We need to find how many gallons the hose fills in 26 minutes at a constant rate. First, calculate the filling rate: 45 gallons ÷ 18 minutes = 2.5 gallons per minute. Then multiply by the new time: 2.5 gallons/minute × 26 minutes = 65 gallons. Common errors include dividing 26 by 45 (which gives a meaningless ratio) or treating 45 as the rate without dividing by 18. Always set up rates as quantity per unit time, then multiply by the desired time.

We need to find how long it takes to drain 63 gallons when the pump removes 18 gallons every 4 minutes. First, find the drainage rate: 18 gallons ÷ 4 minutes = 4.5 gallons per minute. Then find the time: 63 gallons ÷ 4.5 gallons/minute = 14 minutes. The key is recognizing this as a rate problem where we need gallons per minute first. A common mistake is setting up the proportion backwards, which would give minutes per gallon instead.

We need to find how much flour is needed for 32 muffins when 2.5 cups make 20 muffins. First, find the rate of flour per muffin: 2.5 cups ÷ 20 muffins = 0.125 cups per muffin. Then multiply by 32 muffins: 0.125 cups/muffin × 32 muffins = 4.0 cups. This is a scaling problem where we find the unit rate first. Watch out for the temptation to use mental math shortcuts that might introduce rounding errors.

The question asks for the time to paint 1 room when two workers collaborate. Worker A's rate is 1/6 room per hour, and Worker B's rate is 1/8 room per hour. When working together, add their rates: 1/6 + 1/8 = 4/24 + 3/24 = 7/24 rooms per hour. To find time for 1 room, divide: 1 room ÷ (7/24 rooms/hour) = 1 × 24/7 = 24/7 hours. This equals 3 3/7 hours. The key insight is that rates add when workers collaborate, not times. Always work with rates (rooms per hour) rather than times (hours per room) when combining efforts.

The question asks for the car’s speed in miles per hour, given it travels 96 miles in 1 hour 36 minutes at a constant speed. The rate relationship is distance in miles per hour of time. First, convert 1 hour 36 minutes to hours: 36 minutes ÷ 60 minutes per hour = 0.6 hours, so total time = 1.6 hours. Then, speed = 96 miles ÷ 1.6 hours = 60 miles per hour, with units miles over hours. A key error is forgetting to convert minutes to hours, leading to incorrect time like 1.36 hours. Another mistake is treating 36 minutes as 0.36 hours without proper division. For test-taking, always convert all times to the same unit before calculating rates to ensure accurate unit analysis.

We need to find how long it takes to ride 30 miles at the same speed the cyclist used to ride 18 miles in 1.5 hours. First, calculate the cyclist's rate: rate = 18 miles ÷ 1.5 hours = 12 miles per hour. To find the time for 30 miles, use time = distance ÷ rate: time = 30 miles ÷ 12 miles/hour = 2.5 hours. A common error is to set up a proportion incorrectly or forget to find the rate first. When dealing with rate problems, always identify what quantity per what unit you're working with.

We need to find how many words the student types in 25 minutes when typing 540 words in 12 minutes. First, find the typing rate: 540 words ÷ 12 minutes = 45 words per minute. Then multiply by 25 minutes: 45 words/minute × 25 minutes = 1125 words. The key is maintaining consistent units throughout the calculation. A common error is trying to set up a proportion without first finding the unit rate, which can lead to inverted fractions.

The question asks how many flyers the printer can produce in 45 minutes, given it produces 180 flyers in 12 minutes at a constant rate, with the answer in flyers. The rate relationship is flyers per minute. Calculate the rate: 180 flyers ÷ 12 minutes = 15 flyers per minute, with units flyers over minutes. Then, for 45 minutes: 15 flyers per minute × 45 minutes = 675 flyers, where minutes cancel out leaving flyers. A key error is setting up the proportion incorrectly, like dividing time by flyers instead. Another mistake could be confusing minutes with hours without conversion. For test-taking, set up the rate first and multiply by the new time, checking units to ensure they cancel properly.

The question asks how many minutes it will take to run 8 kilometers at the same constant pace, rounded to the nearest whole minute. The rate is pace, in minutes per kilometer or equivalently kilometers per minute. First, find the pace: 24 minutes / 5 kilometers = 4.8 minutes per kilometer. Then, for 8 kilometers: 8 km * 4.8 min/km = 38.4 minutes, which rounds to 38 minutes since 0.4 is less than 0.5. Unit analysis shows kilometers cancel, leaving minutes, emphasizing the importance of consistent units. A key error is setting up the proportion backwards, like using kilometers per minute incorrectly, leading to times like 15 minutes. For rounding in rate problems, always apply standard rules after calculating to ensure accuracy.

When you encounter average speed problems, remember that average speed isn't simply the arithmetic mean of the speeds—it's total distance divided by total time.

Let's calculate the total distance and time for this trip. In the first part, the driver travels for 2 hours at 40 mph, covering $$2 \times 40 = 80$$ miles. In the second part, they travel for 1 hour at 60 mph, covering $$1 \times 60 = 60$$ miles. The total distance is $$80 + 60 = 140$$ miles, and the total time is $$2 + 1 = 3$$ hours.

Therefore, the average speed is $$\frac{140 \text{ miles}}{3 \text{ hours}} = 46.67$$ mph, which rounds to 47 mph. This confirms answer choice B.

Now let's see why the other answers are wrong. Choice A (45 mph) might tempt you if you incorrectly weighted the speeds by time: $$\frac{(40 \times 2) + (60 \times 1)}{2 + 1} = \frac{140}{3}$$, but then made an arithmetic error. Choice C (50 mph) is the simple arithmetic mean of 40 and 60 mph, which ignores the fact that the driver spent more time at the slower speed. Choice D (55 mph) has no clear mathematical basis but might appeal if you mistakenly thought the average should be closer to the higher speed.

The key insight: average speed problems require you to find total distance and total time separately, then divide. Never just average the individual speeds unless the time periods are equal.