Proportionally decrease 3 numbers by n percent

Question

I have 3 numbers 5, 90, 60, that gives = 155, I need to decrease them by n percent, so they together give 100, but to keep the ratio between them. Not sure is it even possible, any feedback would be great. So

a = 5
b = 90
c = 60

How much in percent, I should decrease each number so they give 100, but to keep ration as it is now.

Thank you.

Answer 1

First keep in mind that if you decrease each number by the same percentage, their ratio remains same.

$$a \rightarrow a- \frac{na}{100}$$ and $$b \rightarrow b- \frac{nb}{100}$$ $$\implies \frac ab=\frac{a-\frac{na}{100}}{b- \frac{nb}{100}}=\frac ab$$

Now, for your problem, you must have, $$\left(a- \frac{na}{100}\right)+\left(b- \frac{nb}{100}\right)+\left(c- \frac{nc}{100}\right)=100$$ plug in the values of $a,b,c$ and you can find $n$

Answer 2

$$\frac{155-100}{155} = 0.3548$$ As the overall sum needs to reduce by 34.58%, we need to reduce each number by 35.48%.