ISEE Middle Level Quantitative Reasoning Quiz: Ratios And Rates In Context

What this quiz covers

This quiz focuses on Ratios And Rates In Context, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Quantitative Reasoning.

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.

Question 1

1. $0.30 (correct answer)
2. $0.90
3. $1.20
4. $1.80

Explanation: First, calculate the cost of buying 6 individual pens: 6 * $1.50 =$9.00. The cost of the pack is $7.20. The total savings is$9.00 - $7.20 =$1.80. The question asks for the savings per pen. To find this, divide the total savings by the number of pens in the pack: $1.80 / 6 =$0.30. So, you save $0.30 per pen.

Question 2

1. 9.0 miles
2. 11.25 miles (correct answer)
3. 12.5 miles
4. 22.5 miles

Explanation: Set up a proportion to solve this problem. Let x be the actual distance in miles. The ratio of inches to miles is constant: 2 inches / 5 miles = 4.5 inches / x miles. Cross-multiply: 2x = 5 * 4.5. 2x = 22.5. Divide by 2: x = 11.25. The actual distance is 11.25 miles.

Question 3

1. 5:9 (correct answer)
2. 5:12
3. 1:2
4. 5:18

Explanation: Let the number of cows be 5x and the number of sheep be 6x. The number of brown cows is (1/3) of the total cows, so (1/3) * 5x = (5/3)x. The number of brown sheep is (1/2) of the total sheep, so (1/2) * 6x = 3x. The ratio of brown cows to brown sheep is (5/3)x : 3x. The 'x' cancels out, leaving the ratio (5/3) : 3. To eliminate the fraction, multiply both sides of the ratio by 3: (5/3)3 : 33, which simplifies to 5:9.

Question 4

1. 11.25
2. 15
3. 20 (correct answer)
4. 25

Explanation: The number of rotations of two connected gears is inversely proportional to their number of teeth. The ratio of the number of teeth is 48:36, which simplifies to 4:3. Therefore, the ratio of their rotations (larger gear to smaller gear) will be the inverse, 3:4. Let R_L be the rotations of the large gear and R_S be the rotations of the small gear. R_L / R_S = 3 / 4. We are given R_L = 15. So, 15 / R_S = 3 / 4. Cross-multiply: 3 * R_S = 15 * 4 = 60. R_S = 60 / 3 = 20. The smaller gear makes 20 rotations.

Question 5

1. 1/6 (correct answer)
2. 1/5
3. 1/4
4. 1/3

Explanation: First, calculate the number of gallons of gas used on the trip. Gallons used = Total miles / miles per gallon = 350 miles / 28 miles/gallon = 12.5 gallons. The tank started with 15 gallons. The amount of gas remaining is 15 - 12.5 = 2.5 gallons. The question asks for the fraction of the tank that is full. This is the remaining gas divided by the tank's capacity: 2.5 gallons / 15 gallons. To simplify this fraction, you can write it as 25/150. Divide both the numerator and the denominator by 25: 25/25 = 1 and 150/25 = 6. The fraction is 1/6.

Question 6

1. 2
2. 3 (correct answer)
3. 4
4. 5

Explanation: First, convert the total leakage to milliliters: 12 liters * 1000 mL/liter = 12,000 mL. Next, find the total time in hours for this leakage: 12,000 mL / 200 mL/hour = 60 hours. Finally, convert the hours to days. There are 24 hours in a day. 60 hours / 24 hours/day = 2.5 days. Since the question asks for the number of full days it will take, the leak will be completed during the 3rd day. So, it will take 3 full days to ensure at least 12 liters have leaked.

Question 7

1. 4:3
2. 8:9
3. 8:3
4. 16:9 (correct answer)

Explanation: When you encounter problems comparing volumes and surface areas of similar shapes, remember that these measurements scale differently. Volume scales with the cube of the linear dimension, while surface area scales with the square. Since the volume ratio of Cube A to Cube B is 64:27, you can find the ratio of their edge lengths. If the edge of Cube A is $s_A$ and the edge of Cube B is $s_B$, then: $\frac{s_A^3}{s_B^3} = \frac{64}{27}$ Taking the cube root of both sides: $\frac{s_A}{s_B} = \frac{4}{3}$ Now you can find the surface area ratio. Since each cube has 6 faces and surface area is $6s^2$: $\frac{\text{Surface Area of A}}{\text{Surface Area of B}} = \frac{6s_A^2}{6s_B^2} = \frac{s_A^2}{s_B^2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$ Choice A (4:3) gives you the ratio of edge lengths, not surface areas. Choice B (8:9) appears to mix up the relationship and gets the wrong values entirely. Choice C (8:3) incorrectly applies the relationship between volume and surface area scaling. Choice D (16:9) correctly squares the edge length ratio to get the surface area ratio. The key insight is recognizing the scaling relationships: when linear dimensions change by a factor of $k$, areas change by $k^2$ and volumes by $k^3$. Always work backward from the given ratio to find the linear scale factor first.

Question 8

1. 300
2. 375
3. 450 (correct answer)
4. 500

Explanation: First, find the rate in hours per room. The rate is 5 hours / 2 rooms = 2.5 hours per room. To paint 3 rooms, it will take 3 rooms * 2.5 hours/room = 7.5 hours. The question asks for the time in minutes. Convert hours to minutes: 7.5 hours * 60 minutes/hour = 450 minutes.

Question 9

1. $285
2. $330
3. $360 (correct answer)
4. $420

Explanation: The ratio of red to blue is 3:4. The total number of parts is 3 + 4 = 7. The total amount of paint needed is 21 gallons. The amount of red paint needed is (3/7) * 21 = 9 gallons. The amount of blue paint needed is (4/7) * 21 = 12 gallons. Now calculate the cost. Cost of red paint = 9 gallons * $20/gallon =$180. Cost of blue paint = 12 gallons * $15/gallon =$180. The total cost is $180 +$180 = $360.

Question 10

1. 1 to 35
2. 31 to 1260 (correct answer)
3. 1 to 40
4. 3 to 140

Explanation: First, find the original number of security guards. Let S be the number of guards. The ratio is S / 1260 = 3 / 140. To solve for S, multiply both sides by 1260: S = (3 * 1260) / 140. S = (3 * 9) = 27. So there were originally 27 guards. Then, 4 additional guards are hired, making the new total 27 + 4 = 31 guards. The number of fans remains 1,260. The new ratio is 31 guards to 1,260 fans, or 31:1260. This ratio cannot be simplified.

Question 11

1. 15.0
2. 16.5 (correct answer)
3. 18.0
4. 22.5

Explanation: The ratio of lemon juice : sugar : water is 2:1:8. The total number of 'parts' in the ratio is 2 + 1 + 8 = 11. You are using 12 cups of water, which corresponds to the 8 parts in the ratio. To find the value of one 'part', set up the equation 8x = 12, so x = 12/8 = 1.5. The total amount of lemonade corresponds to 11 parts. So, the total volume is 11 * x = 11 * 1.5 = 16.5 cups.

Question 12

1. 25
2. 50 (correct answer)
3. 75
4. 100

Explanation: First, find the initial amount of salt. 20% of 50 liters is 0.20 * 50 = 10 liters of salt. The amount of salt does not change when water is added. Let w be the amount of water added. The new total volume will be 50 + w. We want the salt concentration to be 10%. So, the amount of salt (10 liters) divided by the new total volume (50 + w) must equal 10% (or 0.10). The equation is 10 / (50 + w) = 0.10. Multiply both sides by (50 + w): 10 = 0.10 * (50 + w). 10 = 5 + 0.10w. Subtract 5 from both sides: 5 = 0.10w. Divide by 0.10: w = 50. So, 50 liters of water must be added.

Question 13

1. $126,000
2. $210,000 (correct answer)
3. $336,000
4. $420,000

Explanation: The ratio of research spending to marketing spending is 3:8. This means for every $11 spent in total (3+8),$3 goes to research and $8 to marketing. The difference between marketing and research spending for every$11 is $8 -$3 = $5. The fraction of the total spending that represents this difference is 5/11. To find the total difference, multiply this fraction by the total amount spent: (5/11) *$462,000. $462,000 / 11 =$42,000. $42,000 * 5 =$210,000.

Question 14

1. 3:2
2. 4:5
3. 8:1
4. 6:5 (correct answer)

Explanation: When you encounter population density problems, remember that density equals population divided by area. You'll need to find missing values using this relationship before making comparisons. For City A, you're given the population (12,000) and area (30 square kilometers) directly. For City B, you have the population density (500 people per square kilometer) and area (20 square kilometers), so you need to calculate the population: $500 \times 20 = 10,000$ people. Now you can find the ratio of City A's population to City B's population: $12,000 : 10,000$. To simplify this ratio, divide both numbers by their greatest common factor of 2,000: $12,000 ÷ 2,000 = 6$ and $10,000 ÷ 2,000 = 5$. The ratio is 6:5. Looking at the wrong answers: Choice A (3:2) would result from incorrectly using the area ratio (30:20 simplified). Choice B (4:5) might come from calculation errors or mixing up which city goes first in the ratio. Choice C (8:1) doesn't match any logical relationship between the given numbers and suggests a major computational mistake. The key strategy here is to organize your work systematically: first identify what information you have versus what you need, then calculate any missing values using the density formula, and finally set up your ratio carefully. Always double-check which quantity the question asks you to compare first, as ratios depend on order.

Question 15

1. 28 (correct answer)
2. 39
3. 45
4. 63

Explanation: The ratio of an object's height to its shadow's length is constant at the same time of day. Set up a proportion: (man's height) / (man's shadow) = (tree's height) / (tree's shadow). Let h be the tree's height. 6 / 9 = h / 42. The ratio 6/9 simplifies to 2/3. So, 2/3 = h / 42. To solve for h, multiply both sides by 42: h = (2 * 42) / 3 = 84 / 3 = 28. The height of the tree is 28 feet.

Question 16

1. 27
2. 55
3. 65 (correct answer)
4. 90

Explanation: First, find the unit rate for each machine. Machine A's rate is 15 widgets / 6 minutes = 2.5 widgets per minute. Machine B's rate is 12 widgets / 3 minutes = 4 widgets per minute. Their combined rate is 2.5 + 4 = 6.5 widgets per minute. In 10 minutes, they can produce 6.5 widgets/minute * 10 minutes = 65 widgets.

Question 17

1. 36
2. 45
3. 81 (correct answer)
4. 90

Explanation: Let the original number of boys be 4x and the original number of girls be 5x. The total number of students originally was 4x + 5x = 9x. After the change, the number of boys is 4x + 9 and the number of girls is 5x - 9. The new ratio is (4x + 9) / (5x - 9) = 5/4. Cross-multiply to solve for x: 4(4x + 9) = 5(5x - 9) -> 16x + 36 = 25x - 45 -> 81 = 9x -> x = 9. The original number of students was 9x, so 9 * 9 = 81 students.

Question 18

1. 10 miles
2. 35 miles (correct answer)
3. 40 miles
4. 50 miles

Explanation: The car travels 10 miles per hour faster than the truck (50 mph - 40 mph). The time is 3 hours and 30 minutes, which is 3.5 hours. To find the difference in distance, multiply the difference in speed by the time: 10 miles/hour * 3.5 hours = 35 miles. Alternatively, calculate the distance each travels: Car travels 50 * 3.5 = 175 miles. Truck travels 40 * 3.5 = 140 miles. The difference is 175 - 140 = 35 miles.

Question 19

1. 30
2. 49
3. 50 (correct answer)
4. 58

Explanation: First, find a common ratio for all three fruits. Apples:Oranges = 3:2. Oranges:Bananas = 3:5. To make the 'oranges' term the same, find the least common multiple of 2 and 3, which is 6. Convert the ratios: Apples:Oranges = 9:6. Oranges:Bananas = 6:10. So, the combined ratio Apples:Oranges:Bananas is 9:6:10. We are given there are 18 apples. This corresponds to the 9 in the ratio. The scaling factor is 18/9 = 2. Now find the number of each fruit: Oranges = 6 * 2 = 12. Bananas = 10 * 2 = 20. The total number of fruits is 18 (apples) + 12 (oranges) + 20 (bananas) = 50.