1. **State the problem:** We know 752 tickets were sold with the ratio of adult to children tickets as 3:1. We need to find the number of adult and children tickets sold. 2. **Use the ratio formula:** The total parts in the ratio are $3 + 1 = 4$ parts. 3. **Calculate one part:** Total tickets divided by total parts gives one part: $$\frac{752}{4} = 188$$ 4. **Calculate number of adult tickets:** Adult tickets are 3 parts: $$3 \times 188 = 564$$ 5. **Calculate number of children tickets:** Children tickets are 1 part: $$1 \times 188 = 188$$ --- 6. **Next problem:** An adult ticket costs 2.5 times as much as a children’s ticket. Total income is 17578. Find the cost of a children’s ticket. 7. **Let the cost of a children’s ticket be $x$.** Then adult ticket cost is $2.5x$. 8. **Write income equation:** $$564 \times 2.5x + 188 \times x = 17578$$ 9. **Simplify:** $$1410x + 188x = 17578$$ $$1598x = 17578$$ 10. **Solve for $x$:** $$x = \frac{17578}{1598}$$ 11. **Simplify fraction by canceling common factors:** $$x = \frac{\cancel{17578}}{\cancel{1598}}$$ (Here, 1598 divides into 17578 exactly 11 times) 12. **Calculate:** $$x = 11$$ **Final answers:** - Number of adult tickets = 564 - Number of children tickets = 188 - Cost of a children’s ticket = 11