ISEE Middle Level Quiz: Proportional Relationships
20 questions · exam conditions
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Proportional RelationshipsQuestion 1 of 20

A store sells T-shirts at a price of 3 for $40. Another store sells the same T-shirts at 5 for $65. If a team needs to buy 30 T-shirts, how much money would be saved by using the store with the better price?

$10
$15
$20
$25
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ISEE Middle Level Quiz

ISEE Middle Level Quiz: Proportional Relationships

Practice Proportional Relationships in ISEE Middle Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Proportional Relationships, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A store sells T-shirts at a price of 3 for $40. Another store sells the same T-shirts at 5 for $65. If a team needs to buy 30 T-shirts, how much money would be saved by using the store with the better price?

  1. $10
  2. $15
  3. $20 (correct answer)
  4. $25
Explanation: First, find the price per shirt at each store. Store 1: $40 / 3 shirts ≈ $13.33/shirt. Store 2: $65 / 5 shirts = $13.00/shirt. Store 2 has the better price. Now, calculate the total cost for 30 shirts from each store. Store 1: Buying 30 shirts is like buying 10 sets of 3 shirts. Cost = 10 * $40 = $400. Store 2: Buying 30 shirts is like buying 6 sets of 5 shirts. Cost = 6 * $65 = $390. The savings is the difference between the two total costs: $400 - $390 = $20.

Question 2

In a collection of marbles, the ratio of red to blue is 3:4, and the ratio of blue to green is 5:2. What is the ratio of red marbles to green marbles?

  1. 3:2
  2. 6:5
  3. 12:8
  4. 15:8 (correct answer)
Explanation: We are given Red:Blue = 3:4 and Blue:Green = 5:2. To find the ratio of Red:Green, we need to make the 'Blue' term the same in both ratios. The least common multiple of 4 and 5 is 20. Convert the first ratio by multiplying by 5: Red:Blue = (35):(45) = 15:20. Convert the second ratio by multiplying by 4: Blue:Green = (54):(24) = 20:8. Now that the Blue term is 20 in both, we can combine them: Red:Blue:Green = 15:20:8. The ratio of red to green is 15:8.

Question 3

A plant grows at a constant rate of 5 centimeters every 8 days. If the plant is currently 12 centimeters tall, how many full days from now will it take for the plant to be at least 30 centimeters tall?

  1. 18
  2. 28
  3. 29 (correct answer)
  4. 30
Explanation: First, determine how much more the plant needs to grow: 30 cm - 12 cm = 18 cm. Next, set up a proportion to find the number of days, d, required for this growth: (5 cm / 8 days) = (18 cm / d days). Cross-multiply: 5d = 8 * 18 = 144. Solve for d: d = 144 / 5 = 28.8 days. Since the question asks for the number of full days it will take to be at least 30 cm tall, we must round up to the next whole day. After 28 days, it will not have reached the required height. Therefore, it will take 29 full days.

Question 4

Compare the quantity in Column A to the quantity in Column B.

Column A: The time it takes a train to travel 250 miles at an average speed of 60 miles per hour. Column B: The time it takes a car to travel 220 miles at an average speed of 50 miles per hour.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater. (correct answer)
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Explanation: Calculate the time for Column A: Time = Distance / Speed = 250 miles / 60 mph = 25/6 hours. 25/6 = 4 and 1/6 hours. 1/6 of an hour is (1/6)*60 = 10 minutes. So, Column A is 4 hours and 10 minutes. Calculate the time for Column B: Time = Distance / Speed = 220 miles / 50 mph = 22/5 hours. 22/5 = 4 and 2/5 hours. 2/5 of an hour is (2/5)*60 = 24 minutes. So, Column B is 4 hours and 24 minutes. Since 4 hours and 24 minutes is longer than 4 hours and 10 minutes, the quantity in Column B is greater.

Question 5

A runner goes nine miles in 72 minutes. At this pace, how many minutes for 15 miles?

  1. 96 minutes
  2. 108 minutes
  3. 120 minutes (correct answer)
  4. 144 minutes
Explanation: This question tests middle school quantitative reasoning skills related to solving proportional relationships. Proportional relationships involve comparing two ratios or rates and finding a missing value, often using cross-multiplication or setting up equivalent fractions. In this scenario, you are asked to solve a problem involving time for a runner to cover a distance, requiring identification of the correct proportional relationship. Choice C is correct because it accurately applies the proportional relationship by finding the rate 9 / 72 = 0.125 miles per minute, then 15 / 0.125 = 120 minutes. Choice D is incorrect because it results from misproportioning, like 72 × 2 = 144 minutes. To help students, teach them to set up ratios correctly and cross-multiply to find the unknown. Practice identifying key words that signal a proportional relationship, and watch for common errors like flipping ratios or miscalculating units.

Question 6

Compare the quantity in Column A to the quantity in Column B. x and y are positive numbers, and 3x = 7y.

Column A: x Column B: y

  1. The quantity in Column A is greater. (correct answer)
  2. The quantity in Column B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Explanation: From the equation 3x = 7y, we can find the ratio of x to y. Divide both sides by 3y: x/y = 7/3. This means that x is 7/3 times as large as y. Since 7/3 is greater than 1, x must be greater than y. For example, if y=3, then 3x = 7(3), so 3x = 21 and x=7. In this case, 7 > 3. Since x and y must be positive, x will always be greater than y. Therefore, the quantity in Column A is greater.

Question 7

Compare the quantity in Column A to the quantity in Column B.

Column A: The price per ounce of a 20-ounce box of cereal that costs $4.50. Column B: The price per ounce of a 32-ounce box of cereal that costs $7.04.

  1. The quantity in Column A is greater. (correct answer)
  2. The quantity in Column B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Explanation: Calculate the unit price for Column A: $4.50 / 20 ounces. This is $45 / 200 = $9 / 40. As a decimal, 9/40 = 0.225. So, the price is 22.5 cents per ounce. Calculate the unit price for Column B: $7.04 / 32 ounces. This is $704 / 3200 cents. We can simplify by dividing by 32: 704 / 32 = 22. So, the price is 22 cents per ounce. Since 22.5 cents is greater than 22 cents, the quantity in Column A is greater.

Question 8

A recipe for 12 muffins requires 1.5 cups of flour. A baker has a 10-cup bag of flour and needs to make 60 muffins for a school event. After making the 60 muffins, how many cups of flour will be left in the bag?

  1. 1.5 cups
  2. 2.5 cups (correct answer)
  3. 3.5 cups
  4. 7.5 cups
Explanation: First, find the amount of flour needed per muffin: 1.5 cups / 12 muffins = 0.125 cups/muffin. Then, calculate the total flour needed for 60 muffins: 60 muffins * 0.125 cups/muffin = 7.5 cups. Finally, subtract the amount used from the initial amount: 10 cups - 7.5 cups = 2.5 cups remaining. Alternatively, set up a proportion: (1.5 cups / 12 muffins) = (x cups / 60 muffins). Solving for x gives x = (1.5 * 60) / 12 = 7.5 cups. Then, 10 - 7.5 = 2.5 cups.

Question 9

The property tax on a home is proportional to its assessed value. A home with an assessed value of $200,000 has a property tax of $3,500. What is the property tax on a home with an assessed value of $280,000?

  1. $3,900
  2. $4,200
  3. $4,550
  4. $4,900 (correct answer)
Explanation: Set up a proportion relating the tax to the value. Let T be the unknown tax. ($3,500 / $200,000) = (T / $280,000). To simplify, we can divide the dollar amounts in the first ratio: 3,500/200,000 = 35/2000 = 7/400. So, 7/400 = T / 280,000. Solve for T: T = (7/400) * 280,000. T = 7 * (280,000 / 400) = 7 * 700 = $4,900.

Question 10

In a middle school with 600 students, 45% of the students are in the 7th grade. Within the 7th grade, the ratio of students who play a musical instrument to those who do not is 2 to 3. How many 7th-grade students play a musical instrument?

  1. 108 (correct answer)
  2. 162
  3. 180
  4. 270
Explanation: First, find the number of 7th-grade students: 600 students * 0.45 = 270 students. The ratio of students who play an instrument to those who don't is 2:3. This means the total ratio parts are 2 + 3 = 5. The fraction of 7th graders who play an instrument is 2/5. Finally, calculate the number of students who play an instrument: (2/5) * 270. (270 / 5) = 54. Then, 54 * 2 = 108 students.

Question 11

Compare the quantity in Column A to the quantity in Column B.

A recipe calls for 3 cups of sugar for every 5 cups of flour.

Column A: The amount of sugar needed for a batch using 12 cups of flour. Column B: The amount of flour needed for a batch using 8 cups of sugar.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater. (correct answer)
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Explanation: The ratio of sugar (S) to flour (F) is S/F = 3/5. For Column A, we are given F=12. S/12 = 3/5, so S = (3/5) * 12 = 36/5 = 7.2 cups of sugar. For Column B, we are given S=8. 8/F = 3/5. Cross-multiply: 3F = 8 * 5 = 40. So, F = 40/3 = 13.33... cups of flour. Comparing Column A (7.2 cups) and Column B (13.33... cups), the quantity in Column B is greater.

Question 12

Compare the quantity in Column A to the quantity in Column B.

School X has 15 teachers and 240 students. School Y has 18 teachers and 288 students.

Column A: The student-to-teacher ratio at School X. Column B: The student-to-teacher ratio at School Y.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater.
  3. The two quantities are equal. (correct answer)
  4. The relationship cannot be determined from the information given.
Explanation: Calculate the student-to-teacher ratio for School X (Column A). The ratio is 240 students to 15 teachers. To simplify, divide 240 by 15. 240 / 15 = 16. So the ratio is 16 students for every 1 teacher, or 16:1. Calculate the student-to-teacher ratio for School Y (Column B). The ratio is 288 students to 18 teachers. To simplify, divide 288 by 18. 288 / 18 = 16. So the ratio is 16 students for every 1 teacher, or 16:1. Since both ratios simplify to 16:1, the two quantities are equal.

Question 13

On a particular map, the scale is 2 inches to 5 miles. The distance on the map from Centerville to Oakdale is 11 inches. If a car travels at an average speed of 50 miles per hour, how many minutes will it take to drive from Centerville to Oakdale?

  1. 22 minutes
  2. 27.5 minutes
  3. 33 minutes (correct answer)
  4. 55 minutes
Explanation: First, find the actual distance. Set up a proportion: (2 inches / 5 miles) = (11 inches / x miles). Solving for x gives 2x = 55, so x = 27.5 miles. Next, calculate the travel time. Time = Distance / Speed. Time = 27.5 miles / 50 mph = 0.55 hours. Finally, convert hours to minutes: 0.55 hours * 60 minutes/hour = 33 minutes.

Question 14

A special purple paint is made by mixing red and blue paint in a ratio of 4 parts red to 7 parts blue. If a painter needs to make 33 gallons of this purple paint, how many more gallons of blue paint are needed than red paint?

  1. 3 gallons
  2. 9 gallons (correct answer)
  3. 12 gallons
  4. 21 gallons
Explanation: The ratio of red to blue is 4:7. This means for every 4+7 = 11 parts of purple paint, 4 parts are red and 7 are blue. To make 33 gallons, we need to find the size of one 'part'. 33 gallons / 11 parts = 3 gallons per part. Amount of red paint needed: 4 parts * 3 gallons/part = 12 gallons. Amount of blue paint needed: 7 parts * 3 gallons/part = 21 gallons. The question asks for how many more gallons of blue than red are needed: 21 - 12 = 9 gallons.

Question 15

An architect is building a scale model of a building. The model is 60 centimeters tall, and the actual building will be 180 meters tall. If a doorway in the model is 0.7 centimeters wide, what will be the width of the actual doorway in meters?

  1. 1.8 meters
  2. 2.1 meters (correct answer)
  3. 2.4 meters
  4. 210 meters
Explanation: First, establish the scale factor between the model and the actual building. It's important to use the same units. Convert 180 meters to centimeters: 180 m * 100 cm/m = 18,000 cm. The ratio of the actual height to the model height is 18,000 cm / 60 cm = 300. This means the actual building is 300 times larger than the model. Now apply this scale factor to the doorway's width: 0.7 cm * 300 = 210 cm. Finally, convert this width back to meters: 210 cm / 100 cm/m = 2.1 meters.

Question 16

A 6-foot tall person casts a 9-foot long shadow. At the same time of day, a telephone pole casts a 42-foot long shadow. How tall is the telephone pole?

  1. 24 feet
  2. 28 feet (correct answer)
  3. 36 feet
  4. 63 feet
Explanation: The ratio of an object's height to its shadow's length is constant at the same time of day. Let H be the height of the pole. Set up the proportion: (person's height / person's shadow) = (pole's height / pole's shadow). So, 6/9 = H/42. Simplify the ratio 6/9 to 2/3. Now, 2/3 = H/42. Cross-multiply: 3H = 2 * 42 = 84. Divide by 3: H = 28 feet.

Question 17

A car travels 165 miles using 5 gallons of gasoline. At this rate, how many gallons of gasoline would be needed to travel 396 miles?

  1. 10 gallons
  2. 11 gallons
  3. 12 gallons (correct answer)
  4. 13 gallons
Explanation: First, find the car's fuel efficiency in miles per gallon: 165 miles / 5 gallons = 33 miles per gallon. Then, divide the new distance by the fuel efficiency to find the gallons needed: 396 miles / 33 miles per gallon = 12 gallons. Alternatively, set up a proportion: (165 miles / 5 gallons) = (396 miles / x gallons). Cross-multiply: 165x = 396 * 5, which is 165x = 1980. Divide by 165: x = 12.

Question 18

A model uses scale 1:101:10. If the real width is 7 feet, what is the model width?

  1. 0.7 feet (correct answer)
  2. 17 feet
  3. 70 feet
  4. 1.4 feet
Explanation: This question tests middle school quantitative reasoning skills related to solving proportional relationships. Proportional relationships involve comparing two ratios or rates and finding a missing value, often using cross-multiplication or setting up equivalent fractions. In this scenario, you are asked to solve a problem involving a scale model width from real width, requiring identification of the correct proportional relationship. Choice A is correct because it accurately applies the proportional relationship by dividing 7 / 10 = 0.7 feet for the model. Choice C is incorrect because it results from multiplying instead, like 7 × 10 = 70 feet. To help students, teach them to set up ratios correctly and cross-multiply to find the unknown. Practice identifying key words that signal a proportional relationship, and watch for common errors like flipping ratios or miscalculating units.

Question 19

A recipe serves eight and needs 20 strawberries. How many strawberries are needed for 12 servings?

  1. 30 strawberries (correct answer)
  2. 25 strawberries
  3. 15 strawberries
  4. 32 strawberries
Explanation: This question tests middle school quantitative reasoning skills related to solving proportional relationships. Proportional relationships involve comparing two ratios or rates and finding a missing value, often using cross-multiplication or setting up equivalent fractions. In this scenario, you are asked to solve a problem involving scaling strawberries in a recipe for more servings, requiring identification of the correct proportional relationship. Choice A is correct because it accurately applies the proportional relationship by finding 20 / 8 = 2.5 per serving, then 2.5 × 12 = 30 strawberries. Choice B is incorrect because it results from adding instead of proportioning, like 20 + 5 or similar, yielding 25. To help students, teach them to set up ratios correctly and cross-multiply to find the unknown. Practice identifying key words that signal a proportional relationship, and watch for common errors like flipping ratios or miscalculating units.

Question 20

A train goes 90 miles in two hours. At the same rate, how long for 225 miles?

  1. four hours
  2. five hours (correct answer)
  3. six hours
  4. seven hours
Explanation: This question tests middle school quantitative reasoning skills related to solving proportional relationships. Proportional relationships involve comparing two ratios or rates and finding a missing value, often using cross-multiplication or setting up equivalent fractions. In this scenario, you are asked to solve a problem involving time for a train to travel a certain distance, requiring identification of the correct proportional relationship. Choice B is correct because it accurately applies the proportional relationship by finding the rate 90 / 2 = 45 mph, then 225 / 45 = 5 hours. Choice C is incorrect because it results from misapplying the proportion, like 90 / 225 × 2, yielding about 0.8 but rounded wrong to 6. To help students, teach them to set up ratios correctly and cross-multiply to find the unknown. Practice identifying key words that signal a proportional relationship, and watch for common errors like flipping ratios or miscalculating units.