# Thread: Basic Question. Why does 5x/10 = 5/10 * 5 ?

1. ## Basic Question. Why does 5x/10 = 5/10 * 5 ?

So I am doing an equation for straight line graphs and one of the expressions simplified out to:
5x/10.

It then said this could simply again to (5/10)x.

Why does this work?
I understand if I do it in my calculator (substituting x = 3)
5 * 3 = 15
15 / 10 = 1.5

and

5/10 = 0.5
0.5 * 3 = 1.5

They both give the same answer.
But I am trying to UNDERSTAND why this works as I am still confused.

I thought this only worked with terms that started with a + or - (where we could re-arrange them any way we wanted).

Why does it work here?

2. ## Re: Basic Question. Why does 5x/10 = 5/10 * 5 ?

5x/10 equals 1(x)/2 which equals (5/10) x or (1/2)x its just manipulating the numbers, all they did is factor the x out

think of it like this:

if x =2 then

5(2)/10
=(5/10)(2)
=1
right?
so 5x/10 = (5/10) (x)

3. ## Re: Basic Question. Why does 5x/10 = 5/10 * 5 ?

Hi Algebro.

You are asking why $\frac{5x}{10}=\frac{5}{10}\cdot x$

Look at it this way:

$\frac{5x}{10}=\frac{5}{10}\cdot \frac{x}{1}=\frac {5 \cdot x}{10 \cdot 1}$

You see, $x=\frac {x}{1}$

4. ## Re: Basic Question. Why does 5x/10 = 5/10 * 5 ?

Originally Posted by alegbro
But I am trying to UNDERSTAND why this works as I am still confused.
I am not sure I can help you understand, but I can deduce this equality from basic axioms (properties) of real numbers. Usually this is what is meant by "understanding" a mathematical fact. Of course, it is possible not to understand why those axioms hold, but this is a different issue.

First, axioms don't deal with division directly. Instead, they deal with the reciprocal function that maps an x into 1 / x, which is also often denoted by $x^{-1}$. Then x / y is an abbreviation of x * (1 / y). Thus, there is the reciprocal function with one argument and multiplication with two arguments.

Next, the basic properties of multiplication include

associativity: (x * y) * z = x * (y * z) and
commutativity: x * y = y * x.

Then we have

\begin{align*}5x/10&=(5\cdot x)\cdot10^{-1}&&\text{by definition}\\ &=5\cdot(x\cdot10^{-1})&&\text{by associativity}\\ &=5\cdot(10^{-1}\cdot x)&&\text{by commutativity}\\ &=(5\cdot10^{-1})\cdot x&&\text{by associativity}\\ &=(5/10)\cdot x&&\text{by definition}\end{align*}