# Distributive property for seventh grade - Examples, Exercises and Solutions

Solving algebraic equations is made easier by understanding some basic rules and properties. A few examples of properties that we will learn to use in the seventh grade are: the distributive, associative and commutative properties. These properties get learned and relearned throughout our time in school, each time adding new layers to or understanding. Today we will focus on the distributive property. We will go into depth on what it is and how to use it, and we will briefly get to know the associative and commutative properties as well.

## What is the distributive property?

The distributive property is a method to simplify multiplication and division exercises. Essentially, it breaks down expressions into smaller, easier to manage terms.

Let's see some examples:

• $6 \times 26 = 6 \times (20 + 6) = 120 + 36 = 156$
• $7 \times 32 = 7 \times (30 + 2) = 210 + 14 = 224$
• $104:4 = (100+4):4 = 100:4 + 4:4 = 25+1 = 26$

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:

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

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

## Practice Distributive property for seventh grade

### 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

$63-36=$

### 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 #4

$133+30=$

### 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 #5

$140-70=$

### 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 #1

Solve the following exercise

=90:5

### Step-by-Step Solution

We use the distributive property of division and separate the number 90 between the sum of 50 and 40, which makes the division operation easier and gives us the possibility to solve the exercise even without a calculator.

Keep in mind: it is beneficial to choose to separate the number according to your knowledge of multiples. In this case of the figure 5 because it is necessary to divide by 5.

Reminder: the distributive property of division actually allows us to separate the larger term in a division exercise into the sum or the difference of smaller numbers, which makes the division operation easier and gives us the possibility to solve the exercise even without a calculator.

We use the formula of the distributive property

$(a+b):c=a:c+b:c$

$90:5=(50+40):5$

$(50+40):5=50:5+40:5$

$50:5+40:5=10+8$

$10+8=18$

Therefore, the answer is option c: 18

18

### Exercise #2

Solve the exercise:

=74:4

### Step-by-Step Solution

To make the resolution process easier for us, we break down the number 74 into a subtraction exercise:

We choose numbers divisible by:

$(80-6):4=$

Now we divide each of the terms in parentheses by 4:

$80:4=20$

$6:4=1.5$

Now we subtract the result we obtained:

$20-1.5=18.5$

18.5

### Exercise #3

$35\times4=$

### Step-by-Step Solution

To facilitate the resolution process, we divide the number 35 into a smaller addition exercise.

It is easier to choose round whole numbers, therefore we get:

$(30+5)\times4=$

Now, we multiply each of the terms in parentheses by 4:

$(4\times30)+(4\times5)=$

We solve the exercises in parentheses:

$120+20=140$

140

### Exercise #4

$74\times8=$

### Step-by-Step Solution

To facilitate the resolution process, we break down the number 74 into a smaller addition exercise.

It is easier to choose round whole numbers, therefore we get:

$(70+4)\times8=$

Now, we multiply each of the terms in parentheses by 8:

$(8\times70)+(8\times4)=$

We solve the exercises in parentheses:

$560+32=592$

592

### Exercise #5

$480\times3=$

### Step-by-Step Solution

To facilitate the resolution process, we divide the number 480 into a smaller addition exercise:

$(400+80)\times3=$

Now, we multiply each of the terms in parentheses by 3:

$(400\times3)+(80\times3)=$

We solve the exercises in parentheses and obtain:

$1200+240=1440$

1440

### Exercise #1

Which equation is the same as the following?

$13\times29$

### Step-by-Step Solution

We solve each of the options and keep in mind the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

$(10+3)\times(30-1)=13\times29$

b.

$10\times3\times30\times1=30\times30\times1=900$

c.

$(10\times3)\times30=13\times30$

d.

$10\times3+29=30+29=59$

Therefore, the answer is option A.

$(10+3)\times(30-1)$

### Exercise #2

Which equation is the same as the following?

$36\times4$

### Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

$36+4=40$

b.

$4\times(30+6)=4\times36$

c.

$40-4+4=36+4=40$

d.

$4\times30+6=120+6=126$

Therefore, the answer is option B.

$4\times(30+6)$

### Exercise #3

Which equation is the same as the following?

$3\times83$

### Step-by-Step Solution

We solve each of the options and keep in mind the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

$3\times8\times3=24\times3=72$

b.

$(2+1)\times(80+3)=3\times83$

c.

$3+(80+3)=3+83$

d.

$3+83=86$

Therefore, the answer is option B.

$(2+1)\times(80+3)$

### Exercise #4

Which equation is the same as the following?

$14\times42$

### Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

$10+4+40+2=14+42=36$

b.

$(10\times4)+(40\times2)=40+80$

c.

$(10+4)\times(40+2)=14\times42$

d.

$10\times4\times40\times2=40\times40\times2=160\times2=360$

Therefore, the answer is option C.

$(10+4)\times(40+2)$

### Exercise #5

Which equation is the same as the following?

$160\times6$

### Step-by-Step Solution

We solve each of the options and keep in mind the order of operations: calculation of the operation inside parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

a.

$160+6=166$

b.

$(100\times60)\times6=6,000\times6$

c.

$(100+60)+(3+3)=160+6$

d.

$(100+60)\times(3+3)=160\times6$

Therefore, the answer is option D.

$(100+60)\times(3+3)$