1. **State the problem:** We need to find the maximum number of students that can go on a trip with the condition that there is at least one adult for every seven students, and the total number of passengers (students + adults) cannot exceed 52. 2. **Define variables:** Let $S$ be the number of students and $A$ be the number of adults. 3. **Write the constraints:** - The ratio condition: $A \geq \frac{S}{7}$ (at least one adult per seven students). - The capacity condition: $S + A \leq 52$ (maximum passengers). 4. **Express $A$ in terms of $S$ using the ratio condition:** $$A \geq \frac{S}{7}$$ 5. **Substitute $A$ into the capacity condition:** $$S + A \leq 52 \implies S + \frac{S}{7} \leq 52$$ 6. **Combine terms:** $$S + \frac{S}{7} = \frac{7S}{7} + \frac{S}{7} = \frac{8S}{7} \leq 52$$ 7. **Solve for $S$:** $$\frac{8S}{7} \leq 52 \implies 8S \leq 364 \implies S \leq \frac{364}{8} = 45.5$$ 8. **Interpret the result:** Since $S$ must be an integer number of students, the maximum number of students is $45$. 9. **Check the number of adults needed:** $$A \geq \frac{45}{7} = 6.43$$ So at least 7 adults are needed. 10. **Check total passengers:** $$45 + 7 = 52$$ which fits the coach capacity. **Final answer:** The maximum number of students that can go on the trip is **45**.