\section*{Task 1 - V11121}

In soil investigations in agrochemistry, the so-called stepwise dilution is applied. One washes \(1 \mathrm{~cm}^{3}\) of a soil sample \((x)\) with \(10 \mathrm{~cm}^{3}\) of chemically pure water \((y)\). From the resulting mixture, one takes \(1 \mathrm{~cm}^{3}\) and washes it again with \(10 \mathrm{~cm}^{3}\) of pure water!

a) How many times must this washing be performed to achieve a dilution ratio of approximately \(1: 2000000\)?

b) How many bacteria are present on average in \(1 \mathrm{~cm}^{3}\) of the wash solution if \(1 \mathrm{~cm}^{3}\) of the undiluted soil sample contains about 10 million bacteria?
}

a) With \(x=1 \mathrm{~cm}^{3}, y=10 \mathrm{~cm}^{3}, x+y=11 \mathrm{~cm}^{3}\) the ratio is 1:10 and for the mixing ratio \(1: 2000000\) we get:

\[
\left(\frac{1}{11}\right)^{n}=\frac{1}{2000000} \quad ; \quad n=\frac{\log (2000000)}{\log (11)} \approx 6
\]

b)

\[
n_{B a k}=\frac{10000000}{2000000}=5
\]