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Math Help - Basic Question. Why does 5x/10 = 5/10 * 5 ?

  1. #1
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    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?
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  2. #2
    Senior Member sakonpure6's Avatar
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    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)
    Last edited by sakonpure6; February 9th 2013 at 10:51 AM.
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  3. #3
    Senior Member Paze's Avatar
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    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}
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  4. #4
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    Re: Basic Question. Why does 5x/10 = 5/10 * 5 ?

    Quote Originally Posted by alegbro View Post
    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*}

    Does this answer your question?
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