5. 

In a 6-liter vessel, 4 liters of a 70% (by volume) sulfuric acid solution are poured, and in a second vessel of the same capacity, 3 liters of a 90% solution of sulfuric acid are poured. A certain amount of the solution is transferred from the second vessel to the first so that it becomes an $r-\%$ solution of sulfuric acid. Find the greatest integer value of $\mathrm{r}$ for which the problem has a solution.
# Solution.

Let $x$ liters of the solution be transferred from the second vessel to the first. Since it follows from the condition that $0 \leq x \leq 2$, to find the amount of pure acid in the new solution, we obtain the equation $2.8 + 0.9x = (4 + x) \frac{r}{100}$, from which $x = \frac{4r - 280}{90 - r}$. Now, considering that $x \in [0; 2]$, we conclude that the problem has a solution only when $r \in \left[70; \frac{230}{3}\right]$. Therefore, the largest integer $r$ for which the problem has a solution is 76.

Answer: 76