10. The number of students in mathematies class is four fifths the number of students in English class, The total number of students in both classes doesn't exceed 54 . What is the greatest possible number of students in the English class?

Let \( E \) be the number of students in the English class and \( M \) be the number of students in the mathematics class. According to the problem, we have: 1. \( M = \frac{4}{5}E \). 2. The total number of students does not exceed 54: \[ E + M \leq 54. \] Substitute \( M \) into the total: \[ E + \frac{4}{5}E \leq 54. \] Combine like terms: \[ \frac{9}{5}E \leq 54. \] Multiply both sides by \(\frac{5}{9}\) to solve for \( E \): \[ E \leq 54 \times \frac{5}{9} = 30. \] Since the number of students must be an integer and \( M = \frac{4}{5}E \) must also be an integer, \( E \) must be a multiple of 5. The greatest multiple of 5 that does not exceed 30 is \( 30 \) itself. Thus, the greatest possible number of students in the English class is \( 30 \).