The Distributive Property in the Case of Multiplication - Examples, Exercises and Solutions

There are several ways to solve the following exercise,

We will present two of them.

In both ways, we begin by decomposing the number 84 into smaller units; 80 and 4.

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

Subsequently we are left with only the 80.

Continuing on with the first method, we will then further decompose 80 into smaller units; $10\times8$

We know that:$\frac{8}{4}=2$

And therefore, we are able to reduce the exercise as follows: $\frac{10}{4}\times8$

Eventually we are left with$2\times10$

which is equal to 20

In the second method, we decompose 80 into the following smaller units:$40+40$

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 it in to our above answer:

$20+1=21$

Thus we are left with the following solution:$\frac{84}{4}=21$

Apply the distributive property of division and proceed to split the number 24 into a sum of 12 and 12, which ultimately renders the division operation easier and allows us to solve the exercise without a calculator.

Note - it's best to choose to split the number based on knowledge of multiples. In this case of the number 12 because we need to divide by 12.

Reminder - The distributive property of division actually allows us to split the larger term in a division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator

We will use the formula of the distributive property

 (a+b):c=a:c+b:c 

$24:12=(12+12):12$

$(12+12):12=12:12+12:12$

$12:12+12:12=1+1$

$1+1=2$

Therefore the answer is section a - 2.

We will use the distributive property of division and split the number 93 into a sum of 90 and 3. This ultimately renders the division operation easier and allows us to solve the exercise without a calculator.

Note - it's best to choose to split the number based on knowledge of multiples. In this case, we use 3 because we need to divide by 3. Additionally splitting by tens and ones is suitable and makes the division operation easier.

Reminder - The distributive property of division essentially allows us to split the larger term in the division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator

We will use the formula of the distributive property

 (a+b):c=a:c+b:c 

$93:3=(90+3):3$

$(90+3):3=90:3+3:3$

$90:3+3:3=30+1$

$30+1=31$

Therefore, the answer is option B - 31.

In order to solve the question, we first use the distributive property for 133:

$(100+33)+30=$

We then use the distributive property for 33:

$100+30+3+30=$

We rearrange the exercise into a more practical form:

$100+30+30+3=$

We solve the middle exercise:

$30+30=60$

Which results in the final exercise as seen below:

$100+60+3=163$

In order to simplify the resolution process, we begin by using the distributive property for 140:

$100+40-70=$

We then rearrange the exercise using the substitution property into a more practical form:

$100-70+40=$

Lastly we solve the exercise from left to right:

$100-70=30$

$30+40=70$

We will use the distributive law and split the number 143 into a sum of 100 and 43.

The distributive law allows us to distribute, meaning, to split a number into two or more numbers. This actually allows us to work with smaller numbers and simplify the operation.

$(100+43)-43=$

We will operate according to the order of operations

We can remove parentheses and perform addition and subtraction operations in any order since there are only addition and subtraction operations in the equation

$100+43-43=100+0=100$

Therefore, the answer is option C - 100.

And now let's see the solution to the exercise in a centered format:

$143-43= (100+43)-43= 100+43-43=100+0=100$

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

In order to simplify the calculation , we first break down 94 and 72 into smaller and preferably round numbers.

We obtain the following exercise:

$90+4+70+2=$

Using the associative property, we then rearrange the exercise to be more functional.

$90+70+4+2=$

We solve the exercise in the following way, first the round numbers and then the small numbers.

$90+70=160$

$4+2=6$

Which results in the following exercise:

$160+6=166$

Apply the distributive property of division and proceed to split the number 88 into the sum of 80 and 8. Simplifying he division operation allows us to solve the exercise without a calculator

Reminder - The distributive property of division actually allows us to split the larger term in the division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator

$(80+8):4=$

Apply the formula of the distributive property $(a+b):c=a:c+b:c$

$(80:4)+(8:4)=$

Continue to solve the problem according to the order of operations

$20+2=22$

Therefore the answer is option C - 22.

Shown below are the various steps of our solution:

$88:4=(80+8):4=80:4+80:4=20+2=22$

Let's simplify this expression while maintaining the order of operations.

Let's start by solving what's in the parentheses:

$29-4=25$

Now we get the expression:

$25:5=$

In the next step, to make the division easier, we'll break down 25 into two smaller factors that are divisible by 5:

$(20+5):5=$

Let's divide each factor in the parentheses by 5:

$(20:5)+(5:5)=$

We'll solve each expression in the parentheses and obtain:

$4+1=5$

Apply the distributive property of division and proceed to split the number 186 into the sum of 180 and 6. This ultimately makes the division operation easier and allows us to solve the exercise without a calculator

Reminder - The distributive property of division actually allows us to split the larger number in a division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator

$(180+6):6=$

Apply the formula of the distributive property $(a+b):c=a:c+b:c$

$180:6+6:6=$

Continue to solve the problem according to the order of operations

$30+1=31$

Therefore the answer is option D - 31.

Below are the various steps of our solution:

$186:6= (180+6):6= 180:6+6:6= 30+1=31$

Apply the distributive property of multiplication in order to break down the number 13 into a subtraction exercise with smaller numbers. This allows us to work with smaller numbers and ultimately simplify the operation

Reminder - The distributive property of multiplication actually allows us to break down the larger term in the multiplication exercise into a sum or difference of smaller numbers, which makes the multiplication operation easier and gives us the ability to solve the exercise even without a calculator

$13\times(10-2)=$

Apply the distributive property formula $a(b+c)=ab+ac$

$13\times10-13\times2=$

Proceed to solve the problem according to the order of operations

$130-26=$

Therefore the answer is option D - 104.

Shown below are the various stages of the solution:

$13\times8=13\times(10-2)=13\times10-13\times2=130-26=104$

Apply the distributive property of multiplication in order to split the number 17 into the sum of numbers 10 and 7. This ultimately allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication essentially allows us to split the larger term in a multiplication problem into a sum or difference of smaller numbers, which makes multiplication easier and gives us the ability to solve the problem even without a calculator

$(10+7)\times7=$

Apply the distributive property formula $a(b+c)=ab+ac$

$10\times7+7\times7=$

Proceed to solve according to the order of operations

$70+49=$

Therefore the answer is option C - 119.

Shown below are the various stages of the solution

$17\times7=(10+7)\times7=(10\times7)+(7\times7)=70+49=119$

Apply the distributive property and proceed to multiply each term inside of the parentheses by 187:

$187\times8-187\times5=$

Solve the first multiplication problem vertically, making sure to solve it in the correct order (ones multiplied by ones, ones multiplied by tens, ones multiplied by hundreds )

$187\\\times8$

We should obtain the following result: 1496

Proceed to solve the second multiplication problem vertically, once again making sure to solve it in the correct order (ones multiplied by ones, ones multiplied by tens, ones multiplied by hundreds )

$187\\\times5$

We should obtain the following result: 935

Now to tackle the next problem:

$1496-935=$

We should once again solve this vertically. Make sure to align the digits properly, ones under ones, tens under tens, etc.:

$1496\\-935$

Subtract ones from ones, tens from tens, etc., to obtain the final result: $561$

Apply the distributive property of multiplication and proceed to split the number 36 into the sum of the numbers 30 and 6. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication essentially allows us to split the larger term in a multiplication problem into a sum or difference of smaller numbers, which makes multiplication easier and gives us the ability to solve the problem without a calculator

$3×(30+6)=$

Apply the distributive property formula $a(b+c)=ab+ac$

$3×30+3×6=$

Proceed to solve the problem according to the order of operations

$90+18= 108$

Therefore the answer is option D - 108.

Shown below are the various stages of our solution:

$3\times36=3\times(30+6)=(3\times30)+(3\times6)=90+18=108$