10000 korona increased to 17800 korona over 10 years with compound interest. What is the interest rate if the rate for the first 5 years was $\frac{1}{2}$ higher than for the last 5 years.

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10000 korona increased to 17800 korona over 10 years with compound interest. What is the interest rate if the rate for the first 5 years was $\frac{1}{2}$ higher than for the last 5 years.
Suppose that in the last 5 years the interest rate was $x$, then in the first 5 years it was $x+\frac{1}{2}$. At the end of the fifth year, the compounded capital

$$
C=10000\left(1+\frac{x+\frac{1}{2}}{100}\right)^{5}
$$

became, by the end of the 10th year:

$$
C\left(1+\frac{x}{100}\right)^{5}=10000\left(1+\frac{x+\frac{1}{2}}{100}\right)^{5}\left(1+\frac{x}{100}\right)^{5}=17800
$$

From this,

$$
\left(1+\frac{x+\frac{1}{2}}{100}\right)\left(1+\frac{x}{100}\right)=\sqrt[5]{1.78}=1.12223
$$

and thus

$$
x=5.685 \%
$$

(Enyedi Béla, Budapest.)

The problem was also solved by: Bartók I., Beck P., Blau A., Deutsch I., Haar A., Hirschfeld Gy., Kelemen M., Kertész G., König D., Mixich P., Pintér M., Pivnyik I., Raab R., Schmidl I., Tóbiás J. L., Weisz P., Wohlstein S.