-1
$\begingroup$

I have been doing multiply and divide but never really tried to understand it.

E.g. (4 x 12 / 4) could be written as: (12, 4 x 3, 48 / 4) but yet the result is always same, can anyone prove or explain why?

  • 1
    @user3231: Dividing by 4 is *by definition* the inverse operation to multiplying by 4.2011-03-12

2 Answers 2

2

It may help to think of the numbers as products of smaller numbers. It may make it easier to understand what's going on. $4\times \frac{12}{4} = 2\times 2 \times \frac{2\times 2\times 3}{2\times 2}=2\times 2\times 3 = 4 \times 3 = 4\times 3 \times \frac{4}{4}=12$ Notice how you can always multiply a number by a fraction $\frac{a}{a}$ for any $a \neq 0$ and get the same number (because $\frac{a}{a}=1$): $ b = 1 \times b= \frac{a}{a}\times b = \frac{a\times b}{a}=a\times \frac{b}{a}$ We can for instance use it on your example: $ 12 = 12 \times \frac{4}{4} = \frac{12\times 4}{4} = \frac{48}{4}$

  • 0
    Yes, this helped me realizing about what i was thinking, thanks.2011-03-12
2

You want to interpret the expression $a\times b\,\,\, /\,\,\, c$ We could first multiply $a$ and $b$, and then divide the result by $c$, i.e. $(a\times b)\,\,\, /\,\,\, c\hskip0.5in (1)$ or we could multiply $a$ by the quantity $b/c$, i.e. $a\times(b/c)\hskip0.5in (2)$ Here is the reason expressions (1) and (2) are equal: dividing by a number $c$ is the same thing as multiplying by its reciprocal $1/c$. This is because, for any number $x$, the number $x/c$ is the unique solution to the equation $\_\_\_\times c = x\hskip0.5in (\ast)$ Anything that can go in the spot $\_\_\_$ is equal to $x/c$. But we also have $\left(x\times (1/c)\right)\times c = x\times\left((1/c)\times c\right)=x\times 1 =x$ where the first equality is justified by the associative law of multiplication. This means that $x\times (1/c)$ is also the solution to the equation $(\ast)$, and hence must be equal to $x/c$.

So, how does this help? We can rewrite (1) and (2) as $(a\times b)\times (1/c)\hskip0.5in (1)$ and $a\times (b\times (1/c))\hskip0.5in (2)$ respectively, and the associative law of multiplication tells us that these two quantities must be equal.

  • 0
    This really should be the proper answer to this question. :$-$)2011-09-13