Question: Question 6: Two sump pumps working at the same rate drain a flooded basement in 69 hours. How long does it take 3 pumps working at the same rate to drain the basement? Question 7: A solution contains $\frac{1}{4}$ ounce acid and $8 \frac{1}{2}$ ounces of water. For the same strength solution, how much acid should be mixed with $12 \frac{3}{4}$ ounces of water? Question 8: A piece of lumber $2.8$ meters long weighs $24.5$ kilograms. A piece $0.8$ meter long is cut from the $2.8$-meter length. Determine the weight of the $0.8$-meter piece. Question 9: The volume of gas decreases as pressure increases. When pressure is $45.0$ units, the volume is $9.0$ units. Find the proportionality constant for this situation.

1. **Problem 6:** Two sump pumps working at the same rate drain a flooded basement in $69$ hours. Find how long $3$ pumps take. 2. **Formula:** Work rate is additive. If $2$ pumps take $69$ hours, their combined rate is $\frac{1}{69}$ basements per hour. 3. **Calculate rate per pump:** Rate per pump $= \frac{1}{2} \times \frac{1}{69} = \frac{1}{138}$ basements per hour. 4. **Calculate time for 3 pumps:** Combined rate for $3$ pumps $= 3 \times \frac{1}{138} = \frac{3}{138} = \frac{1}{46}$ basements per hour. 5. **Time taken by 3 pumps:** $\text{Time} = \frac{1}{\text{rate}} = \frac{1}{\frac{1}{46}} = 46$ hours. --- 6. **Problem 7:** A solution has $\frac{1}{4}$ ounce acid and $8 \frac{1}{2}$ ounces water. Find acid amount for $12 \frac{3}{4}$ ounces water at same strength. 7. **Convert mixed numbers:** $8 \frac{1}{2} = \frac{17}{2}$, $12 \frac{3}{4} = \frac{51}{4}$. 8. **Strength ratio:** $\text{acid} : \text{water} = \frac{1}{4} : \frac{17}{2} = \frac{1}{4} \div \frac{17}{2} = \frac{1}{4} \times \frac{2}{17} = \frac{2}{68} = \frac{1}{34}$. 9. **Find acid for new water amount:** $\text{acid} = \text{strength} \times \text{water} = \frac{1}{34} \times \frac{51}{4} = \frac{51}{136} = 0.375$ ounces. --- 10. **Problem 8:** Lumber $2.8$ m weighs $24.5$ kg. Find weight of $0.8$ m piece. 11. **Weight per meter:** $\frac{24.5}{2.8} = 8.75$ kg/m. 12. **Weight of $0.8$ m piece:** $8.75 \times 0.8 = 7.0$ kg. --- 13. **Problem 9:** Volume decreases as pressure increases. At pressure $45.0$, volume $9.0$. Find proportionality constant $k$ where $P \times V = k$. 14. **Calculate $k$:** $k = P \times V = 45.0 \times 9.0 = 405.0$ units. **Final answers:** - Q6: $46$ hours - Q7: $0.375$ ounces acid - Q8: $7.0$ kg - Q9: $405.0$ units