How much do we need to pay at the beginning of each year for 10 years, so that 5 years after the last payment, we can receive $1500 \mathrm{~K}-\mathrm{t}$ at the beginning of each year for 15 years, if $p=4$?
If for 10 years we pay in $x$ crowns at the beginning of each year and the interest factor is $e$, then the value of the money we have paid in at the beginning of the 10th year is

$$
x \frac{e^{10}-1}{e-1}
$$

and the present value of this is:

$$
\frac{x}{e^{9}} \frac{e^{10}-1}{e-1}
$$

If for 15 years we receive $1500 \mathrm{~K}-\mathrm{t}$ at the beginning of each year, then the value of our annuity at the beginning of the 29th year is

$$
1500 \frac{e^{15}-1}{e-1}
$$

and the present value of this amount is

$$
\frac{1500}{e^{28}} \frac{e^{15}-1}{e-1}
$$

Thus,

$$
\frac{x}{e^{9}} \frac{e^{10}-1}{e-1}=\frac{1500}{e^{28}} \frac{e^{15}-1}{e-1}
$$

from which

$$
x=\frac{1500}{e^{19}} \frac{e^{15}-1}{e-1}
$$

Performing the calculations:

$$
x=1188 \mathrm{~K}
$$

(Fried Ernö, Budapest.)