ISEE Middle Level Quantitative Reasoning Quiz
Practice Real World Scaling in ISEE Middle Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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An artist is creating two scale drawings of the same statue.
Column A: The height of the statue in a drawing with a scale of 1 inch = 4 feet.
Column B: The height of the statue in a drawing with a scale of 1 inch = 40 inches.
This quiz focuses on Real World Scaling, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Quantitative Reasoning.
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.
An artist is creating two scale drawings of the same statue.
Column A: The height of the statue in a drawing with a scale of 1 inch = 4 feet.
Column B: The height of the statue in a drawing with a scale of 1 inch = 40 inches.
Explanation: To compare the scales, they must be in the same units. In Column A, the scale is 1 inch = 4 feet. Since 1 foot = 12 inches, 4 feet = 4 * 12 = 48 inches. So, the scale in Column A is 1 inch = 48 inches (a 1:48 ratio). In Column B, the scale is 1 inch = 40 inches (a 1:40 ratio). A larger scale ratio results in a larger drawing. Since 1/40 is greater than 1/48, the drawing with the 1:40 scale will be larger. Therefore, the quantity in Column B is greater.
A cleaning solution is made by mixing 2 parts concentrate with 7 parts water. A total of 36 liters of the solution is prepared.
Column A: The number of liters of water used.
Column B: 28 liters
Explanation: The ratio of concentrate to water is 2:7. The total number of parts in the mixture is 2 + 7 = 9 parts. The fraction of the solution that is water is 7/9. To find the amount of water in 36 liters of solution, multiply the total volume by this fraction: (7/9) * 36 liters = 7 * 4 = 28 liters. The quantity in Column A is 28 liters, and the quantity in Column B is 28 liters. Therefore, the two quantities are equal.
A miniature replica of a statue is 5 inches tall. A larger version of the statue is created by scaling up the replica's dimensions by a factor of 2.5. What is the difference in height between the larger version and the miniature replica?
Explanation: First, calculate the height of the larger version by multiplying the replica's height by the scale factor: 5 inches * 2.5 = 12.5 inches. The question asks for the difference in height, so subtract the original height from the new height: 12.5 inches - 5 inches = 7.5 inches.
A photograph in a book shows a coin at 3 times its actual diameter. In the photo, the coin's diameter is 1.5 inches. A large poster shows the same coin at 15 times its actual diameter. What is the diameter of the coin on the poster?
Explanation: First, find the actual diameter of the coin. The photo shows it at 3 times its actual size, and the photo diameter is 1.5 inches. So, Actual Diameter * 3 = 1.5 inches. This means the Actual Diameter = 1.5 / 3 = 0.5 inches. Next, find the diameter on the poster, which is 15 times the actual diameter: 0.5 inches * 15 = 7.5 inches.
A square has a side length of 6 cm. The dimensions of the square are scaled up to create a new, larger square with a side length of 18 cm.
Column A: The ratio of the new square's area to the original square's area.
Column B: The ratio of the new square's perimeter to the original square's perimeter.
Explanation: First, find the linear scaling factor, which is the ratio of the new side length to the original side length: 18 cm / 6 cm = 3. For Column B, the perimeter scales by the same factor as the side lengths. The ratio of the perimeters is 3. For Column A, the area scales by the square of the linear scaling factor. The ratio of the areas is 3^2 = 9. Since 9 is greater than 3, the quantity in Column A is greater.
A recipe that yields 12 muffins requires 1.5 cups of flour and 0.75 cups of milk. If a baker uses the same recipe to make 42 muffins, how much more flour than milk will be needed?
Explanation: First, find the scaling factor for the recipe. The baker is making 42 muffins instead of 12, so the scaling factor is 42 / 12 = 3.5. Now, calculate the new amounts of flour and milk. New flour amount: 1.5 cups * 3.5 = 5.25 cups. New milk amount: 0.75 cups * 3.5 = 2.625 cups. Finally, find the difference: 5.25 - 2.625 = 2.625 cups. Alternatively, find the original difference (1.5 - 0.75 = 0.75 cups) and multiply it by the scaling factor (0.75 * 3.5 = 2.625 cups).
A banner is 50 inches long. Its length is increased by 40% to create a new, proportionally larger banner.
Column A: The new length of the banner, in inches.
Column B: 70 inches
Explanation: When you encounter percentage increase problems, you're working with the concept that a percentage increase means adding that percentage of the original value to the original value itself. To find the new length, you need to calculate what 40% of 50 inches equals, then add it to the original 50 inches. First, convert 40% to a decimal: 40% = 0.40. Then multiply: 0.40×50=20 inches. The new length is 50+20=70 inches. Alternatively, you can use the shortcut method: a 40% increase means the new value is 140% of the original, or 1.40×50=70 inches. Since Column A equals 70 inches and Column B is also 70 inches, the quantities are equal. Looking at why the other answers are wrong: Choice (A) suggests Column A is greater than 70, which would mean you either calculated the percentage increase incorrectly or added an extra step. Choice (B) suggests 70 is greater than the new length, which would happen if you mistakenly calculated a percentage decrease or used the wrong base number. Choice (C) implies insufficient information, but you have everything needed: the original length and the percentage increase. Remember that "increased by X%" means the new value equals the original value times (1 + X%), where X is the percentage as a decimal. This formula works for any percentage increase problem and helps you avoid calculation errors.
A store sells rope by the foot. The price for 4 feet of a certain rope is $6.
Column A: The cost of 10 feet of the same rope.
Column B: The cost of 3 feet of a different rope that costs $5.50 per foot.
Explanation: For Column A, first find the price per foot of the rope: 6/4feet=1.50 per foot. Then, calculate the cost of 10 feet: 10 feet * 1.50/foot=15.00. For Column B, calculate the cost directly: 3 feet * 5.50/foot=16.50. Comparing the two quantities, 15.00islessthan16.50. Therefore, the quantity in Column B is greater.
A rectangular park is 4 miles long and 3 miles wide. It has a population of 240 rabbits. The park's area is then expanded by doubling both its length and width. If the rabbit population also doubles, what is the new population density in rabbits per square mile?
Explanation: First, find the original area: 4 miles * 3 miles = 12 square miles. The original density is 240 rabbits / 12 sq mi = 20 rabbits/sq mi. Now, find the new dimensions and area. New length = 4 * 2 = 8 miles. New width = 3 * 2 = 6 miles. New area = 8 * 6 = 48 square miles. The new rabbit population is 240 * 2 = 480 rabbits. The new population density is 480 rabbits / 48 square miles = 10 rabbits per square mile.
A certain car can travel 168 miles on 6 gallons of gasoline. At this same rate of fuel consumption, how many gallons of gasoline are needed for a trip of 406 miles?
Explanation: First, find the car's fuel efficiency in miles per gallon (mpg). Efficiency = 168 miles / 6 gallons = 28 mpg. Next, to find the number of gallons needed for the new trip, divide the trip distance by the car's efficiency: 406 miles / 28 mpg = 14.5 gallons.
A map has a scale where 2 inches represents 70 miles. The distance between two cities on this map is 5 inches. What is the actual distance between the two cities?
Explanation: First, find the unit rate of the scale. If 2 inches = 70 miles, then 1 inch = 70 / 2 = 35 miles. Next, multiply this unit rate by the map distance to find the actual distance: 5 inches * 35 miles/inch = 175 miles.
A small gift box is a cube with a side length of 4 inches. A larger, proportionally shaped gift box has a side length of 12 inches. The volume of the larger box is how many times the volume of the smaller box?
Explanation: The linear scaling factor from the small box to the large box is the ratio of their side lengths: 12 inches / 4 inches = 3. Volume scales by the cube of the linear scaling factor. Therefore, the ratio of the volumes is 3^3 = 3 * 3 * 3 = 27. The volume of the larger box is 27 times the volume of the smaller box. Alternatively, one could calculate the volumes: Small Volume = 4^3 = 64 cubic inches. Large Volume = 12^3 = 1728 cubic inches. The ratio is 1728 / 64 = 27.
The dimensions of Rectangle P are scaled by a factor of 5 to create Rectangle Q.
Column A: The perimeter of Rectangle Q divided by the perimeter of Rectangle P.
Column B: 5
Explanation: When you see scaling problems involving rectangles, focus on how different measurements change when dimensions are multiplied by a scale factor. Let's work with concrete numbers first. Suppose Rectangle P has length 4 and width 2, giving it a perimeter of 2(4+2)=12. When scaled by a factor of 5, Rectangle Q has length 20 and width 10, with a perimeter of 2(20+10)=60. The ratio of perimeters is 1260=5. This pattern holds for any rectangle. If Rectangle P has length l and width w, its perimeter is 2(l+w). Rectangle Q has dimensions 5l and 5w, so its perimeter is 2(5l+5w)=2⋅5(l+w)=5⋅2(l+w). Therefore, the ratio of perimeters is 2(l+w)5⋅2(l+w)=5. Since Column A equals 5 and Column B equals 5, choice D is correct—the quantities are equal. Choice A claims Column A is greater than 5, which we've shown is false. Choice B claims Column B (which is 5) is greater than Column A (which is also 5), also false. Choice C suggests we can't determine the relationship, but we can—perimeter always scales by the same factor as the linear dimensions. Remember: when linear dimensions are scaled by factor k, perimeter scales by k, but area scales by k2. Don't confuse these scaling relationships.
On a construction blueprint, the scale is set at ( rac{1}{4}) inch = 1 foot.
Column A: The actual length, in feet, of a wall that is 4.5 inches long on the blueprint.
Column B: 18 feet
Explanation: The scale is ( rac{1}{4}) inch = 1 foot. This means that for every inch on the blueprint, the actual length is 4 feet (since there are four quarter-inches in one inch). To find the actual length of the wall in Column A, multiply the blueprint length by 4: 4.5 inches * 4 feet/inch = 18 feet. The quantity in Column A is 18 feet, and the quantity in Column B is 18 feet. Therefore, the two quantities are equal.
At a certain time of day, a 6-foot-tall person casts an 8-foot-long shadow. At the same time, a nearby tree casts a 32-foot-long shadow.
Column A: The height of the tree, in feet.
Column B: 24 feet
Explanation: The ratio of an object's height to its shadow's length is constant. For the person, the ratio is 6 feet / 8 feet = 3/4. Let the tree's height be H. The proportion is H / 32 = 6 / 8. To solve for H, simplify the ratio: H / 32 = 3 / 4. Multiply both sides by 32: H = (3 / 4) * 32 = 3 * 8 = 24 feet. The quantity in Column A is 24 feet, and the quantity in Column B is 24 feet. Thus, the two quantities are equal.
A rectangular photograph measuring 5 inches by 7 inches is enlarged proportionally so that its shorter side is 12.5 inches. What is the area of the enlarged photograph?
Explanation: First, determine the scaling factor. The shorter side of the original photograph is 5 inches, and the new shorter side is 12.5 inches. The scaling factor is 12.5 / 5 = 2.5. Now, find the length of the new longer side by applying the same scaling factor: 7 inches * 2.5 = 17.5 inches. Finally, calculate the area of the enlarged photograph by multiplying its new dimensions: 12.5 inches * 17.5 inches = 218.75 square inches.
A blueprint for a rectangular patio has a scale of 0.5 inches = 3 feet. On the blueprint, the patio is 2 inches wide and 4.5 inches long. What is the actual area of the patio in square feet?
Explanation: First, determine the scale factor in feet per inch. If 0.5 inches = 3 feet, then 1 inch = 6 feet. Next, convert the blueprint dimensions to actual dimensions. Actual width = 2 inches * 6 feet/inch = 12 feet. Actual length = 4.5 inches * 6 feet/inch = 27 feet. Finally, calculate the actual area: Area = 12 feet * 27 feet = 324 square feet.
A model car is built using a 1:24 scale. The model is 8 inches long. What is the actual length of the car in feet? (Note: 12 inches = 1 foot)
Explanation: First, find the actual length of the car in inches by multiplying the model's length by the scale factor: 8 inches * 24 = 192 inches. Next, convert this length from inches to feet by dividing by 12: 192 inches / 12 inches/foot = 16 feet.