The Distributive Property

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:

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

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

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.

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!

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:

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

$8\times 500=4000$

$+$

$8\times 30=240$

$+$

$8\times2=16$

$=$

$4000+240+16=4256$

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:

$76:4=19$

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:

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

$=$

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

$=$

$35+10+56+16$

$=$

$117$

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

For example:

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

$=$

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

$=$

$3X²-5X+6X-10$

$=$

$3X²+X-10$

Task:

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: $84:4=21$

Answer:

$21$

Task:

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.

$60+12=72$

$72:6=\text{?}$

$(60+12):6=$

$+60:6=10$

$+12:6=2$

$72:6=$

Answer:

$12$

Task:

Solve the following:

$?=65:13$

Solution:

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

Then, we divide each of them by 13:

$65:13=\text{?}$

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

$+26:13=2$

$+26:13=2$

$+13:13=1$

$(2+2+1)=5$

$65:13=5$

Answer:

$5$

Task:

Solve the following:

$742:4=$

Solution:

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

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

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

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

Answer:

$185\frac{1}{2}$

Task:

Solve the following:

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

Solution:

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

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

Answer:

$368$

Task:

Solve the following:

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

Solution:

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

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

$=60\times13=780$

Answer:

$780$

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

• Equivalent expressions / Equivalent algebraic expressions.
• The Associative Property
• The distributive property in the case of divisions
• The distributive property in the case of multiplication
• The distributive property: extension
• The commutative property of addition
• The commutative property of multiplication
• The associative property
• Associative property of addition
• Associative property of multiplication

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Task:

Solve the following exercises by using the distributive property:

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

Solutions:

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

Task:

$351$ 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:

$351:9=(360−9):9=360:9−9:9=40−1=39$

Answer:

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

Task:

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:

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

Answer:

Dani bought 135 pieces of candy in total.

Task:

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:

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

Answer:

Laura packed $41$ notebooks in each package.

Task:

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:

$894:3=(900−6):3=900:3−6:3=300−2=298$

Answer:

Each of the children received $$298$.

Task:

Use the distributive property to solve the following exercises:

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