The Distributive Property for 7th Grade: Addition, subtraction, multiplication and division

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

To make solving easier, we break down 53 into more comfortable numbers, preferably round ones.

We obtain:

$4\times(50+3)=$

We multiply 2 by each of the terms inside the parentheses:

$(4\times50)+(4\times3)=$

We solve the exercises inside the parentheses and obtain:

$200+12=212$

To make solving easier, we break down 11 into more comfortable numbers, preferably round ones.

We obtain:

$(10+1)\times34=$

We multiply 34 by each of the terms in parentheses:

$(34\times10)+(34\times1)=$

We solve the exercises in parentheses and obtain:

$340+34=374$

To make solving easier, we break down 29 into more comfortable numbers, preferably round ones.

We obtain:

$6\times(30-1)=$

We multiply 6 by each of the terms in parentheses:

$(6\times30)-(6\times1)=$

We solve the exercises in parentheses and obtain:

$180-6=174$

To solving easier, we break down 39 into more convenient numbers, preferably round ones.

We obtain:

$30\times(40-1)=$

We multiply 30 by each of the terms in parentheses:

$(30\times40)-(30\times1)=$

We solve the exercises in parentheses and obtain:

$1,200-30=1,170$

In order to simplify our calculation, we first break down 93 into smaller, more manageable parts. (Preferably round numbers )

We obtain the following:

$3\times(90+3)=$

We then use the distributive property in order to find the solution.

We multiply each of the terms in parentheses by 3:

$(3\times90)+(3\times3)=$

Lastly we solve each of the terms in parentheses and obtain:

$270+9=279$

In order to facilitate the resolution process, we break down 56 into an exercise with smaller, preferably round, numbers.

$3\times(50+6)=$

We use the distributive property and multiply each of the terms in parentheses by 3:

$(3\times50)+(3\times6)=$

We then solve each of the exercises inside of the parentheses and obtain the following result:

$150+18=168$

In order to facilitate the resolution process, we first break down 33 into a smaller addition exercise with more manageable and preferably round numbers:

$9\times(30+3)=$

Using the distributive property we then multiply each of the terms in parentheses by 9:

$(9\times30)+(9\times3)=$

Finally we solve each of the exercises inside of the parentheses:

$270+27=297$

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

Keep in mind: it is beneficial to choose to split the number according to your knowledge of multiples. In this case into multiples of 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 facilitates the division operation and gives us the possibility to solve the exercise without a using 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

To make solving easier, we break down 560 into more comfortable numbers, preferably round ones.

We obtain:

$3\times(500+60)=$

We multiply 3 by each of the terms in parentheses:

$(3\times500)+(3\times60)=$

We solve the exercises in parentheses and obtain:

$1,500+180=1,680$

To make solving easier, let's break down 12 and 19 into more convenient numbers, preferably round numbers.

We get:

$(10+2)\times(10+9)=$

We'll use the distributive property to solve.

First, we'll multiply the first term in the left parentheses by the first term in the right parentheses.

We'll multiply the first term in the left parentheses by the second term in the right parentheses.

We'll multiply the second term in the left parentheses by the first term in the right parentheses.

We'll multiply the second term in the left parentheses by the second term in the right parentheses.

We get:

$(10\times10)+(10\times9)+(2\times10)+(2\times9)=$

Let's solve what's in the parentheses and we get:

$100+90+20+18=$

Let's solve from left to right:

$190+20+18=210+18=228$

To make it easier for us to solve, let's break down 12 and 19 into more convenient numbers, preferably round ones.

We get:

$(20-2)\times(70-1)=$

We'll use the distributive property to solve.

First, we'll multiply the first term in the left parentheses by the first term in the right parentheses.

We'll multiply the first term in the left parentheses by the second term in the right parentheses.

We'll multiply the second term in the left parentheses by the first term in the right parentheses.

We'll multiply the second term in the left parentheses by the second term in the right parentheses.

We get:

$(20\times70)-(20\times1)-(2\times70)-(2\times-1)=$

Let's solve what's in the parentheses and we get:

$1,400-20-140+2=$

We'll use the associative property and first solve:

$20-140=160$

Now we get:

$1,400-160+2=$

Let's solve from left to right:

$1,400-160=1,240$

$1,240+2=1,242$

To make it easier for us to solve, let's break down 33 and 16 into more convenient numbers, preferably round ones.

We get:

$(30+3)\times(10+6)=$

We'll use the distributive property to solve.

First, we'll multiply the first term in the left parentheses by the first term in the right parentheses.

We'll multiply the first term in the left parentheses by the second term in the right parentheses.

We'll multiply the second term in the left parentheses by the first term in the right parentheses.

We'll multiply the second term in the left parentheses by the second term in the right parentheses.

We get:

$(30\times10)+(30\times6)+(3\times10)+(3\times6)=$

Let's solve what's in the parentheses and we get:

$300+180+30+18$

Let's solve from left to right:

$300+180=480$

$480+30=510$

$510+18=528$

First, we solve the exercise from left to right.

We apply the associative property in order to simplify the exercise making it easier to solve:

$2\times3=6$

Resulting in the following calculation:

$6\times43=$

We then decompose 43 into smaller parts rendering the equation easier to solve:

$6\times(40+3)=$

We apply the distributive property in order to solve the equation.

We then multiply 6 by each of the terms inside the parentheses:

$(6\times40)+(6\times3)=$

Resulting in the following solution:

$240+18=258$

First, we break down 122 into smaller numbers and write the division exercise in the form of a fraction:

$(100+20+2)\times\frac{12}{4}=$

We solve the fraction exercise:

$(100+20+2)\times3=$

We then multiply the terms inside the parentheses by 3 and obtain the following result:

$(100\times3)+(20\times3)+(2\times3)=$

Lastly we solve each of the exercises inside of the parentheses:

$300+60+6=366$

In order to simplify our calculation, we first separate the addition exercise into two smaller multiplication exercises:

$\frac{(3\times3)+(40\times3)}{3}=$We then split the resulting equation into an addition exercise between fractions:

$\frac{3\times3}{3}+\frac{40\times3}{3}=$

Lastly we reduce the 3 in both the numerator and denominator, and obtain:

$3+40=43$

To make it easier for ourselves in the solving process, we'll break down 12 and 33 into exercises with smaller and more convenient numbers, preferably round numbers.

$(10+2)\times(30+3)=$

We'll solve the exercise using the distribution law:

We'll multiply the first term in the left parentheses by the first term in the right parentheses.

We'll multiply the first term in the left parentheses by the second term in the right parentheses.

We'll multiply the second term in the left parentheses by the first term in the right parentheses.

We'll multiply the second term in the left parentheses by the second term in the right parentheses.

And we get:

$(10\times30)+(10\times3)+(2\times30)+(2\times3)=$

We'll solve each of the expressions in parentheses and get:

$300+30+60+6=$

We'll solve the exercise from left to right:

$300+30=330$

$330+60=390$

$390+6=396$

In order to simplify the resolution process, we begin by breaking down the number 65 into a smaller addition exercise.

We choose numbers that are divisible by 13:

$(26+26+13):13=$

We then divide each of the terms within parentheses by 13:

$26:13=2$

$26:13=2$

$13:13=1$

To finish we add up all of the results that we obtained:

$2+2+1=4+1=5$

Simplify this expression by paying attention to the order of arithmetic operations which states that exponentiation precedes multiplication, division precedes addition and subtraction and that parentheses precede all of the above.

Thus, let's begin by simplifying the expressions within the parentheses, and following this, the multiplication between them.

$(12+2)\cdot(3+5)= \\ 14\cdot8=\\ 112$Therefore, the correct answer is option C.

We simplify this expression by observing the order of arithmetic operations which states that exponentiation precedes multiplication, division precedes addition and subtraction, and that parentheses precede everything else.

Therefore, we first start by simplifying the expression within the parentheses. We then multiply the result of the expression within the parentheses by the term that multiplies it:

$(40+70+35-7)\cdot9= \\ 138\cdot9=\\ 1242$Therefore, the correct answer is option C.