4. There is 25 ml of a $70\%$ acetic acid solution and 500 ml of a $5\%$ acetic acid solution. Find the maximum volume of a $9\%$ acetic acid solution that can be obtained from the available solutions (no dilution with water is allowed).

#
# Solution

Let's take $x$ ml of the 70% solution, and $\mathrm{V}$ ml - the volume of the resulting 9% solution. Then we have the equation

$$
0.7 x + 0.05(V - x) = 0.09 V
$$

from which we find

$$
\frac{x}{V} = \frac{4}{65}
$$

that is, we take 4 parts of the 70% solution and 61 parts of the 5% solution.

The largest volume of the 9% solution will be obtained if and only if at least one of the available solutions is taken completely.

If we take all of the 5% solution, then $1/65$ of the resulting solution will be 500/61 ml. Then we need to take $\frac{4 \times 500}{61}$ ml of the 70% solution. This is impossible, as $\frac{4 \times 500}{61} > 25$. Therefore, we take all of the 70% solution, and in this case, $1/65$ of the resulting solution will be $25/4$ ml. Then the largest volume is $\frac{25}{4} \times 65 = \frac{1625}{4} = 406.25$ ml.

## Grading Criteria

Incorrect equation set up according to the problem statement - 0 points.

Only the equation is correctly set up according to the problem statement - 1 point.

The equation is correctly set up according to the problem statement. However, a computational error is made in the subsequent solution, although the solution approach is correct - 4 points. The equation is correctly set up according to the problem statement. However, there is an error in the subsequent reasoning, making the solution approach incorrect - 1 point.

The equation is correctly set up according to the problem statement. The subsequent solution is correct, but there are no justifications (only calculations) - 4 points.

The problem is solved correctly, completely, and the solution is presented with justifications - 7 points.

If it is not specified that to obtain the largest volume of the 9% solution, at least one of the available solutions must be taken completely, but the subsequent solution is based on this consideration, no points are deducted.