# The Distributive Property of Division

🏆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:

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

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.

## Test yourself on distributive property for seventh grade!

$$140-70=$$

Here are some more examples using the distributive property of divisions

Other examples:

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

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

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## Exercises for the distributive property of division:

### Exercise 1

Figure:

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 $14$

$15x^2+2x$

$15x^2+2x$

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### Exercise 2

What expression is $19:8$ equal to ?

Solution:

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

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

$2.5-0.125=2.375$

$2.375$

### Exercise 3

$742:4=$

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.

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

$185\frac{1}{2}$

$185\frac{1}{2}$

Do you know what the answer is?

### Exercise 4

$74:8=$

Solution:

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

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

$9\frac{1}{4}$

### Exercise 5

$354:3=$

Solution:

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

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

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

$118$

## 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:

$36:9$

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

Solve the following division:

$120:5$

Solution:

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

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

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

$=10+10+4=24$

$24$

Example 2

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:

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

$132$

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:

$9:1=9$

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=

### 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.

$\frac{4}{4}=1$

And thus we are left with only the 80.

From the first method, we will decompose 80 into$10\times8$

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

And therefore, we reduce the exercise $\frac{10}{4}\times8$

In fact, we will be left with$2\times10$

which is equal to 20

In the second method, we decompose 80 into$40+40$

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

And therefore: $\frac{40+40}{4}=\frac{80}{4}=20=10+10$

which is also equal to 20

Now, let's remember the 1 from the first step and add them:

$20+1=21$

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

21

### Exercise #2

$94+72=$

### Step-by-Step Solution

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

We obtain:

$90+4+70+2=$

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

$90+70+4+2=$

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

$90+70=160$

$4+2=6$

Now we obtain the exercise:

$160+6=166$

166

### Exercise #3

$140-70=$

### Step-by-Step Solution

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

$100+40-70=$

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

$100-70+40=$

We solve the exercise from left to right:

$100-70=30$

$30+40=70$

70

### Exercise #4

$63-36=$

### 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

27

### Exercise #5

$133+30=$

### Step-by-Step Solution

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

$(100+33)+30=$

Now we use the distributive property for 33:

$100+30+3+30=$

We arrange the exercise in a more comfortable way:

$100+30+30+3=$

We solve the middle exercise:

$30+30=60$

Now we obtain the exercise:

$100+60+3=163$