Additional Arithmetic Rules: Using variables

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:

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

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:

$5y-3z$

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:

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

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:

$13z+2m$

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.

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:

$-4m-11n$

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:

$\frac{x}{12y}$

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:

$4x-4b$

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:

$0+2b=2b$

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:

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

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 write the expression as a fraction:

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

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

$\frac{1}{y}$

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:

$2x-y$

Let's write the exercise as a fraction:

$\frac{34a}{8a\times4b}=$

We'll reduce between the a in the numerator and the denominator of the fraction:

$\frac{34}{8\times4b}=$

Let's write the 34 in the numerator of the fraction as a smaller multiplication exercise:

$\frac{17\times2}{8\times4b}=$

Let's write the 4 in the denominator of the fraction as a smaller multiplication exercise:

$\frac{17\times2}{8\times2\times2b}=$

We'll reduce between the 2 in the numerator and denominator of the fraction:

$\frac{17}{8\times2b}=\frac{17}{16b}$

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:

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

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

$\frac{12m}{3n}\times\frac{5m}{36n}=$

We'll combine the fractions into one exercise:

$\frac{12m\times5m}{3n\times36n}=$

Let's factor 36 into a smaller multiplication exercise:

$\frac{12m\times5m}{3n\times12\times3n}=$

We'll reduce the 12 in both the numerator and denominator of the fraction:

$\frac{m\times5m}{3n\times3n}=\frac{5m^2}{9n^2}$

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

$(x\cdot y\cdot z)\times\frac{z}{xy}=$

We'll add the multiplication exercise in parentheses to the numerator of the fraction:

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

We'll simplify the x and y in the numerator and denominator of the fraction:

$\frac{z\times z}{1}=z^2$

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, 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:

$12x-13x-4y=$

Now we group the x factors:

$12x-13x=-1x$

Now we obtain:

$-1x-4y$

To begin with 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.

Thus we obtain the following:

$13x+4y-6x+3y=$

We then join the x coefficients:

$13x-6x=7x$

We join the y coefficients:

$4y+3y=7y$

Lastly we obtain:

$7x+7y$