\section*{Task 3 - 301013}

If the production of a company decreases by \(50 \%\) (e.g., due to the failure of part of the plant), it must subsequently be doubled, i.e., increased by \(100 \%\) to return to the initial value.

Determine a formula that allows you to calculate the percentage \(b\) from a given percentage \(a\) for which the following statement (1) holds!

(1) If production decreases by \(a\) percent, it must subsequently be increased by \(b\) percent to return to the initial value.
}

When decreasing by \(a\) percent, the initial value is multiplied by \(\left(1-\frac{a}{100}\right)\), and when subsequently increasing by \(b\) percent, the resulting value is multiplied by \(\left(1+\frac{b}{100}\right)\). For the final value to be equal to the initial value, it must be that

\[
\left.\left(1-\frac{a}{100}\right)\left(1+\frac{b}{100}\right)\right) 1
\]

This implies

\[
\begin{aligned}
1+\frac{b}{100} & =\frac{1}{1-\frac{a}{100}}=\frac{100}{100-a} \\
\frac{b}{100} & =\frac{100}{100-a}-1=\frac{a}{100-a} \\
b & =\frac{100 a}{100-a}
\end{aligned}
\]

as the desired formula.