Additional Arithmetic Rules: Using variables

First, we address the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

We obtain the following:

$x-y+x=$

We join the x coefficients:

$x+x=2x$

Lastly we obtain:

$2x-y$

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by minus will give us negative numbers:

$7x-4b-3x=$

Now we group the X factors:

$7x-3x=4x$

Now we obtain:

$4x-4b$

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by the minus will give us negative numbers:

$10y-5y-3z=$

Now we group the Y factors:

$10y-5y=5y$

Now we obtain:

$5y-3z$

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers within the parentheses, multiplying by a negative will give us negative numbers:

$x-3x-4y=$

Now we group the X factors:

$x-3x=-2x$

We obtain:

$-2x-4y$

Let's begin by writing the exercise as a fraction:

$\frac{12a}{7x\times4b}=$

Next we'll factor the numerator of the fraction into a smaller multiplication exercise:

$\frac{4\times3a}{7x\times4b}=$

Let's now reduce the 4 in both the numerator and denominator of the fraction:

$\frac{3a}{7xb}$

Let's flip the fraction to get a multiplication exercise:

$a\times\frac{a}{4}=$

We'll add the a to the numerator of the fraction:

$\frac{a\times a}{4}=\frac{a^2}{4}$

Let's write the exercise as a fraction:

$\frac{3m\times12n}{7m\times4n}=$

Let's reduce between the m in the numerator and the n in the denominator:

$\frac{3\times12}{7\times4}=$

Let's write the 12 in the numerator of the fraction as a smaller multiplication exercise:

$\frac{3\times4\times3}{7\times4}=$

Let's reduce between the 4 in the numerator and the denominator:

$\frac{3\times3}{7}=\frac{9}{7}=1\frac{2}{7}$

Let's write the exercise as a fraction:

$\frac{5x^2}{3y\times20x}=$

We'll factor the numerator of the fraction into a multiplication exercise:

$\frac{5x\times x}{3y\times20x}=$

Let's write the 20 in the denominator of the fraction as a smaller multiplication exercise:

$\frac{5x\times x}{3y\times4\times5x}=$

We'll cancel out the 5x in both the numerator and denominator of the fraction:

$\frac{x}{3y\times4}=$

Let's multiply the denominator of the fraction:

$\frac{x}{12y}$

Let's write the expression as a fraction:

$\frac{x}{x\times y}=$

We'll reduce between the x in the numerator and denominator and get:

$\frac{1}{y}$

The problem asks us to simplify the expression $\frac{14a}{2a \cdot 3a}$.

First, simplify the expression in the denominator:

• The denominator $2a \cdot 3a$ can be expanded by multiplying the constants and variables separately: $2 \cdot 3 \cdot a \cdot a = 6a^2$.

Now, the expression becomes:

• $\frac{14a}{6a^2}$

Next, simplify the fraction by canceling out common factors in the numerator and the denominator:

• Both the numerator and the denominator contain the variable $a$. We can divide both by $a$, resulting in:
• $\frac{14}{6a}$

Further simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

• $\frac{14 \div 2}{6 \div 2a} = \frac{7}{3a}$

Thus, the simplified form of the expression is $\frac{7}{3a}$.

Therefore, the solution to the problem is $\frac{7}{3a}$.

To solve this problem, follow these steps:

• Step 1: Simplify the inner division $8t : 3m$. This can be rewritten as $\frac{8t}{3m}$.
• Step 2: Find the reciprocal of $\frac{8t}{3m}$. The reciprocal is $\frac{3m}{8t}$.
• Step 3: Multiply $32m$ by the reciprocal of the inner division result, which is $\frac{3m}{8t}$.

Carrying out the multiplication:

$32m \times \frac{3m}{8t} = \frac{32m \times 3m}{8t} = \frac{96m^2}{8t}$

Simplify the expression $\frac{96m^2}{8t}$ by dividing the numerator and the denominator by 8:

$= \frac{12m^2}{t}$

Thus, the final simplified solution is $12\frac{m^2}{t}$.

The correct choice from the given options is $12\frac{m^2}{t}$.

To solve this problem, we'll simplify the expression $2x : (7y : 4x)$ using properties of division:

• Step 1: Recognize $a : (b : c) = a \times \frac{c}{b}$. This changes the nested division into a singular operation.
• Step 2: Substitute into our expression: $2x : (7y : 4x) = 2x \times \frac{4x}{7y}$.
• Step 3: Simplify the expression $= 2 \times \frac{4x^2}{7y}$.
• Step 4: Calculate the product: $= \frac{8x^2}{7y}$.
• Step 5: Convert to a mixed number if necessary: $= 1\frac{1}{7}\frac{x^2}{y}$.

Therefore, the solution to the problem is $\frac{8x^2}{7y}$, which can be expressed as a mixed number $1\frac{1}{7}\frac{x^2}{y}$.

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

Hence we obtain the following calculation:

$2a+3b-4b+3a=$

We join together the a coefficients:

$2a+3a=5a$

We then join together the b coefficients:

$3b-4b=-b$

We obtain the following:

$5a-b$

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number the result will be negative.

When we multiply a negative number by a negative number the result will be positive.

Hence we obtain the following calculation:

$a+b-a+b=$

Next we join together the a coefficients:

$a-a=0$

We then join together the b coefficients:

$b+b=2b$

We obtain the following:

$0+2b=2b$

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number the result will be negative.

When we multiply a negative number by a negative number the result will be positive.

Thus we obtain the following equation:

$3m-14n-7m+3n=$

Next we join the m coefficients:

$3m-7m=-4m$

We then join the n coefficients:$-14n+3n=-11n$

Finally we obtain:

$-4m-11n$

We begin by addressing the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

Hence we obtain the following calculation:

$12z+3m-m+z=$

Next we join together the z coefficients:

$12z+z=13z$

We then join together the m coefficients:

$3m-m=2m$

Finally we obtain the following:

$13z+2m$

Begin by writing the exercise as a fraction:

$(a+b):\frac{3}{4}=$

Proceed to write the fraction in reverse order for the purpose of obtaining a multiplication exercise:

$(a+b)\times\frac{4}{3}=$

Finally we obtain the following result:

$\frac{4}{3}(a+b)$

First let's rewrite the expression as a fraction:

$\frac{3x}{2t\times4z}=$

We should note that since there is only a multiplication operation in the numerator of the fraction, we will multiply the coefficients together:

$\frac{3x}{2\times4\times t\times z}=\frac{3x}{8tz}$

First let's write the exercise as a fraction:

$\frac{x\times y\times z}{3x\times4y\times5z}=$

Then we'll cancel out the x, the Y, and the z from both the numerator and denominator of the fraction:

$\frac{1}{3\times4\times5}=$

Then, we'll multiply the expression in the denominator from left to right:

$\frac{1}{3\times4\times5}=\frac{1}{12\times5}=\frac{1}{60}$

Firstly we need to look at the expression inside of parentheses.

Remember:

When we multiply a negative number by a negative number, the result will be positive.

When we multiply a positive number by a negative number, the result will be negative.

Therefore:

$a+b+c-a+b+c=$

Now let's combine the $a$ terms:

$a-a=0$

Then we can combine the $b$ terms:

$b+b=2b$

Finally let's combine the $c$ terms:

$c+c=2c$

This leaves us with:

$0+2b+2c=2b+2c$