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Measurement With Unit Rates - ISEE Lower Level Mathematics Achievement

  1. A bathtub is being filled with water at a rate of 4 gallons per minute. The tub has a capacity of 48 gallons. If the tub is already one-fourth full, how many more minutes will it take to fill it completely?
    Answer: 9 minutes First, find out how much water is already in the tub. One-fourth of 48 gallons is 48 / 4 = 12 gallons. Next, find out how many more gallons are needed to fill the tub: 48 gallons (capacity) - 12 gallons (current) = 36 gallons. Finally, divide the remaining volume by the fill rate: 36 gallons / 4 gallons per minute = 9 minutes.
  2. A certain type of metal weighs 5 grams per cubic centimeter. A block of this metal has a volume of 40 cubic centimeters. If this metal costs $2 per gram, what is the total cost of the block?
    Answer: $400 This is a two-step problem. First, calculate the total weight of the metal block. Weight = Density × Volume = 5 grams/cm³ × 40 cm³ = 200 grams. Second, calculate the total cost using the price per gram. Cost = Total weight × Price per gram = 200 grams × $2/gram = $400.
  3. If paint covers 150 square feet per gallon, how many gallons are needed for 450 square feet?
    Answer: 3 gallons This question tests the ability to solve measurement problems using unit rates on the ISEE Lower Level. Understanding unit rates involves calculating how one unit relates to another, such as miles per hour or cost per item. In this specific problem, students must apply the given unit rate of 150 square feet per gallon to determine how many gallons are needed for 450 square feet. The correct answer, Choice B (3 gallons), is calculated by dividing the total area (450 square feet) by the coverage per gallon (150 square feet per gallon): 450 ÷ 150 = 3 gallons, which shows a correct understanding of applying unit rates. Choice C (300 gallons) is incorrect because it reflects a common student error of subtracting the numbers (450 - 150), while Choice D (0.30 gallons) results from dividing in the wrong order (150 ÷ 450 ≈ 0.33). To help students, encourage them to practice identifying unit rates in various contexts and performing calculations step-by-step. Teach students to double-check units and ensure each calculation step logically follows from the last, remembering that number of units needed = total amount ÷ amount per unit.
  4. If paint covers 200 square feet per gallon, how many gallons are needed for 600 square feet?
    Answer: 3 gallons This question tests the ability to solve measurement problems using unit rates on the ISEE Lower Level. Understanding unit rates involves calculating how one unit relates to another, such as miles per hour or cost per item. In this specific problem, students must apply the given unit rate of 200 square feet per gallon to determine how many gallons are needed for 600 square feet. The correct answer, Choice B (3 gallons), is calculated by dividing the total area (600 square feet) by the coverage per gallon (200 square feet per gallon): 600 ÷ 200 = 3 gallons, which shows a correct understanding of applying unit rates. Choice C (800 gallons) is incorrect because it reflects a common student error of adding the numbers instead of dividing (600 + 200), while Choice D (0.33 gallons) results from dividing in the wrong order (200 ÷ 600). To help students, encourage them to practice identifying unit rates in various contexts and performing calculations step-by-step. Teach students to double-check units and ensure each calculation step logically follows from the last, remembering that number of units needed = total amount ÷ amount per unit.
  5. A seedling is 10 centimeters tall. It grows at a steady rate of 2 centimeters every 3 days. How tall will the seedling be in 15 days?
    Answer: 20 cm First, calculate the total growth over 15 days. The rate is 2 cm per 3 days. The number of 3-day periods in 15 days is 15 / 3 = 5. So, the total growth is 5 × 2 cm = 10 cm. Then, add this growth to the seedling's initial height: 10 cm (initial) + 10 cm (growth) = 20 cm.
  6. Jamal is paid $12 per hour for raking leaves. He wants to buy a video game that costs $75. If he has already saved $27, how many hours must he work to afford the game?
    Answer: 4 hours First, determine how much more money Jamal needs to save: $75 (cost) - $27 (saved) = $48. The unit rate for his work is $12 per hour. To find the number of hours he needs to work, divide the remaining amount by his hourly wage: $48 / $12 per hour = 4 hours.
  7. An office printer can print 6 pages every 15 seconds. At this rate, how many minutes will it take to print a 120-page document?
    Answer: 5 minutes First, find the unit rate in seconds per page: 15 seconds / 6 pages = 2.5 seconds per page. Next, calculate the total time in seconds to print 120 pages: 2.5 seconds/page × 120 pages = 300 seconds. Finally, convert the total seconds into minutes by dividing by 60: 300 seconds / 60 seconds/minute = 5 minutes.
  8. A snail moves at a rate of 2 feet per minute. There are 3 feet in a yard. How many yards will the snail travel in one hour?
    Answer: 40 yards First, find the total distance the snail moves in one hour (60 minutes) in feet. Distance in feet = 2 feet/minute × 60 minutes = 120 feet. Next, convert this distance from feet to yards using the given conversion factor. Distance in yards = 120 feet / 3 feet/yard = 40 yards.
  9. A train travels 90 miles in 2 hours. Maintaining this constant speed, how many hours will it take for the train to travel 225 miles?
    Answer: 5.0 hours First, find the train's speed, which is the unit rate of distance per time. Speed = Distance / Time = 90 miles / 2 hours = 45 miles per hour. Now, use this speed to find the time it takes to travel 225 miles. Time = Distance / Speed = 225 miles / 45 miles per hour = 5 hours.
  10. A baker can decorate 15 cupcakes in 30 minutes. The baker needs to decorate 120 cupcakes for a party. After working for 1 hour, how many cupcakes are left to decorate?
    Answer: 90 cupcakes First, determine the baker's decoration rate. Rate = 15 cupcakes / 30 minutes = 0.5 cupcakes per minute. In 1 hour (60 minutes), the baker decorates: 0.5 cupcakes/minute × 60 minutes = 30 cupcakes. Finally, subtract the number of decorated cupcakes from the total needed: 120 total - 30 decorated = 90 cupcakes left.
  11. A juice machine makes 18 liters of fruit punch every 6 minutes. The recipe requires 2 parts cranberry juice for every 1 part orange juice. How many liters of cranberry juice does the machine use in 10 minutes?
    Answer: 20 liters First, find the rate of punch production: 18 liters / 6 minutes = 3 liters per minute. In 10 minutes, the machine makes 3 L/min × 10 min = 30 liters of punch. The recipe has 2+1=3 total parts. The fraction of cranberry juice is 2/3. So, the amount of cranberry juice is (2/3) × 30 liters = 20 liters.
  12. A company uses 3 pounds of ink to print 600 posters. How many pounds of ink would be needed to print 1,000 posters?
    Answer: 5 pounds First, find the unit rate of ink used per poster. Rate = 3 pounds / 600 posters = 0.005 pounds per poster. Then, multiply this rate by the new number of posters. Ink needed = 0.005 pounds/poster × 1,000 posters = 5 pounds. Alternatively, find posters per pound: 600/3 = 200 posters per pound. For 1,000 posters, you need 1,000/200 = 5 pounds.
  13. Maria reads at a constant rate of 25 pages every 30 minutes. If her book is 350 pages long and she has already read 100 pages, how many more hours will it take her to finish the book?
    Answer: 5 hours First, find the number of pages Maria still needs to read: 350 total pages - 100 pages read = 250 pages left. Next, find her reading rate in pages per hour. Since she reads 25 pages in 30 minutes (0.5 hours), her rate is 25 pages / 0.5 hours = 50 pages per hour. Finally, divide the remaining pages by her reading rate to find the time needed: 250 pages / 50 pages per hour = 5 hours.
  14. A long-distance runner can run 12 miles in 90 minutes. At this pace, how many minutes would it take the runner to complete a 20-mile race?
    Answer: 150 minutes This is a rate problem where you need to find how long it takes to travel a certain distance at a constant pace. When you see questions about consistent rates or speeds, set up a proportion to compare the known information with what you're trying to find. First, find the runner's pace. If they run 12 miles in 90 minutes, their rate is 12 miles90 minutes=2 miles15 minutes\frac{12 \text{ miles}}{90 \text{ minutes}} = \frac{2 \text{ miles}}{15 \text{ minutes}}. You can also think of this as 90 minutes12 miles=7.5 minutes per mile\frac{90 \text{ minutes}}{12 \text{ miles}} = 7.5 \text{ minutes per mile}. Now apply this rate to 20 miles: 20 miles×7.5 minutes per mile=150 minutes20 \text{ miles} \times 7.5 \text{ minutes per mile} = 150 \text{ minutes}. Alternatively, you can set up the proportion: 90 minutes12 miles=x minutes20 miles\frac{90 \text{ minutes}}{12 \text{ miles}} = \frac{x \text{ minutes}}{20 \text{ miles}}. Cross-multiplying gives you 90×20=12x90 \times 20 = 12x, so x=180012=150x = \frac{1800}{12} = 150 minutes. Choice A (120 minutes) likely comes from incorrectly assuming the runner's pace is exactly 6 minutes per mile instead of 7.5. Choice B (135 minutes) might result from miscalculating the proportion or making an arithmetic error. Choice C (160 minutes) could come from rounding 7.5 up to 8 minutes per mile. Remember that rate problems often involve three pieces: distance, time, and rate. When you know two of these, you can always find the third. Setting up proportions or using unit rates (like minutes per mile) are both reliable methods for solving these consistently.
  15. A leaky faucet drips 5 milliliters of water every 10 seconds. How many liters of water will it waste in one hour? (1 liter = 1,000 milliliters)
    Answer: 1.8 liters First, find the drip rate in milliliters per second: 5 mL / 10 seconds = 0.5 mL per second. Next, find the total seconds in an hour: 60 seconds/minute × 60 minutes/hour = 3,600 seconds. Calculate the total milliliters wasted: 0.5 mL/second × 3,600 seconds = 1,800 milliliters. Finally, convert milliliters to liters: 1,800 mL / 1,000 mL/liter = 1.8 liters.
  16. At a farmer's market, 5 pounds of apples cost $7.50. Based on this rate, what would be the cost of 8 pounds of apples?
    Answer: $12.00 First, calculate the cost per pound (the unit rate). Cost per pound = Total cost / Number of pounds = $7.50 / 5 pounds = $1.50 per pound. Then, multiply this unit rate by the desired number of pounds to find the total cost. Cost for 8 pounds = $1.50/pound × 8 pounds = $12.00.
  17. A factory machine produces 240 toy cars in an 8-hour shift. If the machine runs for 3 hours, how many toy cars will it produce?
    Answer: 90 cars First, find the production rate of the machine in toy cars per hour. Rate = 240 cars / 8 hours = 30 cars per hour. Then, multiply this rate by the new time period to find the total production. Production = 30 cars/hour × 3 hours = 90 cars.
  18. A car's gas tank holds 15 gallons and it can travel 450 miles on a full tank. The car currently has 4 gallons of gas. How many miles can the car travel before it runs out of gas?
    Answer: 120 miles First, find the car's fuel efficiency in miles per gallon (the unit rate). Efficiency = Total miles / Total gallons = 450 miles / 15 gallons = 30 miles per gallon. Then, multiply this rate by the amount of gas currently in the tank. Distance = 30 miles/gallon × 4 gallons = 120 miles.
  19. A recipe for 12 muffins requires 3 cups of flour. How many cups of flour are needed to make 20 muffins?
    Answer: 5 cups First, find the unit rate of flour per muffin. Rate = 3 cups / 12 muffins = 0.25 cups per muffin. Then, multiply this rate by the desired number of muffins. Flour needed = 0.25 cups/muffin × 20 muffins = 5 cups. Alternatively, notice that 3/12 simplifies to 1/4, meaning 1 cup of flour makes 4 muffins. To make 20 muffins, you need 20 / 4 = 5 cups.
  20. If a car travels at 45 miles per hour, how long will it take to travel 90 miles?
    Answer: 2 hours This question tests the ability to solve measurement problems using unit rates on the ISEE Lower Level. Understanding unit rates involves calculating how one unit relates to another, such as miles per hour or cost per item. In this specific problem, students must apply the given unit rate of 45 miles per hour to determine how long it takes to travel 90 miles. The correct answer, Choice B (2 hours), is calculated by dividing the total distance (90 miles) by the speed (45 miles per hour): 90 ÷ 45 = 2 hours, which shows a correct understanding of applying unit rates. Choice C (45 hours) is incorrect because it reflects a common student error of using the speed as the answer without performing any calculation, while Choice A (1 hour) might result from incorrectly halving the distance without considering the speed. To help students, encourage them to practice identifying unit rates in various contexts and performing calculations step-by-step. Teach students to double-check units and ensure each calculation step logically follows from the last, remembering that time = distance ÷ speed.

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