1. **State the problem:** The original ratio of students to teachers is 5:1. After 10 more teachers join, the ratio changes to 5:2. We need to find the original number of students. 2. **Define variables:** Let the number of students be $5x$ and the number of teachers be $x$ (since the ratio is 5:1). 3. **Set up the equation for the new ratio:** After 10 more teachers join, the number of teachers becomes $x + 10$. The new ratio is given as 5:2, so: $$\frac{5x}{x + 10} = \frac{5}{2}$$ 4. **Solve the equation:** Cross-multiply: $$5x \times 2 = 5 \times (x + 10)$$ $$10x = 5x + 50$$ Subtract $5x$ from both sides: $$10x - 5x = 50$$ $$5x = 50$$ Divide both sides by 5: $$x = 10$$ 5. **Find the original number of students:** Since students = $5x$, substitute $x=10$: $$5 \times 10 = 50$$ **Final answer:** There were originally 50 students.