## Task 4 - 160734

In the context of the Hans-Beimler competitions at school, Fritz participated in distance estimation.

a) With his estimate of 350 meters, he learns that it was too small, and exactly by $12.5\%$ of the true distance. Determine the true distance!

b) How great would the true distance be if Fritz's estimate had been too large, and exactly by 12.5% of the true distance?
a) Let the true distance be $x$ meters. The estimate was $12.5\%$ less than $x$ meters, i.e., $\frac{1}{8} x$ meters too small. This means the estimate was exactly $\frac{7}{8} x$ meters. Therefore, $\frac{7}{8} x = 350$, so $x = 400$. The true distance is thus $400 \mathrm{~m}$.

b) In this case, let the true distance be $y$ meters. The estimate would have been $\frac{1}{8} y$ meters too large, i.e., it would have been $\frac{9}{8} y$ meters. Consequently, $\frac{9}{8} y = 350$, so $y = 311 \frac{1}{9}$. In this case, the true distance would be $311 \frac{1}{9} \mathrm{~m}$.