Percentage Reverse??

Question

I need help for one question, I don't know if this is a percentage reverse, sorry if is the wrong name. The question is: I need increase a specific value in a percentage that I don't know to reach another value then apply exactly $30$ percent to reduce to original value, like this example: $950$ is original value I need increase it to a value, in this case I know is $1358.50$ then I apply a discount of and back to $950$ The question is I tried a lot of times until reach this. I need a formula to do the right way and don't make mistakes with the exactly value even cents. Thank you

Answer 1

Basically you want $(1 - .30)*(1 + x) ORIGINAL = ORIGINAL$

Or $.7(1+x) = 1$

Or $(1+x) = \frac 1{.7} = 1.4286$

Or $x = .4286 = 42.9\%$

So the price you raise it to is $1.4286*950 = 1357.55$

And $70% of 1357.55 = 950.285$

Which is off but $28 \frac 12$ cents.

To be exact you want $\frac 1{.7} = \frac {10}{7} = 1 \frac 37 = 1.42 + \frac 6{700}$ so you want $42 \frac 67\%$.

Then $(1.42 + \frac 6{700}) 950 = 1357.14 \frac {3}{100}$

$.7*135 *1357.14 \frac {3}{100}=950$

But you'll go crazy chasing pennies.

Answer 2

If the starting price is $X$ then you seek a new price $Y$ such that a $30\%$ discount from $Y$ gets you back to $X$. Thus you are asking $$(1-.3)\times Y = X\implies Y= \frac X{1-.3}=\frac X{.7}\approx 1.42857\times X$$

Note that starting with $X=950$ we get $Y\approx 1357.14$ which is close to the value you wrote.

Note too that if you had a discount amount other than $30\%$ you would just change the $.3$ accordingly. Thus if the desired discount was $17\%$ you'd just have $Y=\frac X{1-.17}$.