The Distributive Property of Division

Here are some more examples using the distributive property of divisions

Other examples:

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

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

If you are interested in this article you may also be interested in the following articles:

• Equivalent expressions / Equivalent algebraic expressions.
• The associative property
• The distributive property for students of 1st grade of ESO (ESO)
• The distributive property in the case of multiplication
• The distributive property: extension

For a wide range of mathematics articles visit Tutorela's blog

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).

Solution:

First we calculate the area of the fence.

$(30x+4)\times7x=$

$7x\times30x+7x\times4=$

$210x^2+28x$

Now we calculate Ivan's painting speed

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

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

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

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

We reduce by $14$

$15x^2+2x$

Answer:

$15x^2+2x$

Task:

What expression is $19:8$ equal to ?

Solution:

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

$20:8$ Then we subtract $1:8$

$2.5-0.125=2.375$

Answer:

$2.375$

Task:

$742:4=$

Solution:

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

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

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

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

We convert the numbers into simple fractions.

$\frac{400}{4}+\frac{200}{4}+\frac{100}{4}+\frac{40}{4}+\frac{2}{4}=$

We solve the exercise from left to right.

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

We add everything together.

$185\frac{1}{2}$

Answer:

$185\frac{1}{2}$

Task:

$74:8=$

Solution:

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

$=\frac{72}{8}+\frac{2}{8}=9+\frac{1}{4}=9.25$

Answer:

$9\frac{1}{4}$

Task:

$354:3=$

Solution:

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

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

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

Answer:

$118$

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:

$36:9$

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.

Example:

$9:1=9$