Unit Rate Problems
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ISEE Middle Level: Mathematics Achievement › Unit Rate Problems
Two cars drive at steady speeds on a highway. Car A travels 150 miles in 3 hours, and Car B travels 180 miles in 4 hours. A passenger mistakenly says the faster car is 3/150 hours/mile. What is the unit rate of Car A in miles/hour?
0.50 miles/hour
50 miles/hour
45 miles/hour
0.02 hours/mile
Explanation
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).
Two cyclists ride at steady speeds on a trail. Cyclist A rides 84 miles in 6 hours, and Cyclist B rides 90 miles in 5 hours. A student says 6/84 is miles/hour for Cyclist A. What is the unit rate of Cyclist A in miles/hour?
0.14 miles/hour
14 miles/hour
16 miles/hour
0.07 hours/mile
Explanation
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).
A train and a bus travel at constant speeds. The train goes 240 miles in 4 hours, and the bus goes 210 miles in 3 hours. Someone claims the train’s speed is 4/240 miles/hour. Which of the following is the correct unit rate for the train?
0.06 miles/hour
60 miles/hour
70 miles/hour
0.02 hours/mile
Explanation
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 train travels 240 miles in 4 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 the train going 240 miles in 4 hours. Choice A is correct because it accurately divides the total distance (240 miles) by the time (4 hours), resulting in 240 ÷ 4 = 60 miles per hour, demonstrating an understanding of the concept. Choice B is incorrect because it uses the inverse operation (4 ÷ 240 = 0.0167, not 0.06), which represents hours per mile and shows a calculation error as well. To help students: emphasize the importance of units in setting up calculations—practice converting contexts into mathematical expressions. Use real-world examples like train speeds and encourage students to check if their answer is reasonable (60 mph is a typical train speed).
Two students volunteer to pack boxes for a food drive. Maya packs 48 boxes in 6 hours, and Jordan packs 45 boxes in 5 hours. A friend divides 6/48 and calls it boxes/hour. At what rate does Maya pack boxes per hour?
0.13 hours/box
7 boxes/hour
0.80 boxes/hour
8 boxes/hour
Explanation
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).
Two workers label jars at constant rates. Worker A labels 72 jars in 8 hours, and Worker B labels 63 jars in 7 hours. A coworker adds 72 + 8 to find a rate. At what rate does Worker A label jars per hour?
0.11 hours/jar
9 jars/hour
8 jars/hour
90 jars/hour
Explanation
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 worker labels 72 jars in 8 hours, the unit rate is jars per hour. In this problem, students must determine the unit rate by dividing the total number of jars by the time, using details from the passage such as Worker A labeling 72 jars in 8 hours. Choice B is correct because it accurately divides the total jars (72) by the time (8 hours), resulting in 72 ÷ 8 = 9 jars per hour, demonstrating an understanding of the concept. Choice A is incorrect because it shows 8 jars/hour, which might come from confusing the hours (8) with the rate, or from a calculation error. To help students: emphasize the importance of units in setting up calculations—practice converting contexts into mathematical expressions. Use real-world examples like work rates and encourage students to verify their calculations carefully, especially when numbers are similar.