Commutative, Distributive and Associative Properties

The Distributive Property for 7th Grade

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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:

$54:3= (60-6):3= 60:3-6:3= 20-2=18$

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

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Exercises for the distributive property of division:

Exercise 1

Figure:

Task:

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

He plans to paint it.

Ivan paints at a rate of $7$ 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).

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:

$36:9$

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

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

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

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:

$120:5$

Solution:

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

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

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

$=10+10+4=24$

Answer

$24$

Example 2

Task:

Solve the following division:

$396:3$

Solution:

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

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

We can further break down $90$ as follows:

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

Applying the distributive property of division we get:

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

Answer

$132$

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 $1$

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