\section*{Task 2 - 300912}

A company has increased its production (compared to the previous year) by 8\%, \(11 \%, 9 \%\) and \(12 \%\) over the last four years. Peter thinks that the company has thus achieved a total production increase of \(40 \% \).

Show that this is not correct.

Bernd thinks that the company would have achieved a greater increase if it had increased production four times by \(10 \% \).

Determine whether this is correct!
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The overall increase is calculated by multiplying the initial value with the numbers 1.08 \(\cdot\)1.11. \(1.09 \cdot 1.12\). Because

\[
1.08 \cdot 1.11=1.1988>1.19 ; \quad 1.19 \cdot 1.09=1.2971>1.28 ; \quad 1.29 \cdot 1.12=1.4448>1.4
\]

the overall increase is therefore greater than 40%. Hence, Peter is not correct.

The overall increase with four increases of \(10 \%\), is calculated by multiplying the initial value with the number \(1.1^{4}\). Because

\[
\begin{gathered}
1.08 \cdot 1.12=1.2096<1.21=1.1^{2} ; \quad 1.09 \cdot 1.11=1.2099<1.21=1.1^{2} \quad \text { so } \\
1.08 \cdot 1.11 \cdot 1.09 \cdot 1.12<1.1^{4}
\end{gathered}
\]

a fourfold increase of \(10 \%\), indeed leads to a greater overall increase. Therefore, Bernd's claim is correct.

Hint: With calculator precision, the following results are obtained:

\[
1.08 \cdot 1.11 \cdot 1.09 \cdot 1.12 \approx 1.463495 \quad ; \quad 1.1^{4}=1.4641
\]

If these results are rounded to produce whole percentage values, i.e., to 1.46, then:

1. The overall increase (with increases of \(8 \%, 11 \%, 9 \%, 12 \%\)) is greater than \(40 \%\). However, the proof with the rounding steps performed is not entirely rigorous, as it does not examine whether rounding up steps also occur.

2. The fourfold increase of \(10 \%\), results in only a slightly larger overall increase, so it can also be stated that it is "not significantly larger" and "in this sense," Bernd's claim is not accurate.