

B - The Distributive Property of Division

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Test yourself on distributive property for seventh grade!

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\( 140-70= \)

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Here are some more examples using the distributive property of divisions

Other examples:

85:5=(30+55):5=30:5+55:5=6+11=1785:5= ( 30 + 55):5= 30:5+ 55:5= 6+11=17


104:4=(100+4):4=100:4+4:4=25+1=26104:4 = (100+4):4 = 100:4 + 4:4 = 25+1 = 26


Exercises for the distributive property of division:

Exercise 1

Figure:

Exercise 1 - Figure

Task:

Ivan is building a fence 7X 7X meters high and 30X+4 30X+4 meters long.

He plans to paint it.

Ivan paints at a rate of 7 7 square meters for half an hour. Find the expression for the time it will take Ivan to paint the entire fence (on one side only).

Solution:

First we calculate the area of the fence.

(30x+4)Γ—7x= (30x+4)\times7x=

7xΓ—30x+7xΓ—4= 7x\times30x+7x\times4=

210x2+28x 210x^2+28x

Now we calculate Ivan's painting speed

Speed= 7m212hr=14m2hr \frac{7m^2}{\frac{1}{2}hr}=14\frac{m^2}{hr}

To calculate the time, we will divide the area of the fence by the speed of the paint stroke.

210x2+28x14= \frac{210x^2+28x}{14}=

210x214+28x14= \frac{210x^2}{14}+\frac{28x}{14}=

We reduce by 14 14

15x2+2x 15x^2+2x

Answer:

15x2+2x 15x^2+2x


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Exercise 2

Task:

What expression is 19:8 19:8 equal to ?

Solution:

19:8=(20βˆ’1):8=20:8βˆ’1:8 19:8=\left(20-1\right):8=20:8-1:8

20:8 20:8 Then we subtract 1:81:8

2.5βˆ’0.125=2.3752.5-0.125=2.375

Answer:

2.375 2.375


Exercise 3

Task:

742:4= 742:4=

Solution:

First we break down the number 742 742 into hundreds, tens and ones.

(700+40+2):4= \left(700+40+2\right):4=

After that, we break down 700 700 into hundreds, which we then divide by 4 4

(400+200+100+40+2):4= \left(400+200+100+40+2\right):4=

We convert the numbers into simple fractions.

4004+2004+1004+404+24=\frac{400}{4}+\frac{200}{4}+\frac{100}{4}+\frac{40}{4}+\frac{2}{4}=

We solve the exercise from left to right.

100+50+25+10+12= 100+50+25+10+\frac{1}{2}=

We add everything together.

18512 185\frac{1}{2}

Answer:

18512 185\frac{1}{2}


Do you know what the answer is?

Exercise 4

Task:

74:8= 74:8=

Solution:

74:8=(72+2):8 74:8=\left(72+2\right):8

=728+28=9+14=9.25 =\frac{72}{8}+\frac{2}{8}=9+\frac{1}{4}=9.25

Answer:

914 9\frac{1}{4}


Exercise 5

Task:

354:3=354:3=

Solution:

354:3=(300+54):3 354:3=\left(300+54\right):3

=(300+30+24):3 =\left(300+30+24\right):3

=300:3+30:3+24:3=100+10+8=118 =300:3+30:3+24:3=100+10+8=118

Answer:

118 118


Check your understanding

Review questions

What is the distributive property of division?

The distributive property of division tells us that we can break down the dividend and thus simplify the division with the divisor, let's look at an example:

36:9 36:9

We can break down the first term in the following way:

(18+18):9 \left(18+18\right):9

18:9+18:9=2+2=4 18:9+18:9=2+2=4


How do we apply the distributive property of division, using examples?

Let's see some examples of how to apply the distributive property of division:

Example 1

Task:

Solve the following division:

120:5 120:5

Solution:

To make the division simpler, let's simplify the first term as follows.

(50+50+20):5 \left(50+50+20\right):5

=50:5+50:5+20:5 =50:5+50:5+20:5

=10+10+4=24 =10+10+4=24

Answer

24 24

Example 2

Task:

Solve the following division:

396:3 396:3

Solution:

Let's break down the dividend to simplify the division.

(300+90+6):3 \left(300+90+6\right):3

We can further break down 90 90 as follows:

(300+30+30+30+6):3 \left(300+30+30+30+6\right):3

Applying the distributive property of division we get:

300:3+30:3+30:3+30:3+6:3=100+10+10+10+2=132 300:3+30:3+30:3+30:3+6:3=100+10+10+10+2=132

Answer

132 132


What are the properties of division?

In division there is no commutative property, since in this operation the order of the dividend and the divisor is important, that is, it is not commutative. If we reorder the dividend and the divisor in different ways the result will be different.

For the division there is a neutral element which is 1 1

We can express it as follows: a:1=a a:1=a that is, if we divide a number by the 1 1 it will give us the same number.

Example:

9:1=9 9:1=9


Do you think you will be able to solve it?

examples with solutions for the distributive property of division

Exercise #1

Solve the exercise:

84:4=

Video Solution

Step-by-Step Solution

There are several ways to solve the exercise,

We will present two of them.

In both ways, in the first step we divide the number 84 into 80 and 4.

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

And thus we are left with only the 80.

Β 

From the first method, we will decompose 80 into10Γ—8 10\times8

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

And therefore, we reduce the exercise 104Γ—8 \frac{10}{4}\times8

In fact, we will be left with2Γ—10 2\times10

which is equal to 20

In the second method, we decompose 80 into40+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 them:

20+1=21 20+1=21

And thus we manage to decompose that:844=21 \frac{84}{4}=21

Answer

21

Exercise #2

94+72= 94+72=

Video Solution

Step-by-Step Solution

To facilitate the resolution process, we break down 94 and 72 into smaller numbers. Preferably round numbers

We obtain:

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

Using the associative property, we arrange the exercise in a more comfortable way:

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

Now we obtain the exercise:

160+6=166 160+6=166

Answer

166

Exercise #3

140βˆ’70= 140-70=

Video Solution

Step-by-Step Solution

To facilitate the resolution process, we use the distributive property for 140:

100+40βˆ’70= 100+40-70=

Now we arrange the exercise using the substitution property in a more convenient way:

100βˆ’70+40= 100-70+40=

We solve the exercise from left to right:

100βˆ’70=30 100-70=30

30+40=70 30+40=70

Answer

70

Exercise #4

63βˆ’36= 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 =Β 

27

Answer

27

Exercise #5

133+30= 133+30=

Video Solution

Step-by-Step Solution

To solve the question, we first use the distributive property for 133:

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

Now we use the distributive property for 33:

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

We arrange the exercise in a more comfortable way:

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

We solve the middle exercise:

30+30=60 30+30=60

Now we obtain the exercise:

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

Answer

163

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