- A car travels 480 miles on a full 16-gallon tank of gasoline. If gasoline costs $4.50 per gallon, what is the cost of gasoline for a 120-mile trip?
Answer: $18.00
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.
- Leo is paid $180 for 8 hours of work. If he works for 30 hours during a week, and $115 is deducted from his total pay for taxes, what is his take-home pay for the week?
Answer: $560.00
First, find Leo's hourly wage (unit rate): $180 / 8 hours = $22.50 per hour. Next, calculate his total earnings (gross pay) for 30 hours: 30 hours * $22.50/hour = $675.00. Finally, subtract the tax deduction to find his take-home pay: $675.00 - $115.00 = $560.00.
- A rectangular park measuring 0.5 miles wide and 0.8 miles long has 1,200 trees. A neighboring square-shaped forest preserve with sides of 2 miles has 18,000 trees. How much greater is the tree density, in trees per square mile, of the forest preserve than the park?
Answer: 1,500
First, calculate the area and tree density for each location. Park area: 0.5 miles 0.8 miles = 0.4 square miles. Park density: 1,200 trees / 0.4 sq miles = 3,000 trees per square mile. Forest preserve area: 2 miles 2 miles = 4 square miles. Forest preserve density: 18,000 trees / 4 sq miles = 4,500 trees per square mile. Finally, find the difference in densities: 4,500 - 3,000 = 1,500 trees per square mile.
- A water pump moves 2.5 gallons of water in 2 minutes. The pump is used to fill an empty 80-gallon tank, but the tank has a small leak that loses water at a rate of 0.25 gallons per minute. How many minutes will it take for the pump to fill the tank?
Answer: 80
First, find the pumping rate in gallons per minute: 2.5 gallons / 2 minutes = 1.25 gallons per minute. This is the rate at which water enters the tank. Next, calculate the net fill rate by subtracting the leak rate from the pumping rate: 1.25 gal/min - 0.25 gal/min = 1.0 gallon per minute. Finally, divide the tank's capacity by the net fill rate to find the time it takes to fill: 80 gallons / 1.0 gal/min = 80 minutes.
- A 5-pound bag of apples costs $7.50. After peeling and coring, the edible portion of the apples is 80% of the original weight. What is the cost per pound of the edible portion of the apples?
Answer: $1.88
First, calculate the weight of the edible portion. The original weight is 5 pounds, and 80% is edible, so the edible weight is 5 pounds * 0.80 = 4 pounds. The total cost for the bag is $7.50. To find the cost per pound of the edible portion, divide the total cost by the edible weight: $7.50 / 4 pounds = $1.875 per pound. Rounded to the nearest cent, this is $1.88.
- At a factory, one machine produces 1,800 bolts per hour. A second machine can package 40 bolts per minute. If both machines run for a 3-hour shift, how many finished, packaged bolts will be ready for shipment?
Answer: 5,400
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.
- 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?
Answer: 50 miles/hour
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).
- A cell phone plan costs $25 per month, which includes 2 gigabytes (GB) of data. Any additional data costs $8 per gigabyte. If a customer used 5.5 GB of data in one month, what was their total bill?
Answer: $53.00
First, calculate the amount of additional data used beyond the included 2 GB: 5.5 GB - 2 GB = 3.5 GB. Next, calculate the cost of this additional data using the unit rate of $8 per GB: 3.5 GB * $8/GB = $28.00. Finally, add this extra cost to the base monthly fee to find the total bill: $25.00 + $28.00 = $53.00.
- A train travels at a constant speed, covering 105 miles in 1 hour and 45 minutes. How many miles will the train travel in 3 hours?
Answer: 180
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.
- A faucet fills a 50-gallon tub at a rate of 4 gallons per minute. The drain simultaneously empties the tub at a rate of 1.5 gallons per minute. If the tub is initially empty, how many minutes will it take to fill the tub completely?
Answer: 20.0
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.
- A long-distance runner finished a 26-mile marathon in 4 hours. For the first 10 miles, her average speed was 7.5 miles per hour. What was her average speed, in miles per hour, for the remaining 16 miles of the race?
Answer: 6.0
First, calculate the time spent on the first 10 miles. Time = Distance / Speed = 10 miles / 7.5 mph = 10 / (15/2) = 20/15 = 4/3 hours. Next, find the remaining time for the rest of the race. Total time is 4 hours, so remaining time is 4 - 4/3 = 12/3 - 4/3 = 8/3 hours. The remaining distance is 16 miles. Finally, calculate the average speed for the remaining part of the race: Speed = Distance / Time = 16 miles / (8/3 hours) = 16 * (3/8) = 6 miles per hour.
- A factory machine produces 3 defective lightbulbs for every 500 bulbs it makes. If the factory produces a batch of 12,000 lightbulbs, how many of them can be expected to be defective?
Answer: 72
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.
- A 12-ounce bottle of conditioner costs $4.80. A larger 1.5-pound bottle of the same conditioner is on sale for $8.64. How much is saved per ounce by buying the larger bottle? (Note: 1 pound = 16 ounces)
Answer: $0.04
First, find the unit price of the smaller bottle: $4.80 / 12 ounces = $0.40 per ounce. Next, convert the weight of the larger bottle to ounces: 1.5 pounds * 16 ounces/pound = 24 ounces. Then, find the unit price of the larger bottle: $8.64 / 24 ounces = $0.36 per ounce. Finally, find the difference in the unit prices to determine the savings per ounce: $0.40 - $0.36 = $0.04.
- 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?
Answer: 9 jars/hour
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.
- A scooter uses 2 liters of fuel to travel 80 kilometers. A motorcycle uses 3 liters of fuel to travel 105 kilometers. For a 420-kilometer journey, how many more liters of fuel does the motorcycle require than the scooter?
Answer: 1.5
First, find the fuel consumption rate for each vehicle. Scooter: 80 km / 2 liters = 40 km per liter. Motorcycle: 105 km / 3 liters = 35 km per liter. Next, calculate the fuel needed for the 420 km journey for each. Scooter: 420 km / 40 km/liter = 10.5 liters. Motorcycle: 420 km / 35 km/liter = 12 liters. Finally, find the difference in fuel required: 12 liters - 10.5 liters = 1.5 liters.
- The exchange rate from U.S. dollars (USD) to Japanese yen (JPY) is 1 USD = 110 JPY. A currency exchange service charges a flat fee of 5 USD for each transaction. If a tourist wants to receive exactly 33,000 JPY, how many U.S. dollars must they pay in total, including the fee?
Answer: $305
First, determine the amount of U.S. dollars needed to equal 33,000 JPY before the fee. Divide the yen amount by the exchange rate: 33,000 JPY / 110 JPY/USD = 300 USD. This is the amount of money to be converted. The service charges an additional flat fee of 5 USD. So, the total cost is the converted amount plus the fee: 300 USD + 5 USD = 305 USD.
- Printer A can print 250 pages in 5 minutes. Printer B can print 480 pages in 8 minutes. If both printers are run for 10 minutes, how many more pages does Printer B produce than Printer A?
Answer: 100
First, find the unit rate for each printer. Printer A's rate: 250 pages / 5 minutes = 50 pages per minute. Printer B's rate: 480 pages / 8 minutes = 60 pages per minute. Next, calculate the total pages each printer produces in 10 minutes. Printer A: 50 pages/min 10 min = 500 pages. Printer B: 60 pages/min 10 min = 600 pages. Finally, find the difference: 600 pages - 500 pages = 100 pages.
- 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?
Answer: 60 miles/hour
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).
- Jamal can type 120 words in 90 seconds. At this constant rate, how many words can he type in 15 minutes?
Answer: 1,200
First, establish a consistent unit of time. There are 60 seconds in a minute, so 90 seconds is 1.5 minutes. Now find Jamal's typing rate in words per minute: 120 words / 1.5 minutes = 80 words per minute. Finally, multiply this rate by the total time in minutes: 80 words/minute 15 minutes = 1,200 words. An alternative method is to see that 15 minutes is equal to ten 90-second intervals (15 60 / 90 = 10), so he can type 10 * 120 = 1,200 words.
- A file that is 720 megabytes (MB) in size takes 4 minutes to download. At this same rate, how many seconds will it take to download a file that is 1.2 gigabytes (GB) in size? (Note: 1 gigabyte = 1,024 megabytes)
Answer: 409.6
First, establish a consistent unit for file size by converting gigabytes to megabytes: 1.2 GB 1,024 MB/GB = 1,228.8 MB. Next, find the download rate in MB per second. The first file took 4 minutes, which is 4 60 = 240 seconds. The rate is 720 MB / 240 seconds = 3 MB per second. Finally, calculate the time to download the second file by dividing its size by the download rate: 1,228.8 MB / 3 MB/s = 409.6 seconds.
