Associative Property: Using variables

Upon observing the exercise note that we have two "regular" numbers and one number with a variable.
Given that this is a multiplication exercise, multiplying a number with a variable by a number without a variable doesn't present a problem.

In fact, it's important to remember that a variable attached to a number represents multiplication itself, for example in this case: $11\times x$
Therefore, we can apply the distributive property to separate the variable, and come back to it later.
Proceed to solve the exercise from right to left since it's simpler this way.

$5\times6=30$

We obtain the following:

$11x\times30=$

We'll put aside the x and add it at the end of the exercise.

By solving the exercise in an organized way we simplify the solution process.

It's important to maintain the correct order when solving the problem, meaning first multiply the ones of the first number by the ones of the second number,
then the tens of the first number by the ones of the second number, and so on.

$30\\\times11\\=330$

Don't forget to add the variable at the end. The answer is as follows:

$330x$

Upon observing the exercise note that we have two "regular" numbers and one number with a variable.
Given that this is a multiplication exercise, multiplying a number with a variable by a number without a variable doesn't present a problem.

In fact, it's important to remember that a variable attached to a number represents multiplication by itself, for example in this case: $2\times x$
Therefore, we can apply the distributive property in order to separate the variable, and come back to it later.
Solve the exercise from left to right.

Solve the left exercise by breaking down the decimal number into an addition problem of a whole number and a decimal number as follows:

$2\times(4+0.65)=$

Multiply 2 by each term inside of parentheses:

$(2\times4)+(2\times0.65)=$

Solve each of the expressions inside of the parentheses as follows:

$8+1.3=9.3$

We obtain the following exercise:

$9.3\times6.3=$

Solve the exercise vertically in order to simplify the solution process.

It's important to be careful with the proper placement of the exercise, using the decimal point as an anchor.
Then we can proceed to multiply in order, first the ones digit of the first number by the ones digit of the second number. Then the tens digit of the first number by the ones digit of the second number, and so on.

$9.3\\\times6.3\\=58.59$

Don't forget to add the variable at the end resulting in the following answer:

$58.59x$

Let's solve the exercise from left to right.

We will combine the left expression in the following way:

$\frac{6+8}{7}x=\frac{14}{7}x=2x$

Now we get:

$2x+3\frac{2}{3}x=5\frac{2}{3}x$

We move the fraction to the beginning of the exercise and will place the rest of the exercise in parentheses to make solving the equation easier:

$\frac{a}{4}+(2-2)=$

$\frac{a}{4}+0=\frac{a}{4}$

This exercise can be solved in order, but to make it easier, the associative property can be used

$-5+(3+4)=$

$-5+7=$

$7-5=2$

Upon observing the exercise note that we have two "regular" numbers and one number with a variable.
Given that this is a multiplication exercise, multiplying a number with a variable by a number without a variable doesn't present a problem.

In fact, it's important to remember that a variable attached to a number represents multiplication by itself, for example in this case: $5.2\times x$
Therefore, we can apply the distributive property in order to separate the variable, and come back to it later.
Proceed to solve the exercise from left to right.

Solve the left exercise vertically in order to avoid confusion as shown below:

$~~~~~15.6 \\\times~~~~5.2 \\=~81.12$

It's important to be careful with the correct placement of the exercise, where the decimal point serves as an anchor.
Then we can multiply in order, first the ones digit of the first number by the ones digit of the second number.
Next the tens digit of the first number by the ones digit of the second number, and so on.

We should obtain the following:

$81.21\times0.3=$

Remember that:

$0.3=\text{0}.30$

Calculate:

$24.336$

Let's not forget to add the variable at the end resulting in the following answer:

$24.336 x$

Let's begin by combining the simple fractions into a single multiplication exercise:

$\frac{3\times2}{4\times3}\times2\frac{1}{4}x=$

Let's now proceed to solve the exercise in the numerator and denominator:

$\frac{6}{12}\times2\frac{1}{4}x=$

Finally we'll simplify the simple fraction in order to obtain the following:

$\frac{1}{2}\times2\frac{1}{4}x=1\frac{1}{8}x$

The first step is factorising each of the terms in the exercise into a whole number and its remainder:

$10x+0.1x+5x+0.2x+2x+0.4x=$

Now we'll combine only the whole numbers:

$10x+5x+2x=15x+2x=17x$

Next, we will calculate the remainder:

$0.1x+0.2x+0.4x=0.3x+0.4x=0.7x$

Finally, we are left with the following:

$17x+0.7x=17.7x$

According to the order of operations, we must solve the exercise from left to right:

$0.2x+8.6x=8.8x$

We'll also break down 8.8 into a smaller addition exercise that will be easier for us to calculate:

$8x+0.8x+0.65x=$

Now we'll use the commutative property since the exercise only involves addition.

Let's focus on the leftmost addition exercise, remembering that:

$0.8=0.80$$$

Next, we'll calculate the exercise below:

$0.80x+0.65x=1.45x$

Finally, we are left with the following:

$8x+1.45x=9.45x$

According to the rules of the order of operations, we first eliminate the parentheses.

Remember that a positive times a positive will give a positive result, and a positive times a negative will give a negative result.

Therefore, we obtain:

$4a+5a-2-4a+10=$

Now, we arrange the exercise in a more comfortable way using the substitution property:

$4a+5a-4a+10-2=$

We solve the exercise from left to right, starting by adding the coefficients a:

$4a+5a-4a=9a-4a=5a$

Now we obtain:

$5a+10-2=5a+8$

First, let's convert all mixed fractions to simple fractions:

$\frac{3\times6+5}{6}\times\frac{5\times6+5}{6}\times\frac{1}{3}x=$

Let's solve the exercises with the eight fractions:

$\frac{18+5}{6}\times\frac{30+5}{6}\times\frac{1}{3}x=$

$\frac{23}{6}\times\frac{35}{6}\times\frac{1}{3}x=$

Since the exercise only involves multiplication, we'll combine all the numerators and denominators:

$\frac{23\times35}{6\times6\times3}x=\frac{805}{108}x$

First, let's find a common denominator for 4, 8, and 6: it's 24.

Now, we'll multiply each fraction by the appropriate number to get:

$\frac{5\times4}{6\times4}x+\frac{7\times3}{8\times3}x+\frac{2\times6}{4\times6}x=$

Let's solve the multiplication exercises in the numerator and denominator:

$\frac{20}{24}x+\frac{21}{24}x+\frac{12}{24}x=$

We'll connect all the numerators:

$\frac{20+21+12}{24}x=\frac{41+12}{24}x=\frac{53}{24}x$

Let's break down the numerator into a smaller addition exercise:

$\frac{48+5}{24}=\frac{48}{24}+\frac{5}{24}=2+\frac{5}{24}=2\frac{5}{24}x$