# The Distributive Property

šPractice distributive property for seventh grade

What is the distributive property?

$a \cdot (b + c) = ab + ac$

$(a+b)(c +d) =ac+ad+bc+bd$

The distributive property is a rule in mathematics that says that multiplying a number by the sum of two or more numbers will give us the same result as multiplying that number by the two numbers separately and then adding them together.

For example, 4x4 will give us the same result as (4x2) + (4x2).

How does this help us? Well, it allows us to distribute, or to split up a number into two or more smaller numbers that are easier to work with. When we're working with large numbers, or expressions with variables, the distributive property can save us time and a headache!

## Test yourself on distributive property for seventh grade!

$$94+72=$$

## The distributive property: the basic version

Using the distributive property, we can break down a number into two or more smaller numbers using addition or subtraction, giving us an expression that is easier to solve without changing its final value.

Here is the formula for the basic distributive property:

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

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

## The distributive property: the extended version

At first, we learn to use the distributive property using expressions with only one pair of parentheses. After we feel comfortable, we can move on to the extended distributive property.

The extended distributive property is used to multiply two sets of parentheses, one by the other.

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## The distributive property in practice

Naturally, the distributive property is useful in school, but it's also helpful in your daily life! Splitting up a check at a restaurant? Planning a group trip? The distributive property can make your real-life calculating and planning less of a headache, and everyone will be impressed by how quickly you can get things done. From basic distribution, to complex algebraic equations, to real world dilemmas - the distributive property is a fundamental part of the math we use every day, and getting a good understanding of how and when to use it can help you go far!

## The distributive property of multiplication

Let's say you have a multiplication exercise with numbers that are too large to calculate in your head.

For example: $532\times8$

By using the distributive property, we will be able to break it down into simpler terms to solve:

$8\times 532=8\times\left(500+30+2\right)$

$8\times 500=4000$

$+$

$8\times 30=240$

$+$

$8\times2=16$

$=$

$4000+240+16=4256$

Do you know what the answer is?

## The distributive property of division

With division exercises, the concept is the same. Again, we will use the distributive property to break down large numbers, and make our work easier. Suppose we are asked to solve the following: $76:4$

First, we will take the large, awkward number and round up to the next integer that is a multiple of the divisor (the number that 76 is being divided by, which is 4).

In our example, the number closest to $76$ that is a multiple of $4$ is $80$.

So,

$76:4=\left(80-4\right):4$

$=$

$80:4-4:4$

$=$

$20-1=19$

And we get:

$76:4=19$

## The extended distributive property

At first, we try to focus on simpler expressions that have only one pair of parentheses. After we have mastered these, we can move on to the extended distributive property. Now, we will start solving exercises that have more than one pair of parentheses.

For example:

$\left(7+2\right)\times\left(5+8\right)$

We will use the extended distributive property to simplify the exercise. How?

We multiply each of the terms in the first pair of parentheses by each of the terms in the second pair of parentheses:

$\left(5+8\right)\times\left(7+2\right)$

$=$

$5\times7+5\times2+8\times7+8\times2$

$=$

$35+10+56+16$

$=$

$117$

## The extended distributive property: with variables

Similarly, we can use the extended distributive property to solve equations with variables.

For example:

$\left(X+2\right)\cdot\left(3X-5\right)$

$=$

$\left(X\cdot3X\right)+\left(X\cdot-5\right)+\left(2\cdot3X\right)+\left(2\cdot-5\right)$

$=$

$3XĀ²-5X+6X-10$

$=$

$3XĀ²+X-10$

## Example exercises

### Exercise 1

Solve the following:

$?=84:4$

Solution:

We distribute the number 84 into several smaller numbers.

We recommend to distribute into numbers that are easy to divide by 4, like 40, then we will divide by 4.

Therefore:

$40+40+4=84$

$84:4=\text{?}$

$(40+40+4):4=\text{?}$

$+40:4=10$

$+40:4=10$

$+4:4=1$

Therefore: $84:4=21$

$21$

Do you think you will be able to solve it?

### Exercise 2

Solve the following:

$72:6=\text{?}$

Solution:

We distribute the number 72 into two numbers.

We choose numbers that are easier to divide by 6.

$60+12=72$

$72:6=\text{?}$

$(60+12):6=$

$+60:6=10$

$+12:6=2$

$72:6=$

$12$

### Exercise 3

Solve the following:

$?=65:13$

Solution:

We distribute 65 into 3 numbers: $26+26+13=65$

Then, we divide each of them by 13:

$65:13=\text{?}$

$(26+26+13):13=\text{?}$

$+26:13=2$

$+26:13=2$

$+13:13=1$

$(2+2+1)=5$

$65:13=5$

$5$

### Exercise 4

Solve the following:

$742:4=$

Solution:

$742:4=(700+40+2):4$

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

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

$=100+50+25+10+\frac{1}{2}=185\frac{1}{2}$

$185\frac{1}{2}$

### Exercise 5

Solve the following:

$(3+20)\times(12+4)=\text{?}$

Solution:

$(3+20)\times(12+4)=3\times12+3\times4+20\times12+20\times4$

$=36+12+240+80=48+320=368$

$368$

Do you know what the answer is?

### Exercise 6

Solve the following:

$(7+2+3)(7+6)(12-3-4)=\text{?}$

Solution:

$(7+2+3)(7+6)(12-3-4)=\text{?}$

$(7+2+3)\times13\times5=12\times13\times5=12\times5\times13$

$=60\times13=780$

$780$

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## More practice for seventh graders

### Exercise 1

Solve the following exercises by using the distributive property:

• $=294:3=$
• $105\times4=$
• $505:5=$
• $207\times5=$
• $168:8=$

Solutions:

• $294:3=(300ā6):3=300:3ā6:3=100ā2=98$
• $105\times4=(100+5)\times4=100\times4+5\times4=400+20=420$
• $505:5=(500+5):5=500:5+5:5=100+1=101$
• $207\times5=(200+7)\times5=200\times5+7\times5=1000+35=1035$
• $168:8=(160+8):8=160:8+8:8=20+1=21$

### Exercise 2

$351$ students in a school were divided equally into $9$ classes.

How many students are there in each class?

Answer by making use of the distributive property.

Solution:

First, let's write the expression:

$351:9=(360ā9):9=360:9ā9:9=40ā1=39$

In each of the classes there are $39$ students.

### Exercise 3

Dani bought $15$ packages. In each package there were $9$ pieces of candy.

How many pieces of candy did Dani buy in total?

Find the answer using the distributive property.

Solution:

Let's write out the exercise:

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

Dani bought 135 pieces of candy in total.

Do you think you will be able to solve it?

### Exercise 4

Laura packed $246$ notebooks into $6$ equal packages.

How many notebooks did Laura put in each package?

Use the distributive property.

Solution:

Let's write the exercise:

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

Laura packed $41$ notebooks in each package.

### Exercise 5

A mother had $$894$. She divided the money equally among her three children. How much money did each child receive? Find the answer by using the distributive property. Solution: Let's write out the exercise: $894:3=(900ā6):3=900:3ā6:3=300ā2=298$ Answer: Each of the children received$$298$.

Use the distributive property to solve the following exercises:

• $187\cdot(8-5)=187ā (8ā5)=$
• $(10+5+18)\cdot4=(10+5+18)ā 4=$
• $(5.5-0.8)\cdot5=(5.5ā0.8)ā 5=$
• $340:(12-7)=340:(12ā7)=$
• $(29-4):5=(29ā4):5=$
• $15:(6+1-5)=15ā:(6+1ā5)=$
• $18:(5+7+4)=18:(5+7+4)=$
• $(71-31):4=(71āā31ā):4=$
• $97\cdot12=97ā 12=$
• $3\cdot36=3ā 36=$
• $120:97=120:97=$
• $8:21=8:21ā=$
• $151\cdot23=151ā 23=$

## examples with solutions for distributive property for seventh grade

### Exercise #1

$94+72=$

### 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 #2

$63-36=$

### 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 =Ā

27

27

### Exercise #3

$133+30=$

### 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:

$100+60+3=163$

163

### Exercise #4

$140-70=$

### 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 #5

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.

Ā

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:$\frac{84}{4}=21$