Why do we switch the denominator and numerator when we divide fractions? [duplicate]

Question

Why do we switch the denominator and numerator when we divide fractions?

I've been trying to find out why and I've asked several people and checked many websites but none that give me a good answer. Help?

Answer 1

For $B\not=0,C\not=0$, we have $$A\div\frac BC=\frac{A}{\frac BC}=\frac{A}{\frac BC}\times 1=\frac{A}{\frac BC}\times\frac{C}{C}=\frac{A\times C}{\frac{B}{C}\times C}=\frac{AC}{B}=A\times \frac{C}{B}.$$

Answer 2

Start observing that $\frac ab=a\cdot\frac1b$.

What does $\frac1b$ mean? It is the inverse of $b$ namely that (unique) number that multiplied by $b$ gives $1$, namely $b\cdot\frac1b=1$ (for, mind that you need $b\neq0$).

Given that, we may now contend that $\frac1{\frac xy}=\frac yx$.

Why? Because $\frac xy\cdot\frac yx=\frac{xy}{yx}=1$.

Putting everything together we get the rule $$ \frac{\frac ab}{\frac cd}=\frac ab\cdot\frac dc= \frac{a\cdot d}{b\cdot c}. $$

Answer 3

because $\frac ab\cdot\frac ba=\frac{ab}{ba}=\frac11=1$.

Answer 4

Dividing a number $x$ by another number $y$ is actually taking a part of $x$, a fraction of it which corresponds to $1/y$. So, when $y = \frac{a}{b}$, we have : $$ \frac{x}{y} = \frac{x}{\frac{a}{b}} = x \times \frac{1}{y} = x \times \frac{b}{a} $$

If you have troube understanding this in the purely mathematical way, think of a pizza : that is your number. Imagine now that you want to divide your pizza in $8$ parts. The size of each part will be equal to the total size divided by $8$. to get it, you can also proceed by saying that each part is $1/8$ of the total size.
We've seen here that dividing by $8$ was the same as multiplying by the reciproc of $8$, which is $1/8$. When it is not 8 but a fraction, we just remember that the reciproc of $a/b$ is $b/a$ $(= \frac{1}{a/b})$, hence the switch between numerator and denominator !