1. **State the problem:** We have two pipes filling a swimming pool. The larger pipe runs for 4 hours, the smaller pipe for 9 hours, and together they fill half the pool. The smaller pipe takes 10 hours longer than the larger pipe to fill the pool alone. We need to find the time each pipe takes to fill the pool separately. 2. **Define variables:** Let $x$ be the time (in hours) the larger pipe takes to fill the pool alone. Then the smaller pipe takes $x + 10$ hours. 3. **Write rates:** The rate of the larger pipe is $\frac{1}{x}$ pool per hour. The rate of the smaller pipe is $\frac{1}{x+10}$ pool per hour. 4. **Use the given condition:** The amount filled by the larger pipe in 4 hours is $4 \times \frac{1}{x} = \frac{4}{x}$. The amount filled by the smaller pipe in 9 hours is $9 \times \frac{1}{x+10} = \frac{9}{x+10}$. Together, they fill half the pool: $$\frac{4}{x} + \frac{9}{x+10} = \frac{1}{2}$$ 5. **Solve the equation:** Multiply both sides by $2x(x+10)$ to clear denominators: $$2x(x+10) \left( \frac{4}{x} + \frac{9}{x+10} \right) = 2x(x+10) \times \frac{1}{2}$$ Simplify: $$2(x+10) \times 4 + 2x \times 9 = x(x+10)$$ $$8(x+10) + 18x = x^2 + 10x$$ Expand: $$8x + 80 + 18x = x^2 + 10x$$ Combine like terms: $$26x + 80 = x^2 + 10x$$ Bring all terms to one side: $$0 = x^2 + 10x - 26x - 80$$ $$0 = x^2 - 16x - 80$$ 6. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{16 \pm \sqrt{(-16)^2 - 4 \times 1 \times (-80)}}{2} = \frac{16 \pm \sqrt{256 + 320}}{2} = \frac{16 \pm \sqrt{576}}{2} = \frac{16 \pm 24}{2}$$ Two solutions: $$x = \frac{16 + 24}{2} = 20$$ $$x = \frac{16 - 24}{2} = -4$$ (not valid since time cannot be negative) 7. **Find time for smaller pipe:** $$x + 10 = 20 + 10 = 30$$ **Answer:** The larger pipe takes 20 hours to fill the pool alone. The smaller pipe takes 30 hours to fill the pool alone.