First, convert the time to a single unit, hours. 1 hour and 45 minutes is 1 + 45/60 = 1.75 hours. Next, calculate the train's speed (unit rate): 105 miles / 1.75 hours = 60 miles per hour. Finally, use this speed to find the distance traveled in 3 hours: 60 miles/hour * 3 hours = 180 miles.

This is a bottleneck problem. We must find the maximum output of each machine over the 3-hour period. Production machine: 1,800 bolts/hour * 3 hours = 5,400 bolts produced. Packaging machine: First, convert its rate to bolts per hour: 40 bolts/minute * 60 minutes/hour = 2,400 bolts/hour. Then, find its 3-hour capacity: 2,400 bolts/hour * 3 hours = 7,200 bolts packaged. The number of finished bolts is limited by the slower process, which is production. The factory can only package the bolts that have been produced. Therefore, 5,400 finished, packaged bolts will be ready.

First, calculate the actual driving time by subtracting the stop time from the total time: 5 hours - 0.5 hours = 4.5 hours. Next, find the average speed while moving: 270 miles / 4.5 hours = 60 miles per hour. Finally, calculate the time required to travel the remaining 180 miles at this speed: 180 miles / 60 mph = 3 hours.

First, determine the total number of muffins to be made. 4 dozen muffins is 4 * 12 = 48 muffins. Next, find how many batches of 6 muffins are needed to make 48 muffins: 48 / 6 = 8 batches. Since each batch requires 1/4 cup of sugar, the total amount of sugar is the number of batches multiplied by the sugar per batch: 8 batches * (1/4 cup/batch) = 2 cups.

This question tests middle school mathematics skills, specifically solving unit-rate problems by finding the rate at which a certain quantity is used or compared (e.g., ISEE standard for quantitative reasoning). Unit rate is the ratio of two measurements in which the second term is 1. For example, if a car travels 300 miles in 5 hours, the unit rate is 60 miles per hour. In this problem, students must determine the unit rate by dividing the total distance by the time, using details from the passage such as Car A traveling 150 miles in 3 hours. Choice A is correct because it accurately divides the total distance (150 miles) by the time (3 hours), resulting in 150 ÷ 3 = 50 miles per hour, demonstrating an understanding of the concept. Choice C is incorrect because it uses the inverse operation (3 ÷ 150 = 0.02 hours/mile), a common error when students confuse which quantity should be the numerator. To help students: emphasize the importance of units in setting up calculations—practice converting contexts into mathematical expressions. Use real-world examples like speed or price per item and encourage students to check their units (miles/hour means miles divided by hours).

This question tests middle school mathematics skills, specifically solving unit-rate problems by finding the rate at which a certain quantity is used or compared (e.g., ISEE standard for quantitative reasoning). Unit rate is the ratio of two measurements in which the second term is 1. For example, if a cyclist rides 84 miles in 6 hours, the unit rate is miles per hour. In this problem, students must determine the unit rate by dividing the total distance by the time, using details from the passage such as Cyclist A riding 84 miles in 6 hours. Choice A is correct because it accurately divides the total distance (84 miles) by the time (6 hours), resulting in 84 ÷ 6 = 14 miles per hour, demonstrating an understanding of the concept. Choice C is incorrect because it uses the inverse operation (6 ÷ 84 = 0.071 ≈ 0.07 hours/mile), which represents time per mile rather than miles per hour. To help students: emphasize the importance of units in setting up calculations—practice converting contexts into mathematical expressions. Use real-world examples like cycling speeds and encourage students to think about what makes sense (14 mph is a reasonable cycling speed).

This question tests middle school mathematics skills, specifically solving unit-rate problems by finding the rate at which a certain quantity is used or compared (e.g., ISEE standard for quantitative reasoning). Unit rate is the ratio of two measurements in which the second term is 1. For example, if someone packs 48 boxes in 6 hours, the unit rate is boxes per hour. In this problem, students must determine the unit rate by dividing the total number of boxes by the time, using details from the passage such as Maya packing 48 boxes in 6 hours. Choice A is correct because it accurately divides the total boxes (48) by the time (6 hours), resulting in 48 ÷ 6 = 8 boxes per hour, demonstrating an understanding of the concept. Choice C is incorrect because it uses the inverse operation (6 ÷ 48 = 0.125 ≈ 0.13 hours/box), which represents time per box rather than boxes per hour. To help students: emphasize the importance of units in setting up calculations—practice converting contexts into mathematical expressions. Use real-world examples like productivity rates and encourage students to verify their answer makes sense (8 boxes per hour is reasonable for packing work).

There are two common ways to solve this. Method 1: Find the car's fuel efficiency: 480 miles / 16 gallons = 30 miles per gallon. Then, find how many gallons are needed for the 120-mile trip: 120 miles / 30 mpg = 4 gallons. Finally, calculate the cost: 4 gallons * $4.50/gallon = $18.00. Method 2: Determine the fraction of the tank used for the trip: 120 miles / 480 miles = 1/4 of a tank. Calculate the cost of a full tank: 16 gallons * $4.50/gallon = $72.00. The cost for the trip is 1/4 of the cost of a full tank: (1/4) * $72.00 = $18.00.

First, calculate the net fill rate by subtracting the drain rate from the fill rate: 4 gallons/minute - 1.5 gallons/minute = 2.5 gallons/minute. This is the rate at which the water level actually rises. Next, divide the total capacity of the tub by the net fill rate to find the time to fill: 50 gallons / 2.5 gallons/minute = 20 minutes.

First, determine the unit rate of defects. The rate is 3 defects per 500 bulbs. We can set up a proportion to solve for the number of defects (x) in a batch of 12,000 bulbs: (3 defects / 500 bulbs) = (x defects / 12,000 bulbs). To solve for x, cross-multiply: 500 * x = 3 * 12,000, which gives 500x = 36,000. Divide both sides by 500: x = 36,000 / 500 = 72. So, 72 defective bulbs are expected.