# Formula for inverse relationship between 'scale' and 'physical size'

I'm having trouble solving the following problem. Hopefully someone can help!

I have some drawing plans to prepare: the orginal drawing is a scale 1:200 (printed at 100%)

• if i print at 50% it will be 1:400
• if i print at 25% it will be 1:800
• and so on...

I can see there is an inverse relationship between these numbers but i can't at all remember how to express that as a formula.

What would the '%' be for a given scale '1:n'? Can you please also explain how to figure this out?

Thanks!

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Your scale would be $1:\frac{200}{\text{print percentage expressed as a decimal}}$, where 50%=0.5, 25%=0.25, etc. Or given the scale, the print percent is $\frac{200}{\text{scale}}$ again expressed as a decimal.
The simplest way to get the correct formula is generalize from the examples. If the original scale is $1 : X$ and you print at $Y\%$, then the formula will need either to multiply $X$ by $Y/100$ or divide $X$ by $Y/100$. Checking the examples, we see that the correct operation is division.
A more satisfying way is to understand what's going on. A scale of $1:X$ means that $1$ unit on the map corresponds to $X$ units in real life. If you print at $Y\%$, then what was supposed to be $1$ unit is actually $Y/100$ units. So a "new unit" is $1/(Y/100)$ old units, which correspond to $X/(Y/100)$ units in real life. Therefore the new scale is $1 : X / (Y/100)$.