why dividing a number by 1.25 gives back 20 percent less of original?

Question

So i had to takeout the discount from price.

price = 10
discount = 20%

my default method has been: price - price*discount

10 - 10*.20 = 8

but i had a hunch it should be possible to just do this by diving or multiplying.

So after tries i found that this works too, dont remember how i got to the number...

10/1.25 = 8

how does it work. how can i find other denominators for other percentages.

i feel this is stupidly simply question, but my brain is sort of not working so please help.

Answer 1

If $e$ is the expenditure, $p$ the price and $d$ the discount, you are using the relation: $$ e=p(1-d) $$ you want express $e=p/c$ , so you have: $$ \dfrac{p}{c}=p(1-d) $$ and you find: $$ c=\dfrac {1}{1-d} $$

Answer 2

$\frac{10}{1.25} = \frac{10}{\frac{5}{4}} = \frac{4}{5} \times 10 = 0.8 \times 10 = 10(1-0.2) = 10 - (10\times0.2)$

Thus you have your original number minus $20\%$ of it