Is there a word for multiplying and dividing two terms in a product by the same value?

Question

Obviously doing this is a form of /simplifying/, but I'd like to know if there's a specific term for this kind of multiplication. For example, given $k = 2n$, you know $6k = 12n$, but is there a term for this?

Answer 1

The term that you are looking for is probably multiplicative cancelation (or some variant thereof). If the objects that you are working with are "sufficiently nice" (the real numbers, for example), then the property of multiplicative cancellation is the statement \begin{equation} a\ne 0 \text{ and } b = c \iff ab = ac. \end{equation} Given something of the form $ab = ac$, you can cancel the $a$ on each side and not alter the identity (note that this is an idea that is somewhat distinct from "dividing both sides by $a$", though the distinction may not matter until you get into higher maths). In your particular example, you might write \begin{equation} k = \frac{6}{6} k = \frac{1}{6} 6k \quad\text{and}\quad 2n = \frac{6}{6} 2n = \frac{1}{6} 12n. \end{equation} Then \begin{equation} k = 2n \iff \frac{1}{6} 6k = \frac{1}{6} 12n \iff 6k = 12n, \end{equation} where the last identity is obtained by canceling the factor of $1/6$ (or, if you prefer, dividing both sides by 6).

Answer 2

This is known as multiplication property of equality.

Multiplication property of equality states that when both sides of an equation are multiplied by the same number, the remaining expressions are still equal.