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Real-World Scaling - ISEE Middle Level Quantitative Reasoning

  1. 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?
    Answer: 14.5 gallons 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.
  2. Column A: The number of liters of water used. Column B: 28 liters
    Answer: The two quantities are equal. 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.
  3. 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?
    Answer: 175 miles 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.
  4. Column A: The height of the tree, in feet. Column B: 24 feet
    Answer: The two quantities are equal. 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.
  5. Column A: The perimeter of Rectangle Q divided by the perimeter of Rectangle P. Column B: 5
    Answer: The two quantities are equal. 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)=122(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)=602(20 + 10) = 60. The ratio of perimeters is 6012=5\frac{60}{12} = 5. This pattern holds for any rectangle. If Rectangle P has length ll and width ww, its perimeter is 2(l+w)2(l + w). Rectangle Q has dimensions 5l5l and 5w5w, so its perimeter is 2(5l+5w)=2⋅5(l+w)=5⋅2(l+w)2(5l + 5w) = 2 \cdot 5(l + w) = 5 \cdot 2(l + w). Therefore, the ratio of perimeters is 5⋅2(l+w)2(l+w)=5\frac{5 \cdot 2(l + w)}{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 kk, perimeter scales by kk, but area scales by k2k^2. Don't confuse these scaling relationships.
  6. 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.
    Answer: The quantity in Column A is greater. 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 323^2 = 9. Since 9 is greater than 3, the quantity in Column A is greater.
  7. 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?
    Answer: 7.5 inches 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.
  8. Column A: The new length of the banner, in inches. Column B: 70 inches
    Answer: The two quantities are equal. 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: 0.40Ă—50=200.40 \times 50 = 20 inches. The new length is 50+20=7050 + 20 = 70 inches. Alternatively, you can use the shortcut method: a 40% increase means the new value is 140% of the original, or 1.40Ă—50=701.40 \times 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.
  9. 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.
    Answer: The quantity in Column B is greater. For Column A, first find the price per foot of the rope: $6 / 4 feet = $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, $15.00 is less than $16.50. Therefore, the quantity in Column B is greater.
  10. 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?
    Answer: 7.5 inches 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.
  11. 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?
    Answer: 218.75 square inches 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.
  12. 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?
    Answer: 27 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 333^3 = 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 = 434^3 = 64 cubic inches. Large Volume = 12312^3 = 1728 cubic inches. The ratio is 1728 / 64 = 27.
  13. 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?
    Answer: 324 square feet 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.
  14. 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?
    Answer: 2.625 cups 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).
  15. 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?
    Answer: 10 rabbits per square mile 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.
  16. 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)
    Answer: 16 feet 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.
  17. 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.
    Answer: The quantity in Column B is greater. 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.
  18. Column A: The actual length, in feet, of a wall that is 4.5 inches long on the blueprint. Column B: 18 feet
    Answer: The two quantities are equal. The scale is rac14 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.

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