How can I transform this expression into the desired one?

Question

I know that (x/119)*19 is equal to x - (x/1.19), for every value of x.

However, I am trying to figure out, for curiosity, what steps do I take from the expression (x/119)*19 to reach the expression x - (x/1.19).

Can you show me, despite their equivalence they seem so different.

Thanks

Answer 1

$x-\cfrac{x}{1.19}=x-\cfrac{100x}{119}=\cfrac{119x-100x}{119}=\cfrac{19x}{119}$

Answer 2

Since x - (x/1.19) = x*(1 - 1/1.19), it suffices to show that 19/119 = (1 - 1/1.19),

Note that 1 = 119/119, and that 1/1.19 = 100/119. We can then write (1-1/1.19) = (119/119 - 100/119) = 19/119. Done.