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:**

$Z \times (X + Y) = ZX + ZY$

$Z \times (X - Y) = ZX - ZY$

## The distributive property

Sometimes, an expression will require us to perform *both* addition and subtraction within our parentheses. Not to worry! The distributive property can simplify these expressions too.

**Let's see some examples:**

- $(X + 2) \times (X + 3) =$

$X² + 3X + 2X + 6 = X² + 5X + 6$

- $(X - 4) \times (X - 3) =$

$X² - 3X - 4X + 12 = X² - 7X + 12$

How would you go about using the distributive property in an equation with* two sets* of parentheses?

First, we multiply the **first term** of the first parenthesis by the first *and* second terms of the second parenthesis..

Next, we multiply the **second term** of the first parenthesis and multiply it by the first *and* second terms of the second parenthesis.

**Remember** to place the addition and subtraction signs in the correct places.

**Another way to describe the distributive property:**

$(Z + T) \times (X + Y) = ZX + ZY + TX + TY$

$(Z - T) \times (X - Y) = ZX - ZY - TX + TY$

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## The distributive property in elementary school

When first learning about the distributive property, we practice only with known whole numbers (without variables or fractions) in order to understand the idea of breaking down a larger number into smaller numbers. In this first stage, we use the distributive property mainly to simplify calculation, especially with large numbers.

**For example:**

$3\times 102=3\times(100+2)=300+6=306$

$7\times 96=7\times(100-4)=700-28=672$

By this point, most students have already mastered long addition and subtraction, but they might have less experience multiplying large numbers. The distributive property helps them to solve these problems by reducing into simpler multiplications equations.

## The distributive property in middle school

In middle school, the distributive property gets more interesting. Now, we will start to use not just whole numbers, but variables and exponents too!

**For example:**

- $(X + 5) \times (X + 6) =$

$X² + 6X + 5X+ 30 = X² + 11X + 30$

- $(X- 7) \times (X- 8) =$

$X² - 8X - 7X + 56 = X² - 15X + 56$

Do you know what the answer is?

## Other properties

As we mentioned earlier, there are other rules and properties out there that help us to simplify algebraic expressions. In this section we will briefly look at two of them: the **associative** property and the **commutative** property.

### The associative property

The **associative property** allows us to group several terms of an equation together without changing the final results, by moving the parentheses. However, we can *only* use this property to solve addition or multiplication exercises.

**For example:**

- $(10 + 2) + 8 =$

$10 + (2 + 8) = 10 + 2 + 8 = 20$

- $2 \times (3 \times 6) =$

$(2 \times 3) \times 6 = 2 \times 3 \times 6 = 36$

To learn more about the associative property, you can read the following: The Associative Property.

### The commutative property

The **commutative property** allows us to change the order of the terms in an equation without changing the outcome of the equation. Like the associative property, the commutative property can *only* be used for addition and multiplication.

**Let us look at some examples:**

- $2 + 6 = 6 + 2 = 8$
- $3 \times 4 = 4 \times 3 = 12$

To learn more about the commutative property, you can read the following: The Commutative Property.

## Example exercises for seventh graders

### Exercise 1

**Task:**

Using the distributive property, solve the following:

- $294:3=$
- $105\times 4=$
- $505:5=$
- $207\times 5=$
- $168:8=$

**Solutions:**

- $294:3=(300-6):3=300:3-6:3=100-2=98$
- $105\times 4=(100+5)\times 4=100\times 4+5\times 4=400+20=420$
- $505:5=(500+5):5=500:5+5:5=100+1=101$
- $207\times 5=(200+7)\times 5=200\times 5+7\times 5=1000+35=1035$
- $168:8=(160+8):8=160:8+8:8=20+1=21$

Do you think you will be able to solve it?

### Exercise 2

**Task:**

Three hundred fifty-one students from a high school were divided into nine equal groups.

How many students are in each group?

Solve the problem using the distributive property.

**Solution:**

We begin by expressing the problem numerically:

$351:9=(360-9):9=360:9-9:9=40-1=39$

**Answer:**

There are 39 students in each group.

### Exercise 3

**Task:**

Dani bought 15 boxes. In each box there were 9 pieces of candy.

How many pieces of candy did Dani buy in total?

Use the distributive property to solve the problem.

**Solution:**

We begin by expressing the problem numerically:

$15\times9=(10+5)\times9=10\times9+5\times9=90+45=135$

**Answer:**

Dani bought 135 pieces of candy.

### Exercise 4

**Task:**

Isabel has packed 246 notebooks into 6 equal packages.

How many notebooks has Isabel put in each package?

Use the distributive property to solve the problem.

**Solution:**

We begin by expressing the problem numerically:

$246:6=(240+6):6=240:6+6:6=40+1=41$

**Answer:**

Isabel has put 41 notebooks in each packet.

### Exercise 5

**Task:**

A mother had 894 euros. She divided the money equally among her three children.

How much money did each child receive?

Use the distributive property to solve the problem.

**Solution:**

We begin by expressing the problem numerically:

$894:3=(900-6):3=900:3-6:3=300-2=298$

**Answer:**

Each child received 298 euros.

Do you know what the answer is?

### Exercise 6

**Task:**

Solve the following:

$93:3=\text{?}$

**Solution:**

We simplify the number 93 into 4 smaller numers to make it easier for us to divide it by 3.

For example $30:3=10$

$93:3=\text{?}$

$(30+30+30+3):3=$

$+30:3=10$

$+30:3=10$

$+30:3=10$

$+3:3=1$

Then we will add the results and we will get:

$93:3=31$

**Answer:**

$31$

### Exercise 7

**Task:**

Solve the following:

$=90:5$

**Solution:**

We simplify the number 90 into 3 smaller numbers:

(50,20,20)

$90=5-+20+20$

Then we divide each of them by 5 and add the three results.

$+50:5=10$

$+20:5=4$

$+20:5=4$

Which gives us:

$90:5=10+4+4=18$

**Answer:**

$18$

### Exercise 8

**Task:**

Solve the following:

$=72:18$

**Solution:**

We simplify the number 72 into two smaller numbers $\left(36+36\right)$ and then divide each of them by 18.

$72:18=$

$(36+36):18=$

$+36:18=2$

$+36:18=2$

$72:18=4$

$4$

**Answer:**

$4$

### Exercise 9

**Task:**

Solve the following:

$\left(40+70+35−7\right)×9=$

**Solution:**

$\left(40+70+35−7\right)×9=$

$40\times9+70\times9+35\times9-7\times9=$

$=360+630+(30+5)9-63$

$=360+630+270+45-63=1242$

**Answer:**

$1242$

Do you think you will be able to solve it?

### Exercise 10

**Task:**

Solve the following:

$\left(35+4\right)×\left(10+5\right)=$

**Solution:**

$(35+4)(10+5)=35\times10+35\times5+4\times10+4\times5$

$=350+175+40+20=585$

**Answer:**

$585$

### Exercise 11

**Task:**

Solve the following:

$\left(7x+3\right)×\left(10+4\right)=$

**Solution:**

$(7X+3)(10+4)=7X\times10+7X\times4+3\times10+3\times4$

$=70X+28X+30+12=98X+42=238$,$-42$

$98X=196$,$:98$

$X=\frac{196}{98}=2$

**Answer:**

$2$

## More practice

- $187\times (8-5)=$
- ${2\over3}\times (12+0-5)=$
- $5\times (2{1\over2}+1{1\over6}+{3\over4})=$
- $(10+5+18)\times 4=$
- $(5.5-0.8)\times 5=$
- $340:(12-7)=$
- $(29-4):5=$
- ${5\over1}:(6+1-5)=$
- $18:(5+7+4)=$
- $({1\over7}-{1\over3}):4=$
- $97\times 12=$
- $3\times 36=$
- $120:97=$
- $8:{1\over2}=$
- $151\times 23=$

## FAQs about the distributive property

**What is the distributive property?**

The distributive property is a method used to simplify expressions into smaller, more manageable pieces.

**How is the distributive property used?**

In an equation, we use the distributive property to break down a large number into two or more smaller numbers (using addition and subtraction), and then by distributing the multiplication.

**Example**

- $20 \times 8\times 7=20+8\times 7=20\times 7+8\times 7=140+56=196$

**Can we use the distributive property in division?**

Of course we can! In an expression with division, we break down the numerator into smaller numbers (using addition and subtraction), and then the division is distributed.

**Example**

- $150:6=120+30:6=120:6+30:6=20+5=25$

**If you found this article helpful, you may also be interested in the following:**

**For a wide range of mathematics articles visit the** **Tutorela****blog**.

Do you know what the answer is?