Multiplication of Integers by a Fraction and a Mixed number: Applying the formula

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:

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

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.