why does the rule "turn the second fraction upside down then multiply" works?

Answer 1

Division is the same as multiplying the reciprocal of the divisor (except for zero of course). If your divisor is a fraction, its reciprocal is just turning the fraction updise down. Then you multiply

Answer 2

Consider 4 ÷ (1/2):-
Method 1
4 / (1 / 2) = 4 x 2 /1 = 8 using your method
Method 2
4 / (1/2) = (4 x 2) / ((1/2) x 2) = 8 / 1 also works.
The second method involves multiplying top and bottom by the same number (2 in this case).
ie are in fact multiplying by 1,which is of course permissable.
Hope this explanation helps.

Answer 3

(a/b) divided by (c/d) is the same as (a/b) times (d/c).

In order to understand why this works, you have to understand general division between fractions and/or the nature of multiplicative inverses. In high school, you learn to deal with complex fractions ( (a/b) / (c/d) is a complex fraction ) which would verify this, and would also learn about multiplicative inverses (including why 0 does not have a multiplicative inverse).

Answer 4

(a/b) / (a/b) = 1

And it just so happens that
(a/b) * (b/a) = (a*b)/ (a*b) = 1

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Do you agree that dividing by 2 is the same as multiplying by 1/2?

Why? Its because 2 * (1/2) = 1

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By rearranging the equation 2 * (1/2) = 1
I can prove that
1/2 = 1/2
But I can also prove that
2 = 1/(1/2)