The distributive property of multiplication allows us to break down the highest term of the exercise into a smaller number. This simplifies the multiplication operation and we can solve the exercise without the need to use a calculator.

The distributive property of multiplication allows us to break down the highest term of the exercise into a smaller number. This simplifies the multiplication operation and we can solve the exercise without the need to use a calculator.

**Let's assume we have an exercise with a** **multiplication**** that is simple, but with large numbers, for example:**

$8\times 532$

**Thanks to the distributive property, we can break it down into simpler exercises:**

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

$8\times 500=4000$

+

$8\times 30=240$

+

$8\times 2=16$

=

$4000+240+16=4256$

\( 94+72= \)

$37\times 5= ( 30+7) \times 5= 30\times 5 + 7 \times 5= 150+35= 185$

$48\times 6= (50-2) \times 6= 50\times 6-2\times 6= 300-12= 288$

**Assignment:**

$74:8=$

**Solution:**

We break down $72$ into numbers divisible by $8$

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

We arrange the exercise into simple fractions

$\frac{72}{8}+\frac{2}{8}=$

We divide accordingly

$9+\frac{1}{4}=9\frac{1}{4}$

**Answer:**

$9\frac{1}{4}$

Test your knowledge

Question 1

\( 63-36= \)

Question 2

\( 143-43= \)

Question 3

\( 133+30= \)

**Assignment:**

What expression is equivalent to the exercise $14\times3$?

**Solution:**

We break down the exercise into 2 multiplication operations to facilitate the calculation

$(15-1)\times3=$

$15\times3-3\times1=$

$15\times3-3$

**Answer:**

$15\times3$ Then subtract 3

**Assignment:**

$\left(40+70+35ā7\right)Ć9=$

**Solution:**

First, we multiply the element inside the parentheses by $9$

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

To facilitate the calculation, we break down $35$ into $2$ numbers and the rest of the exercise can be multiplied

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

First, we solve the parentheses

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

Now we add and subtract accordingly

$990+270+45-63=$

$1260+45-63=$

$1305-63=1242$

**Answer:**

$1242$

Do you know what the answer is?

Question 1

\( 140-70= \)

Question 2

Solve the exercise:

84:4=

Question 3

Solve the following exercise

?=24:12

**Assignment:**

$74\times8=$

**Solution:**

We break down $74$ into $2$ numbers to make the calculation easier

$(70+4)\times8=$

We solve the exercise accordingly

$70\times8+4\times8=$

$560+32=592$

**Answer:**

$592$

**Assignment:**

$35\times4=$

**Solution:**

We break down $35$ into $2$ numbers to make the calculation easier

$(30+5)\times4=$

We solve the exercise accordingly

$30\times4+5\times4=$

**$120+20=140$**

**Answer:**

$140$

Check your understanding

Question 1

Solve the following exercise

?=93:3

Question 2

Which equation is the same as the following?

\( 13\times29 \)

Question 3

Which equation is the same as the following?

\( 36\times4 \)

The distributive property of multiplication over addition or subtraction is the property that helps us simplify and more easily carry out an operation where it is expressed with grouping symbols and related to the order of operations. We can express it as:

Distributive property of multiplication over addition.

$a\times\left(b+c\right)=a\times b+a\times c$

Distributive property of multiplication over subtraction.

$a\times\left(b-c\right)=a\times b-a\times c$

Just like the distributive property of multiplication, the distributive property of division with respect to addition and subtraction helps us to simplify an operation, and it can be expressed as:

$\left(a+b\right):c=a:c+b:c$

Do you think you will be able to solve it?

Question 1

Which equation is the same as the following?

\( 3\times83 \)

Question 2

Which equation is the same as the following?

\( 14\times42 \)

Question 3

Which equation is the same as the following?

\( 160\times6 \)

P

**Assignment**

$\left(3+8\right)\times5=$

$\left(3+8\right)\times5=3\times5+8\times5$

$3\times5+8\times5=15+40$

$=55$

**Answer**

$=55$

**Assignment** $198\times7=$

We can break down $198$ in the following way:

$\left(100+90+8\right)\times7=$

We apply the distributive property of multiplication

$100\times7+90\times7+8\times7=$

$=700+630+56$

$=1386$

**Answer**

$=1386$

**Assignment** $\left(22+14\right):2=$

Applying the distributive property of division

$=\frac{22}{2}+\frac{14}{2}$

$=11+7=18$

**Result**

$=18$

**Assignment** $250:5$

We break down the $250$ into two numbers

$\left(300-50\right):5$

We apply the distributive law of division with respect to subtraction

$\frac{300}{5}-\frac{50}{5}=60-10$

$=50$

**Answer**

$=50$

Test your knowledge

Question 1

Which equation is the same as the following?

\( 34\times11 \)

Question 2

Which equation is the same as the following?

\( 39\times19 \)

Question 3

\( 94+72= \)

$94+72=$

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

$63-36=$

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

$133+30=$

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

$140-70=$

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

Solve the exercise:

84:4=

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

21

Related Subjects

- Recurrence Relations
- Sequences
- Algebraic Method
- Factorization: Common factor extraction
- The Extended Distributive Property
- Advanced Arithmetic Operations
- The commutative property
- The Commutative Property of Addition
- The Commutative Property of Multiplication
- The Associative Property
- The Associative Property of Addition
- The Associative Property of Multiplication
- Subtracting Whole Numbers with Addition in Parentheses
- Division of Whole Numbers Within Parentheses Involving Division
- Subtracting Whole Numbers with Subtraction in Parentheses
- Division of Whole Numbers with Multiplication in Parentheses
- The commutative properties of addition and multiplication, and the distributive property