The Distributive Property for 7th Grade: Applying the formula

We'll use the distributive property and multiply each term in parentheses by 187:

$187\times8-187\times5=$

Let's solve the first multiplication problem vertically, making sure to solve it correctly, meaning units times units, units times tens, units times hundreds.

$187\\\times8$

We get the result: 1496

Let's solve the second multiplication problem vertically, making sure to solve it correctly, meaning units times units, units times tens, units times hundreds.

$187\\\times5$

We get the result: 935

Now we'll get the problem:

$1496-935=$

We'll solve this vertically as well. We'll make sure to align the digits properly, units under units, tens under tens, etc.:

$1496\\-935$

We'll subtract units from units, tens from tens, etc., and get the result: $561$

In order to simplify the resolution process, we divide the number 35 into a smaller addition exercise.

It is easier to choose round whole numbers, hence the following calculation:

$(30+5)\times4=$

We then multiply each of the terms inside of the parentheses by 4:

$(4\times30)+(4\times5)=$Lastly we solve the exercises inside of the parentheses:

$120+20=140$

In order to simplify the resolution process, we begin by breaking down the number 74 into a smaller addition exercise.

It is easier to choose round whole numbers, hence the following calculation:

$(70+4)\times8=$

We then multiply each of the terms within the parentheses by 8:

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

Lastly we solve the exercises within the parentheses:

$560+32=592$

In order to simplify the resolution process, we begin by breaking down the number 480 into a smaller addition exercise:

$(400+80)\times3=$

We then multiply each of the terms within the parentheses by 3:

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

Lastly we solve the exercises inside the parentheses and obtain the following:

$1200+240=1440$

In order to simplify the resolution process, we begin by breaking down the number 354 into a smaller addition exercise.

It is easier to choose round whole numbers, and also to consider numbers that are easily divisible by 3.

Hence the following calculation:

$(300+54):3=$

Once again, for the purpose of facilitating the resolution process, we break down 54 into a smaller addition exercise.

Just as in the previous calculation we choose round numbers and numbers divisible by 3.

We obtain the following:

$(300+30+24):3=$

We then divide each of the terms within the parentheses by 3:

$300:3=100$

$30:3=10$

$24:3=8$

We finish by adding up all the results we obtained:

$100+10+8=110+8=118$

In order to simplify the resolution process, we begin by breaking down the number 12345 into a smaller addition exercise:

$(10000+2000+300+40+5)\times6=$

We multiply each term inside the parentheses by 6:

$(10000\times6)+(2000\times6)+(300\times6)+(40\times6)+(5\times6)=$

We then solve each of the exercises inside of the parentheses:

$60000+12000+1800+240+30=$

Lastly we solve the exercise from left to right:

$60000+12000=72000$

$72000+1800=73800$

$73800+240=74040$

$74040+30=74070$

In order to simplify the resolution process, we begin by breaking down the number 74 into a smaller addition exercise with numbers divisible by 8:

$(72+2):8=$

We then divide each of the terms within the parentheses by 8:

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

We solve each of the exercises inside of the parentheses:

$9+\frac{2}{8}=$

Lastly we reduce the numerator and the denominator of the fraction by 2:

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

In order to simplify the resolution process, we begin by breaking down 30 into a smaller addition exercise:

$(30+5)\times20=$

We then multiply each of the terms inside of the parentheses by 20:

$(30\times20)+(5\times20)=$

Lastly we solve the exercises inside of the parentheses as follows:

$600+100=700$

In order to simplify the resolution process, we first separate 458 into a smaller addition exercise and choose numbers that are divisible by 7:

$(420+38):7=$

We then further separate 38 into a smaller addition exercise and choose numbers that are divisible by 7:

$(420+35+3):7=$

We divide each of the terms inside of the parentheses by 7:

$\frac{420}{7}+\frac{35}{7}+\frac{3}{7}=$

Finally we solve the fractions as follows:

$60+5+\frac{3}{7}=65\frac{3}{7}$

In order to simplify the resolution process, we begin by breaking down the number 742 into a smaller addition exercise:

$(700+42):4$

We then divide the two numbers within the parentheses into smaller numbers. The numbers should be more manageable for us to divide by 4:

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

Following this we divide each number inside of the parentheses by 4:

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

We then solve all the fractions:

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

Lastly we solve the exercise from left to right:

$100+50=150$

$150+25=175$

$175+10=185$

$185+\frac{1}{2}=185\frac{1}{2}$

We will use the distributive property of multiplication and break down the fraction into a subtraction exercise between a whole number and a fraction. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication allows us to break down 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

$9\times(4-\frac{1}{9})=$

We will use the distributive property formula $a(b+c)=ab+ac$

$(9\times4)-(9\times\frac{1}{9})=$

Let's solve what's in the left parentheses:

$9\times4=36$

Note that in the right parentheses we can reduce 9 by 9 as follows:

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

$\frac{9}{1}\times\frac{1}{9}=\frac{9\times1}{1\times9}=\frac{9}{9}=\frac{1}{1}=1$

And we get the exercise:

$36-1=35$

And now let's see the solution centered:

$9\times3\frac{8}{9}=9\times(4-\frac{1}{9})=(9\times4)-(9\times\frac{1}{9})=36-1=35$

We will use the distributive property of multiplication and separate the fraction into an addition exercise between fractions. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication 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 without a calculator

$3\times(2+\frac{1}{4})=$

We will use the distributive property formula $a(b+c)=ab+ac$

$(3\times2)-(3\times\frac{1}{4})=$

Let's solve what's in the left parentheses:

$3\times2=6$

Let's solve what's in the right parentheses:

$3=\frac{3}{1}$

$\frac{3}{1}\times\frac{1}{4}=\frac{3\times1}{1\times4}=\frac{3}{4}$

And we get the exercise:

$6+\frac{3}{4}=6\frac{3}{4}$

And now let's see the solution centered:

$3\times2\frac{1}{4}=3\times(2+\frac{1}{4})=(3\times2)+(3\times\frac{1}{4})=6+\frac{3}{4}=6\frac{3}{4}$

We will use the distributive property of multiplication and separate the fraction into an addition exercise between fractions. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication actually allows us to separate 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 without a calculator

$5\times(3+\frac{1}{3})=$

We will use the distributive property formula $a(b+c)=ab+ac$

$(5\times3)+(5\times\frac{1}{3})=$

Let's solve what's in the left parentheses:

$5\times3=15$

Let's solve what's in the right parentheses:

$5=\frac{5}{1}$

$\frac{5}{1}\times\frac{1}{3}=\frac{5\times1}{1\times3}=\frac{5}{3}$

And we get the exercise:

$15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}$

And now let's see the solution centered:

$5\times3\frac{1}{3}=5\times(3+\frac{1}{3})=(5\times3)+(5\times\frac{1}{3})=15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}$

We begin by opening the parentheses using the extended distributive property to create a long addition exercise:

We then multiply the first term of the left parenthesis by the first term of the right parenthesis.

We multiply the first term of the left parenthesis by the second term of the right parenthesis.

Now we multiply the second term of the left parenthesis by the first term of the left parenthesis.

Finally, we multiply the second term of the left parenthesis by the second term of the right parenthesis.

In the following way:

$(35\times10)+(35\times5)+(4\times10)+(4\times5)=$

We solve each of the exercises within parentheses:

$350+175+40+20=$

We solve the exercise from left to right:

$350+175=525$

$525+40=565$

$565+20=585$

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

We'll start by simplifying the expression inside the parentheses.

In this expression, there are addition operations between mixed fractions, so in the first step we'll convert all mixed fractions in this expression to improper fractions.

We'll do this by multiplying the whole number by the denominator of the fraction, and adding the result to the numerator.

In the fraction's denominator (which is the divisor) - nothing will change of course.

We'll do this in the following way:

$2\frac{1}{2}=\frac{(2\times2)+1}{2}=\frac{4+1}{2}=\frac{5}{2}$

$1\frac{1}{6}=\frac{(1\times6)+1}{6}=\frac{6+1}{6}=\frac{7}{6}$

Now we'll get the exercise:

$5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)$

We'll continue and perform the addition of fractions in the expression inside the parentheses.

First, we'll expand each fraction to the common denominator, which is 12 (since it is the least common multiple of all denominators in the expression), we'll do this by multiplying the numerator of the fraction by the number that answers the question: "By how much did we multiply the current denominator to get the common denominator?"

Then we'll perform the addition operations between the expanded numerators:

$5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big) =\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{30+14+9}{12} =\\ 5\cdot\frac{53}{12} =\\$We performed the addition operation between the numerators above, after expanding the fractions mentioned.

Note that since multiplication comes before addition, we first performed the multiplications in the fraction's numerator and only then the addition operations,

We'll continue and simplify the expression we got in the last step, meaning - we'll perform the multiplication we got, while remembering that multiplying a fraction means multiplying the fraction's numerator.

In the next step, we'll write the result as a mixed fraction, we'll do this by finding the whole numbers (the answer to the question "How many complete times does the denominator go into the numerator?") and adding the remainder divided by the divisor:

$5\cdot\frac{53}{12}=\\ \frac{5\cdot53}{12}=\\ \frac{265}{12}=\\ 22\frac{1}{12}$

Let's summarize the steps of simplifying the given expression:

$5\cdot\big(2\frac{1}{2}+1\frac{1}{6}+\frac{3}{4}\big)= \\ 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)=\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{53}{12} =\\ 22\frac{1}{12}$

Therefore the correct answer is answer B.