In 1892, the Bajai casino built a house, for which it raised the necessary funds, 20,000 francs, by issuing shares (400 units at 50 francs each), for which it pays a 5% interest. The casino intends to repay this debt over 10 years, in (nearly) equal annual installments, such that part of the annual repayment is used for redeeming shares of a certain value, and the other part is used for paying interest on the outstanding shares. Question: What is the annual repayment, and from this, how much is paid in interest and how much is used for redeeming shares in each year?
Suppose that the casino pays out a fixed amount of $a$ francs at the end of each year as interest and principal repayment. The present values of these payments on the day the shares are issued are as follows:

$$
\frac{a}{k}, \quad \frac{a}{k^{2}}, \quad \frac{a}{k^{3}}, \ldots \frac{a}{k^{10}}
$$

Their sum is therefore

$$
\frac{a}{k} \frac{\frac{1}{k^{10}}-1}{\frac{1}{k}-1}=20000 \mathrm{frt}
$$

Or

$$
\frac{a}{k^{10}} \frac{k^{10}-1}{k-1}=20000 \text { frt }
$$

From this,

$$
\begin{gathered}
a=20000 \frac{k^{10}(k-1)}{k^{10}-1} \text { frt } \\
\log a=\log 20000+10 \log k+ \\
+\log (k-1)+\log \left(k^{10}-1\right) \\
=4.30103+0.21189+ \\
+0.69897-2-0.79858+1= \\
=3.41331 \\
a=2591 \mathrm{frt}
\end{gathered}
$$

Since the principal repayment must always be a multiple of 50 francs, the annual repayment will not be exactly $a$, but can differ by at most $\pm 50$ francs. A review of the amortization table is as follows:

| Year | Interest | Repayment | Total | Diff |
| :---: | :---: | :---: | :---: | :---: |
| 1 | 1000 | 1600 | 2600 | -09 |
| 2 | 920 | 1700 | 2620 | -29 |
| 3 | 835 | 1800 | 2635 | -44 |
| 4 | 745 | 1800 | 2545 | +46 |
| 5 | 655 | 1900 | 2555 | +36 |
| 6 | 560 | 2000 | 2560 | +31 |
| 7 | 460 | 2100 | 2560 | +31 |
| 8 | 355 | 2200 | 2555 | +36 |
| 9 | 245 | 2400 | 2645 | -46 |
| 10 | 125 | 2500 | 2625 | -26 |

Kugel Sándor and Seidner Mihály, Chief Gymnasium VIII. o. t. Losoncz.