## Task 5 - 090835

From 77 percent and 87 percent spirits, and only these, exactly $1000 \mathrm{~g}$ of 80 percent spirits is to be produced by mixing.

Determine the exact masses required!

The percentages refer to the masses.
If there is a solution to the problem, let $x$ be the measure of the mass of the 77% spirit. Then $(1000-x)$ is the measure of the mass of the 87% spirit, and we have:

$$
\frac{77}{100} x+\frac{87}{100}(1000-x)=\frac{80}{100} \cdot 1000
$$

thus $77 x+87000-87 x=80000$, from which we obtain $x=700$.

Therefore, the only masses that can be a solution to the problem are $700 \mathrm{~g}$ of 77% spirit and $300 \mathrm{~g}$ of 87% spirit.

These masses indeed have the required property; for $700 \mathrm{~g}$ of 77% spirit contain exactly $539 \mathrm{~g}$ of pure spirit; $300 \mathrm{~g}$ of 87% spirit contain exactly $261 \mathrm{~g}$ of pure spirit. Together, these are $800 \mathrm{~g}$ of pure spirit.

By definition, $1000 \mathrm{~g}$ of a mixture that contains $800 \mathrm{~g}$ of spirit and $200 \mathrm{~g}$ of water is referred to as 80% spirit, which was to be produced according to the problem.