Problem 997

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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.