When you encounter a multi-step rate problem like this, break it down into phases and track what happens in each one.

First, calculate how much water Hose A adds during its solo 20 minutes: $$15 \text{ gallons/minute} \times 20 \text{ minutes} = 300 \text{ gallons}$$. This leaves $$1350 - 300 = 1050 \text{ gallons}$$ still needed.

Next, determine the combined rate when both hoses run together: $$15 + 20 = 35 \text{ gallons per minute}$$. To find how long it takes to fill the remaining 1050 gallons at this combined rate: $$\frac{1050 \text{ gallons}}{35 \text{ gallons/minute}} = 30 \text{ minutes}$$. This confirms answer B.

Now let's see where the wrong answers come from. Choice A (28 minutes) likely results from a calculation error, perhaps incorrectly computing $$\frac{1050}{35}$$. Choice C (32 minutes) might come from using the wrong remaining volume or making an arithmetic mistake in the division. Choice D (34 minutes) could result from forgetting to add the rates together and instead using just one hose's rate for the second phase.

The key strategy for rate problems is to organize your work in clear steps: identify what happens in each time period, calculate any intermediate amounts, then solve for the unknown time using the formula $$\text{time} = \frac{\text{amount}}{\text{rate}}$$. Always double-check that your rates match the scenario—individual rates for solo work, combined rates when working together.

When you encounter problems involving simultaneous filling and draining, think about the net rate of change. You need to find how much water is actually being added to the pool each minute after accounting for both the inflow and outflow.

The inlet adds 40 gallons per minute while the drain removes 25 gallons per minute, giving you a net rate of $$40 - 25 = 15$$ gallons per minute. This means the pool gains 15 gallons every minute.

Since the pool starts with 200 gallons and needs to reach 800 gallons, it needs $$800 - 200 = 600$$ more gallons. At a net rate of 15 gallons per minute, this will take $$600 \div 15 = 40$$ minutes.

Looking at the wrong answers: Choice A (35 minutes) likely comes from incorrectly calculating the gallons needed or making an arithmetic error. Choice C (45 minutes) might result from using the wrong net rate, perhaps calculating $$40 - 25 = 13.33$$ through sloppy math. Choice D (50 minutes) could come from dividing 600 by 12 instead of 15, suggesting confusion about the net rate calculation.

The correct answer is B: 40 minutes.

Study tip: For rate problems involving opposing forces (filling vs. draining, traveling with vs. against wind, etc.), always identify the net effect first. Subtract the opposing rate from the working rate, then use this net rate for your time calculations. This approach prevents the common mistake of using individual rates instead of the combined effect.

When you encounter work rate problems involving multiple workers or pipes, think in terms of rates of work completion. Each pipe has a rate measured in "fraction of tank filled per hour."

Pipe A fills the tank in 15 hours, so its rate is $$\frac{1}{15}$$ tank per hour. When both pipes work together, they complete the job in 6 hours, giving a combined rate of $$\frac{1}{6}$$ tank per hour.

To find Pipe B's rate: Combined rate = Pipe A's rate + Pipe B's rate, so $$\frac{1}{6} = \frac{1}{15} + \frac{1}{B}$$. Solving: $$\frac{1}{B} = \frac{1}{6} - \frac{1}{15} = \frac{5-2}{30} = \frac{1}{10}$$. Therefore, Pipe B fills the tank in 10 hours working alone.

Now for the actual problem: Pipe B works alone for 4 hours, filling $$4 \times \frac{1}{10} = \frac{4}{10} = \frac{2}{5}$$ of the tank. This leaves $$1 - \frac{2}{5} = \frac{3}{5}$$ of the tank remaining.

When both pipes work together at their combined rate of $$\frac{1}{6}$$ tank per hour, the time needed to fill the remaining $$\frac{3}{5}$$ is: $$\frac{3/5}{1/6} = \frac{3}{5} \times 6 = 3.6$$ hours = 3½ hours.

Choice A (3 hours) incorrectly assumes the remaining work is $$\frac{1}{2}$$ of the tank. Choice C (4 hours) mistakes Pipe B's solo time for the combined working time. Choice D (4½ hours) likely results from calculation errors in finding the combined rate.

Remember: always convert rates to fractions first, then work systematically through each phase of the problem.

When you encounter multi-step work rate problems, break them down into phases and track what each worker accomplishes in each phase.

First, calculate what Belt A accomplished before breaking down. At 45 packages per hour for 3 hours, Belt A moved $$45 \times 3 = 135$$ packages.

Since the total goal is 500 packages, Belt B must handle the remaining work: $$500 - 135 = 365$$ packages.

Now determine how long Belt B needs to move these 365 packages. At Belt B's rate of 36 packages per hour: $$\frac{365}{36} = 10.14$$ hours. Since we need complete hours and Belt B must finish all packages, this rounds up to approximately 11 hours.

Let's verify: Belt B moving for 11 hours produces $$36 \times 11 = 396$$ packages, which exceeds the required 365 packages.

Answer choice A (9 hours) gives Belt B only $$36 \times 9 = 324$$ packages, falling short of the needed 365. Answer choice B (10 hours) produces $$36 \times 10 = 360$$ packages, still 5 packages short. Answer choice D (12 hours) would work but represents more time than necessary, making it inefficient.

Answer choice C (11 hours) is correct because it's the minimum whole number of hours needed for Belt B to complete the remaining work.

Study tip: In work rate problems with breakdowns or interruptions, always calculate the remaining work after the interruption, then determine the time needed based on the continuing worker's rate. Round up when dealing with minimum time requirements.

When you encounter work rate problems involving multiple machines or workers operating for different time periods, you need to track each contributor's output separately and then combine them to reach the total requirement.

Let's call the total time Oven 2 operates $$t$$ hours. Since Oven 1 operates for only 6 hours at 24 loaves per hour, it produces $$6 \times 24 = 144$$ loaves. Oven 2 operates for the full $$t$$ hours at 18 loaves per hour, producing $$18t$$ loaves. Together, they must bake 300 loaves total.

Setting up the equation: $$144 + 18t = 300$$

Solving for $$t$$: $$18t = 300 - 144 = 156$$, so $$t = 156 ÷ 18 = 8.67$$ hours, which rounds to approximately 9 hours when we verify: $$144 + (18 \times 9) = 144 + 162 = 306$$ loaves (slightly over 300, which is acceptable since we can't operate partial hours).

Choice A (8 hours) falls short: $$144 + (18 \times 8) = 288$$ loaves, which is 12 loaves under the requirement. Choice C (10 hours) produces $$144 + 180 = 324$$ loaves, significantly more than needed. Choice D (11 hours) produces $$144 + 198 = 342$$ loaves, which is excessive.

Therefore, B is correct.

Strategy tip: In work rate problems with different operating times, always set up your equation by adding each worker's individual contribution (rate × time = output) and setting the sum equal to the total requirement. Double-check your answer by substituting back into the original scenario.