I own a diner, and just as the breakfast special is about to end, 2 different groups of a dozen people each, walk in & are seated at separate tables. All of those customers order eggs. I look in the restaurant’s refrigerator & see that I have 4 dozen eggs left. If each customer receives an equal number of eggs, how many eggs does each customer get?

4 dozen eggs divided by 2 dozen customers

4 dozen ÷ 2 dozen

(or written with a slash as: 4 dozen / 2 dozen)

Numerically, the statement is:

4(12) ÷ 2(12)

…or…

4(9+3) ÷ 2(9+3)

…which is the monomial division of…

4x ÷ 2x

…when x=12 or x= (9+3).

I deliberately chose to use the word “dozen” because we are all accustomed to buying eggs in that “unit,” which is a box of 12 eggs. It’s easy to understand that when there are 4 full standard packages of eggs, there are 48 eggs in all. That mental image of “4 dozen eggs,” highlights the reason that the coefficient of 4 cannot be separated from its factor of “dozen.” Since multiplication is just a fast way of doing addition, the value of the term “4 dozen” is actually…

1 dozen + 1 dozen + 1 dozen + 1 dozen

...and 2 dozen is actually...

1 dozen + 1 dozen

Also, everyone can easily picture two individual groups of a dozen people, with each group seated as a “unit” of 12 customers at two separate tables in a restaurant, understanding that there are 24 total customers now seated in the restaurant, who are all ordering eggs for breakfast.

The use of “dozen” makes it easy to understand that…

4 dozen divided by 2 dozen = 2

…which can be numerically calculated as…

4(12) / 2(12) =

48 / 24 = 2

…or calculated by canceling out the like factor of “dozen,” which leaves the statement as…

4 / 2

…which, of course, also equals 2.

And if the word “dozen” is replaced with a variable such as “x,” then the statement is…

4x / 2x

…which is one monomial being divided by another monomial, with x=(12) or x=(9+3).

Implied multiplication by juxtaposition indicates that a term is a monomial, such as 4x or 2x (one term with a coefficient & a variable) — never needing parentheses around it, any more than “4 dozen” needs to be completely encased in a set of parentheses to be understood as one term with a single value. Therefore, the “4” in the term “4 dozen,” or the “2” in the term “2 dozen” cannot be detached and used in some other operation in the statement before calculating the total value of the monomial, first. It's the same with monomials like "4x" or "2x," when x=12-- the coefficient can't be "peeled off" and used in some other operation in the statement.

In the monomial division statement 6 ÷ 2(1+2), "6" can be factored out as 2(1+2), making the statement:

2(1+2) ÷ 2(1+2)

Replacing what is inside the parentheses with the variable "x" the statement is:

2x ÷ 2x

...and that monomial division yields a quotient of 1.

Example:

I own a diner, and just as the breakfast special is about to end, 2 different groups of a dozen people each, walk in & are seated at separate tables. All of those customers order eggs. I look in the restaurant’s refrigerator & see that I have 4 dozen eggs left. If each customer receives an equal number of eggs, how many eggs does each customer get?

4 dozen eggs divided by 2 dozen customers

4 dozen ÷ 2 dozen

(or written with a slash as: 4 dozen / 2 dozen)

Numerically, the statement is:

4(12) ÷ 2(12)

…or…

4(9+3) ÷ 2(9+3)

…which is the monomial division of…

4x ÷ 2x

…when x=12 or x= (9+3).

I deliberately chose to use the word “dozen” because we are all accustomed to buying eggs in that “unit,” which is a box of 12 eggs. It’s easy to understand that when there are 4 full standard packages of eggs, there are 48 eggs in all. That mental image of “4 dozen eggs,” highlights the reason that the coefficient of 4 cannot be separated from its factor of “dozen.” Since multiplication is just a fast way of doing addition, the value of the term “4 dozen” is actually…

1 dozen + 1 dozen + 1 dozen + 1 dozen

...and 2 dozen is actually...

1 dozen + 1 dozen

Also, everyone can easily picture two individual groups of a dozen people, with each group seated as a “unit” of 12 customers at two separate tables in a restaurant, understanding that there are 24 total customers now seated in the restaurant, who are all ordering eggs for breakfast.

The use of “dozen” makes it easy to understand that…

4 dozen divided by 2 dozen = 2

…which can be numerically calculated as…

4(12) / 2(12) =

48 / 24 = 2

…or calculated by canceling out the like factor of “dozen,” which leaves the statement as…

4 / 2

…which, of course, also equals 2.

And if the word “dozen” is replaced with a variable such as “x,” then the statement is…

4x / 2x

…which is one monomial being divided by another monomial, with x=(12) or x=(9+3).

Implied multiplication by juxtaposition indicates that a term is a monomial, such as 4x or 2x (one term with a coefficient & a variable) — never needing parentheses around it, any more than “4 dozen” needs to be completely encased in a set of parentheses to be understood as one term with a single value. Therefore, the “4” in the term “4 dozen,” or the “2” in the term “2 dozen” cannot be detached and used in some other operation in the statement before calculating the total value of the monomial, first. It's the same with monomials like "4x" or "2x," when x=12-- the coefficient can't be "peeled off" and used in some other operation in the statement.

In the monomial division statement 6 ÷ 2(1+2), "6" can be factored out as 2(1+2), making the statement:

2(1+2) ÷ 2(1+2)

Replacing what is inside the parentheses with the variable "x" the statement is:

2x ÷ 2x

...and that monomial division yields a quotient of 1.