13.022. On the mathematics entrance exam, $15 \%$ of the applicants did not solve a single problem, 144 people solved problems with errors, and the number of those who solved all problems correctly is to the number of those who did not solve any at all as 5:3. How many people took the mathematics exam that day
Solution.

Let $x$ be the total number of people who took the exam. $0.15x$ people did not solve any problems; $0.25x$ people solved all the problems, which is $\frac{0.15x \cdot 5}{3}=0.25x$ people. According to the problem, $0.15x + 144 + 0.25x = x$, from which we get $x = 240$.

Answer: 240.