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

A recipe calls for 3 cups of flour to make 18 muffins. If Maria wants to make 42 muffins using the same recipe, how many cups of flour will she need?

7 cups
6 cups
8 cups
9 cups
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ISEE Upper Level Quiz

ISEE Upper Level Quiz: Proportional Relationships

Practice Proportional Relationships in ISEE Upper 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 Upper 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 recipe calls for 3 cups of flour to make 18 muffins. If Maria wants to make 42 muffins using the same recipe, how many cups of flour will she need?

  1. 7 cups (correct answer)
  2. 6 cups
  3. 8 cups
  4. 9 cups
Explanation: This is a proportion problem where you need to find how the recipe scales up. When you see questions asking about adjusting recipes or scaling quantities, think about the relationship between the original amounts and what you're trying to find. Start by setting up a proportion comparing flour to muffins. The original recipe uses 3 cups of flour for 18 muffins, so the ratio is 3 cups18 muffins\frac{3 \text{ cups}}{18 \text{ muffins}}. For 42 muffins, you need x cups42 muffins\frac{x \text{ cups}}{42 \text{ muffins}}. Since these ratios must be equal: 318=x42\frac{3}{18} = \frac{x}{42} Cross multiply: 3×42=18×x3 \times 42 = 18 \times x, which gives you 126=18x126 = 18x. Solving for x: x=12618=7x = \frac{126}{18} = 7 cups. You can also think of this as finding the scaling factor. Maria wants 42÷18=2.33...42 ÷ 18 = 2.33... times as many muffins, so she needs 3×2.33...=73 \times 2.33... = 7 cups of flour. Looking at the wrong answers: B) 6 cups would be correct if you mistakenly used 21 muffins instead of 42, or made an arithmetic error in your division. C) 8 cups might result from incorrectly rounding the scaling factor or miscalculating the cross multiplication. D) 9 cups could come from setting up an incorrect proportion or doubling errors in your arithmetic. For proportion problems on the ISEE, always double-check your setup by asking: "Does my ratio make sense?" Here, more muffins should definitely require more flour, and 7 cups for 42 muffins maintains the same flour-per-muffin rate as the original recipe.

Question 2

On a map, 2 inches represents 15 miles. If two cities are 4.8 inches apart on the map, what is the actual distance between them?

  1. 36 miles (correct answer)
  2. 32 miles
  3. 38 miles
  4. 40 miles
Explanation: When you encounter map scale problems, you're working with proportional relationships. The key is setting up a proportion that compares the map distance to the actual distance consistently. Given that 2 inches on the map represents 15 miles in reality, you can set up the proportion: 2 inches15 miles=4.8 inchesx miles\frac{2 \text{ inches}}{15 \text{ miles}} = \frac{4.8 \text{ inches}}{x \text{ miles}} Cross-multiplying: 2x=15×4.8=722x = 15 \times 4.8 = 72 Therefore: x=36 milesx = 36 \text{ miles} You can also think of this as finding the scale factor. Since 152=7.5\frac{15}{2} = 7.5 miles per inch, multiply: 4.8×7.5=364.8 \times 7.5 = 36 miles. Looking at the wrong answers: Choice B (32 miles) likely comes from miscalculating 4.8×7.54.8 \times 7.5 or setting up an incorrect proportion. Choice C (38 miles) could result from arithmetic errors in the cross-multiplication step. Choice D (40 miles) might come from rounding 4.8 to 5 and calculating 5×85 \times 8 (incorrectly using 8 as the scale factor instead of 7.5). The correct answer is A) 36 miles. Strategy tip: Always double-check your scale factor calculation and be careful with decimal multiplication. A quick way to verify: if 2 inches = 15 miles, then 4 inches would be 30 miles, so 4.8 inches should be slightly more than 30 miles. Only answer choice A fits this reasonable estimate.

Question 3

A car travels 240 miles in 4 hours. At this same rate, how long will it take to travel 420 miles?

  1. 7 hours (correct answer)
  2. 6 hours
  3. 8 hours
  4. 6.5 hours
Explanation: When you see a question asking "how long at the same rate," you're dealing with a direct proportion problem. The key insight is that rate (speed) stays constant, so you can set up a ratio. First, find the car's rate: 240 miles4 hours=60 mph\frac{240 \text{ miles}}{4 \text{ hours}} = 60 \text{ mph} Now you can find the time for 420 miles: Time=420 miles60 mph=7 hours\text{Time} = \frac{420 \text{ miles}}{60 \text{ mph}} = 7 \text{ hours} Alternatively, you can set up a proportion: 240 miles4 hours=420 milesx hours\frac{240 \text{ miles}}{4 \text{ hours}} = \frac{420 \text{ miles}}{x \text{ hours}} Cross-multiplying: 240x=420×4=1680240x = 420 \times 4 = 1680, so x=1680240=7 hoursx = \frac{1680}{240} = 7 \text{ hours} Looking at the wrong answers: Choice B (6 hours) might tempt you if you incorrectly think 420 is exactly 1.5 times 240 and assume 4 × 1.5 = 6, but 420240=1.75\frac{420}{240} = 1.75, not 1.5. Choice C (8 hours) could result from rounding errors or miscalculating the ratio as 2:1 instead of 1.75:1. Choice D (6.5 hours) might come from averaging nearby whole numbers or making arithmetic errors in the division. Remember: in rate problems, always identify what stays constant (the rate itself) and what changes (distance and time). Set up your proportion carefully, ensuring the same units are in corresponding positions. Double-check by verifying that your answer makes sense—since 420 miles is less than double 240 miles, the time should be less than double 4 hours.

Question 4

If 5 pounds of apples cost $7.50, how much would 8 pounds of apples cost at the same rate?

  1. $12.00 (correct answer)
  2. $11.25
  3. $13.50
  4. $10.50
Explanation: This is a unit rate problem where you need to find the cost per pound and then scale up. When you see questions asking "at the same rate," you're working with proportional relationships. First, find the cost per pound by dividing the total cost by the weight: $7.505 pounds=$1.50 per pound\frac{\$7.50}{5 \text{ pounds}} = \$1.50 \text{ per pound}. Now multiply this unit rate by 8 pounds: $1.50×8=$12.00\$1.50 \times 8 = \$12.00. You can also solve this using a proportion: 5 pounds$7.50=8 poundsx\frac{5 \text{ pounds}}{\$7.50} = \frac{8 \text{ pounds}}{x}. Cross-multiplying gives you 5x=8×7.50=605x = 8 \times 7.50 = 60, so x=12x = 12. Looking at the wrong answers: Choice B ($11.25) might result from incorrectly calculating the unit rate as $1.40 instead of 1.50,thenmultiplyingby8.ChoiceC(1.50, then multiplying by 8. Choice C (13.50) could come from adding the original 7.50to8poundsinsteadofproperlyscaling,orfromcalculationerrorswiththeproportion.ChoiceD(7.50 to 8 pounds instead of properly scaling, or from calculation errors with the proportion. Choice D (10.50) might result from setting up an incorrect proportion or making arithmetic mistakes when finding the unit rate. The correct answer is A) $12.00. Strategy tip: For rate problems, always find the unit rate first (cost per single item), then multiply by the new quantity. This two-step approach prevents setup errors and makes checking your work easier. Double-check by asking: "Does my answer make sense compared to the original?"

Question 5

A photograph is enlarged so that a 4-inch side becomes 10 inches. If another side of the original photograph was 6 inches, what will be the length of this side in the enlargement?

  1. 15 inches (correct answer)
  2. 12 inches
  3. 14 inches
  4. 16 inches
Explanation: When you encounter enlargement problems, you're dealing with proportional reasoning and scale factors. The key insight is that when a photograph is enlarged, all dimensions change by the same ratio to maintain the original shape. First, find the scale factor by comparing the original and enlarged measurements of the known side. The 4-inch side becomes 10 inches, so the scale factor is 104=2.5\frac{10}{4} = 2.5. This means every dimension is multiplied by 2.5 in the enlargement. Now apply this scale factor to the unknown side: the original 6-inch side becomes 6×2.5=156 \times 2.5 = 15 inches in the enlargement. Looking at the wrong answers: Choice B (12 inches) comes from incorrectly doubling the original measurement, which would be the scale factor if the enlargement went from 4 inches to 8 inches instead of 10. Choice C (14 inches) might result from adding the scale factor increase (6 inches) to the original plus some miscalculation. Choice D (16 inches) could come from incorrectly calculating the scale factor as 2.67 and rounding, or from other computational errors. The correct answer is A) 15 inches. Strategy tip: In any proportional enlargement problem, always establish the scale factor first using the given measurements, then apply that same factor to find unknown dimensions. Set up the proportion: new lengthoriginal length=104=x6\frac{\text{new length}}{\text{original length}} = \frac{10}{4} = \frac{x}{6}. This systematic approach prevents calculation errors and ensures you maintain the correct proportional relationship.

Question 6

If 3 gallons of paint can cover 450 square feet, how many gallons are needed to cover 1,200 square feet?

  1. 8 gallons (correct answer)
  2. 7.5 gallons
  3. 9 gallons
  4. 6.5 gallons
Explanation: This is a classic rate problem where you need to find how much of one quantity corresponds to a given amount of another. When you see "if X amount does Y work, how much X is needed for Z work," you're dealing with proportional relationships. Start by finding the coverage rate per gallon. If 3 gallons cover 450 square feet, then each gallon covers 4503=150\frac{450}{3} = 150 square feet. Now you can find how many gallons are needed for 1,200 square feet: 1,200150=8\frac{1,200}{150} = 8 gallons. Alternatively, you can set up a proportion: 3 gallons450 sq ft=x gallons1,200 sq ft\frac{3 \text{ gallons}}{450 \text{ sq ft}} = \frac{x \text{ gallons}}{1,200 \text{ sq ft}}. Cross-multiplying gives you 3×1,200=450x3 \times 1,200 = 450x, so x=3,600450=8x = \frac{3,600}{450} = 8 gallons. Choice A (8 gallons) is correct. Choice B (7.5 gallons) might result from incorrectly calculating the unit rate or making an arithmetic error in the division. Choice C (9 gallons) could come from rounding errors or miscalculating the proportion. Choice D (6.5 gallons) likely stems from setting up the proportion incorrectly or making significant computational mistakes. For rate problems on the ISEE, always establish the unit rate first (how much per one unit), then multiply by your target amount. This two-step approach is more reliable than setting up proportions if you're prone to cross-multiplication errors, and it helps you catch unreasonable answers quickly.

Question 7

Two quantities, xx and yy, are in the ratio 5:8. If x=35x = 35, what is the value of yy?

  1. 56 (correct answer)
  2. 48
  3. 64
  4. 52
Explanation: When you encounter ratio problems, you're working with proportional relationships between quantities. The key insight is that ratios tell you how many "parts" each quantity represents, and these parts must be equal in size. Given that xx and yy are in the ratio 5:8, this means xx represents 5 equal parts while yy represents 8 equal parts. You can set up the proportion: xy=58\frac{x}{y} = \frac{5}{8} Since x=35x = 35, you need to find what one "part" equals. If xx represents 5 parts and x=35x = 35, then each part equals 35÷5=735 ÷ 5 = 7. Therefore, yy represents 8 parts, so y=8×7=56y = 8 × 7 = 56. Alternatively, you can cross-multiply: 35y=58\frac{35}{y} = \frac{5}{8}, which gives 5y=35×8=2805y = 35 × 8 = 280, so y=56y = 56. Choice A (56) is correct. Choice B (48) likely results from incorrectly thinking the ratio is 5:7 instead of 5:8, or miscalculating 35×8÷535 × 8 ÷ 5. Choice C (64) might come from adding 35 to some incorrect calculation or confusing the ratio setup. Choice D (52) could result from arithmetic errors in the cross-multiplication or incorrectly using ratios. Remember this pattern: in ratio problems, first find the value of one "part" by dividing the known quantity by its ratio number, then multiply by the other ratio number to find the unknown quantity. Always double-check by verifying that your answer maintains the original ratio.

Question 8

The ratio of the perimeter of triangle A to the perimeter of similar triangle B is 3:5. If triangle A has a perimeter of 24 cm, what is the perimeter of triangle B?

  1. 40 cm (correct answer)
  2. 36 cm
  3. 45 cm
  4. 32 cm
Explanation: When you encounter ratio problems involving similar figures, remember that corresponding measurements of similar shapes are proportional. This means if you know one ratio, you can find unknown measurements using cross-multiplication or scaling factors. Here, you're told the ratio of triangle A's perimeter to triangle B's perimeter is 3:5, and triangle A has a perimeter of 24 cm. You can set up a proportion: Perimeter APerimeter B=35\frac{\text{Perimeter A}}{\text{Perimeter B}} = \frac{3}{5} Substituting the known value: 24Perimeter B=35\frac{24}{\text{Perimeter B}} = \frac{3}{5} Cross-multiply: 24×5=3×Perimeter B24 \times 5 = 3 \times \text{Perimeter B} 120=3×Perimeter B120 = 3 \times \text{Perimeter B} Perimeter B=40 cm\text{Perimeter B} = 40 \text{ cm} This confirms answer choice A is correct. Looking at the wrong answers: B) 36 cm results from incorrectly thinking the ratio means triangle B is 1.5 times larger than A (24 × 1.5 = 36), but this ignores the actual 3:5 ratio. C) 45 cm comes from mistakenly adding 21 to triangle A's perimeter, perhaps confusing this with a different type of proportion problem. D) 32 cm might result from incorrectly applying the ratio as an additive relationship rather than multiplicative. Remember this key strategy: when you see ratios involving similar figures, always set up a proportion equation. The ratio tells you the relationship between corresponding parts, so use cross-multiplication to solve for the unknown measurement. Don't try to guess based on "how much bigger" one figure looks.

Question 9

In a certain mixture, the ratio of water to juice is 2:3. If there are 18 ounces of juice in the mixture, how many ounces of water are there?

  1. 12 ounces (correct answer)
  2. 15 ounces
  3. 9 ounces
  4. 24 ounces
Explanation: When you encounter ratio problems, you're working with proportional relationships between quantities. The key is understanding that ratios tell you how parts relate to each other, and you can use this relationship to find unknown quantities. Given that water to juice has a ratio of 2:3, this means for every 2 parts water, there are 3 parts juice. Since you know there are 18 ounces of juice, you can set up a proportion. If 3 parts equals 18 ounces, then each part equals 18÷3=618 ÷ 3 = 6 ounces. Since water represents 2 parts, the amount of water is 2×6=122 × 6 = 12 ounces. You can verify this with a proportion: 23=x18\frac{2}{3} = \frac{x}{18}, where x is the water amount. Cross-multiplying gives 3x=363x = 36, so x=12x = 12. Looking at the wrong answers: (B) 15 ounces likely comes from incorrectly thinking the ratio is 3:2 instead of 2:3, or from adding 3 to 12. (C) 9 ounces might result from dividing 18 by 2 instead of recognizing the proportional relationship. (D) 24 ounces could come from incorrectly multiplying 18 by 4/3 or making an arithmetic error in the proportion. The correct answer is A) 12 ounces. Strategy tip: For ratio problems, always identify what each "part" represents by dividing the known quantity by its ratio number, then multiply by the unknown quantity's ratio number. This systematic approach prevents mix-ups with the ratio order.

Question 10

A gear with 20 teeth is connected to a gear with 35 teeth. If the smaller gear makes 42 rotations, how many rotations does the larger gear make?

  1. 24 rotations (correct answer)
  2. 28 rotations
  3. 30 rotations
  4. 21 rotations
Explanation: When you encounter gear problems, remember that gears with different sizes rotate at different speeds, but they move the same distance along their circumference. This creates an inverse relationship: smaller gears rotate faster, larger gears rotate slower. The key insight is that the total distance traveled along each gear's circumference must be equal. Since distance equals the number of teeth times the number of rotations, you can set up the equation: (teeth on gear 1) × (rotations of gear 1) = (teeth on gear 2) × (rotations of gear 2). For this problem: 20 teeth×42 rotations=35 teeth×x rotations20 \text{ teeth} \times 42 \text{ rotations} = 35 \text{ teeth} \times x \text{ rotations} Solving for x: 840=35x840 = 35x, so x=24x = 24 rotations. Looking at the wrong answers: Choice B (28 rotations) might come from incorrectly using the ratio 3520×24=42\frac{35}{20} \times 24 = 42, but this reverses the relationship. Choice C (30 rotations) could result from using an approximate ratio like 57×42\frac{5}{7} \times 42, which gives about 30. Choice D (21 rotations) might come from incorrectly halving 42 or using 2040×42\frac{20}{40} \times 42. The correct answer is A: 24 rotations. Study tip: For gear problems, always remember the inverse relationship and use the formula: small gear teeth × small gear rotations = large gear teeth × large gear rotations. The gear with fewer teeth always makes more rotations.

Question 11

A spring stretches 4.5 cm when a 12-newton force is applied. Assuming the stretch is proportional to the force, how much will the spring stretch when an 18-newton force is applied?

  1. 6.75 cm (correct answer)
  2. 6.25 cm
  3. 7.5 cm
  4. 5.5 cm
Explanation: When you see a problem stating that one quantity is "proportional to" another, you're dealing with direct variation. This means as one value increases, the other increases at a constant rate, which you can express as a ratio or use to set up a proportion. Here, the spring's stretch is proportional to the applied force. With a 12-newton force causing a 4.5 cm stretch, you can find the constant rate: 4.5 cm12 newtons=0.375 cm per newton\frac{4.5 \text{ cm}}{12 \text{ newtons}} = 0.375 \text{ cm per newton}. For an 18-newton force, multiply: 18×0.375=6.75 cm18 \times 0.375 = 6.75 \text{ cm}. Alternatively, set up a proportion: 4.512=x18\frac{4.5}{12} = \frac{x}{18}. Cross-multiplying gives 12x=8112x = 81, so x=6.75 cmx = 6.75 \text{ cm}. Choice A (6.75 cm) is correct using either method. Choice B (6.25 cm) likely comes from incorrectly calculating the proportion or making an arithmetic error in the cross-multiplication. Choice C (7.5 cm) might result from mistakenly thinking the relationship is 1812×4.5=1.5×4.5=6.75\frac{18}{12} \times 4.5 = 1.5 \times 4.5 = 6.75, but then adding an extra 0.75 through calculation error. Choice D (5.5 cm) could come from various computational mistakes or misunderstanding the proportional relationship. For proportional relationship problems, always identify what stays constant (here, the stretch-per-newton ratio), then either find that rate and multiply, or set up a proportion with the known and unknown values. Double-check your arithmetic, especially when cross-multiplying.

Question 12

The shadow of a 6-foot pole is 8 feet long. At the same time, the shadow of a building is 56 feet long. What is the height of the building?

  1. 42 feet (correct answer)
  2. 48 feet
  3. 36 feet
  4. 45 feet
Explanation: This is a classic similar triangles problem involving shadows and proportional relationships. When the sun creates shadows at the same time of day, the ratio of object height to shadow length remains constant for all objects. Since both the pole and building are casting shadows simultaneously, you can set up a proportion: pole heightpole shadow=building heightbuilding shadow\frac{\text{pole height}}{\text{pole shadow}} = \frac{\text{building height}}{\text{building shadow}} Substituting the known values: 6 feet8 feet=h56 feet\frac{6 \text{ feet}}{8 \text{ feet}} = \frac{h}{56 \text{ feet}} Cross-multiplying: 6×56=8×h6 \times 56 = 8 \times h, which gives you 336=8h336 = 8h Solving for h: h=3368=42 feeth = \frac{336}{8} = 42 \text{ feet} Looking at the wrong answers: Choice B (48 feet) likely comes from incorrectly setting up the proportion as 68=56h\frac{6}{8} = \frac{56}{h} and solving incorrectly. Choice C (36 feet) might result from calculation errors or using an incorrect ratio. Choice D (45 feet) could come from rounding errors or misapplying the proportional relationship. The correct answer is A) 42 feet. Strategy tip: For shadow problems, always remember that the sun creates the same angle for all objects at a given moment, making this a proportion problem. Set up your fraction with heights on top and shadows on bottom (or vice versa), but keep the setup consistent on both sides of the equation.

Question 13

The ratio of cats to dogs at a pet shelter is 7:4. If there are a total of 66 animals (only cats and dogs), how many cats are there?

  1. 42 cats (correct answer)
  2. 38 cats
  3. 35 cats
  4. 48 cats
Explanation: When you encounter ratio problems with a total number of items, you're working with parts of a whole. The key insight is that ratios tell you the relative sizes of groups, and you can use this to find the actual quantities. The ratio 7:4 means that for every 7 cats, there are 4 dogs. Think of this as 7 parts cats and 4 parts dogs, making 11 total parts. Since there are 66 animals total, each "part" represents 6611=6\frac{66}{11} = 6 animals. Since cats make up 7 parts of the ratio, the number of cats is 7×6=427 \times 6 = 42. You can verify this works: if there are 42 cats, then there must be 66−42=2466 - 42 = 24 dogs, and indeed 4224=74\frac{42}{24} = \frac{7}{4}. Looking at the wrong answers: Choice B (38 cats) would leave 28 dogs, giving a ratio of 38:28, which simplifies to 19:14 - not 7:4. Choice C (35 cats) would mean 31 dogs, creating a ratio of 35:31, which doesn't simplify to 7:4. Choice D (48 cats) would leave only 18 dogs, giving 48:18 or 8:3 - again, not our target ratio. The correct answer is A) 42 cats. Study tip: For ratio problems with totals, always add up the ratio parts first, then divide the total by this sum to find the value of one part. This systematic approach prevents calculation errors and works for any ratio problem.

Question 14

A car's fuel efficiency is 28 miles per gallon. At this rate, how many gallons of fuel are needed to travel 350 miles?

  1. 12.5 gallons (correct answer)
  2. 12.25 gallons
  3. 13 gallons
  4. 11.5 gallons
Explanation: This question tests your ability to work with unit rates and set up proportional relationships. When you see fuel efficiency problems, think about the relationship between miles, gallons, and the given rate. Given that the car travels 28 miles per gallon, you need to find how many gallons are required for 350 miles. Set up the problem using the relationship: gallons needed=total milesmiles per gallon\text{gallons needed} = \frac{\text{total miles}}{\text{miles per gallon}} Substituting the values: gallons needed=350 miles28 miles per gallon=12.5 gallons\text{gallons needed} = \frac{350 \text{ miles}}{28 \text{ miles per gallon}} = 12.5 \text{ gallons} You can verify this by checking: 12.5 gallons × 28 miles/gallon = 350 miles ✓ Looking at the wrong answers: Choice B (12.25 gallons) likely results from a calculation error, possibly dividing incorrectly or rounding prematurely. Choice C (13 gallons) might come from rounding 12.5 up unnecessarily or making an arithmetic mistake. Choice D (11.5 gallons) could result from setting up the proportion backwards or making a significant computational error. The correct answer is A) 12.5 gallons. Study tip: For rate problems on the ISEE, always identify what you're looking for first, then set up your division carefully. Remember that "miles per gallon" means you divide total miles by this rate to find gallons needed. Double-check by multiplying your answer by the given rate—you should get back to your starting number.

Question 15

If the cost of 3.5 pounds of cheese is $14.70, what is the cost per pound?

  1. $4.20 per pound (correct answer)
  2. $4.50 per pound
  3. $3.90 per pound
  4. $4.80 per pound
Explanation: Unit rate problems like this one test your ability to find the cost, speed, or amount per single unit when given information about multiple units. When you see "cost per pound," "miles per hour," or similar phrases, you're looking for a rate. To find the cost per pound, you need to divide the total cost by the total weight: total costtotal weight=$14.703.5 pounds\frac{\text{total cost}}{\text{total weight}} = \frac{\$14.70}{3.5 \text{ pounds}} Dividing $14.70 by 3.5 gives you $4.20 per pound. You can verify this by multiplying back: $4.20 × 3.5 = $14.70 ✓ Looking at the wrong answers, choice B ($4.50) represents a common calculation error where students might round 3.5 to 4 pounds, giving $14.70 ÷ 4 = $3.675, which rounds to about 4.50whenstudentsmakeadditionalroundingmistakes.ChoiceC(4.50 when students make additional rounding mistakes. Choice C (3.90) could result from incorrectly setting up the division as 14.70 ÷ 3.77 or similar computational errors. Choice D ($4.80) might come from rounding $14.70 up to $15.00 and then dividing by 3.5, or from other calculation mistakes involving the decimal division. When working with unit rates, always double-check your answer by multiplying the rate back by the original quantity. If your cost per pound times the number of pounds doesn't equal the original total cost, you've made an error. This reverse-check strategy will catch most calculation mistakes on rate problems.

Question 16

A model airplane is built to a scale of 1:48. If the model's wingspan is 9.5 inches, what is the wingspan of the actual airplane?

  1. 38 feet (correct answer)
  2. 42 feet
  3. 35 feet
  4. 40 feet
Explanation: Scale problems test your ability to work with proportional relationships. When you see a scale like 1:48, this means 1 unit on the model represents 48 units on the actual object. To solve this, set up a proportion. The scale 1:48 means that for every 1 inch on the model, there are 48 inches on the real airplane. Since the model wingspan is 9.5 inches, multiply by the scale factor: 9.5×48=456 inches9.5 \times 48 = 456 \text{ inches} Now convert to feet by dividing by 12: 456÷12=38 feet456 \div 12 = 38 \text{ feet} Looking at the wrong answers: Choice B (42 feet) would result from incorrectly using a scale factor of about 53 instead of 48. Choice C (35 feet) comes from using a scale factor of about 44, possibly from misreading the scale ratio. Choice D (40 feet) results from using a scale factor of about 50.5, which might happen if you rounded 48 to 50 for easier calculation. The correct answer is A) 38 feet. Strategy tip: In scale problems, always identify what the ratio means first (1 model unit = 48 real units), then multiply the given measurement by the larger number in the ratio. Don't forget unit conversions at the end—the ISEE often gives measurements in one unit but asks for the answer in another. Double-check by asking yourself if the answer makes sense: a real airplane should definitely be much larger than a 9.5-inch model.

Question 17

A machine produces defective items at a rate of 3 defective items for every 125 items produced. If the machine produces 2,000 items, how many defective items should be expected?

  1. 48 defective items (correct answer)
  2. 45 defective items
  3. 52 defective items
  4. 42 defective items
Explanation: This problem tests your ability to work with ratios and proportional reasoning. When you see a rate or ratio given for one quantity and need to find the corresponding amount for a different quantity, you're looking at a proportion problem. The machine produces 3 defective items for every 125 items total. To find how many defective items to expect from 2,000 items, you need to set up a proportion: 3 defective125 total=x defective2000 total\frac{3 \text{ defective}}{125 \text{ total}} = \frac{x \text{ defective}}{2000 \text{ total}} Cross multiply: 3×2000=125×x3 \times 2000 = 125 \times x, which gives you 6000=125x6000 = 125x. Solving for x: x=6000125=48x = \frac{6000}{125} = 48 defective items. Looking at the wrong answers: Answer B (45) might result from incorrectly using 120 instead of 125 in the denominator, a common arithmetic error. Answer C (52) could come from rounding errors or miscalculating the division. Answer D (42) might result from using an incorrect proportion setup or calculation mistakes in the cross multiplication. The correct answer is A: 48 defective items. For proportion problems like this, always write out the ratio clearly with proper labels (defective/total = defective/total), then cross multiply carefully. Double-check your arithmetic, especially when dividing larger numbers. These problems appear frequently on standardized tests, so practice setting up proportions with consistent units in the same positions.

Question 18

A machine produces 150 widgets in 2.5 hours. At this rate, how many widgets can it produce in 7 hours?

  1. 420 widgets (correct answer)
  2. 400 widgets
  3. 450 widgets
  4. 380 widgets
Explanation: This is a classic rate problem that tests your ability to set up and solve proportional relationships. When you see questions asking "at this rate," you're dealing with a constant rate of production that you can use to predict future output. First, find the machine's rate of production. If it produces 150 widgets in 2.5 hours, the rate is 150 widgets2.5 hours=60 widgets per hour\frac{150 \text{ widgets}}{2.5 \text{ hours}} = 60 \text{ widgets per hour}. Now you can calculate production for any time period: 60 widgets/hour×7 hours=420 widgets60 \text{ widgets/hour} \times 7 \text{ hours} = 420 \text{ widgets}. Alternatively, you can set up a proportion: 150 widgets2.5 hours=x widgets7 hours\frac{150 \text{ widgets}}{2.5 \text{ hours}} = \frac{x \text{ widgets}}{7 \text{ hours}}. Cross-multiplying gives you 150×7=2.5x150 \times 7 = 2.5x, so x=10502.5=420x = \frac{1050}{2.5} = 420. Looking at the wrong answers: B) 400 widgets likely comes from rounding errors or approximating 2.5 as 2.4, giving a rate of 62.5 widgets per hour. C) 450 widgets might result from incorrectly calculating the rate as 64.3 widgets per hour (perhaps from dividing 150 by 2.33 instead of 2.5). D) 380 widgets could come from various computational errors in the division or multiplication steps. For rate problems on the ISEE, always identify what stays constant (the rate of production) and what changes (the time period). Set up your proportion carefully, and double-check your arithmetic since these questions often include answer choices that result from common calculation mistakes.

Question 19

A store sells 4 pencils for $1.20. At this same rate, how much would 15 pencils cost?

  1. $4.50 (correct answer)
  2. $4.20
  3. $3.60
  4. $5.00
Explanation: This is a unit rate problem where you need to find the cost per pencil and then scale up to find the cost of a different quantity. First, find the unit rate by calculating how much one pencil costs. If 4 pencils cost $1.20, then one pencil costs $\frac{\1.20}{4} = $0.30 . Now you can find the cost of any number of pencils by multiplying this unit rate by the desired quantity. For 15 pencils: 15 \times $0.30 = $4.50 . Alternatively, you can set up a proportion: \frac{4 \text{ pencils}}{$1.20} = \frac{15 \text{ pencils}}{x} . Cross-multiplying gives 4x = 15 \times 1.20 = 18 , so x = $4.50 . Choice A (4.50)iscorrectusingeithermethod.ChoiceB(4.50) is correct using either method. Choice B (4.20) might result from incorrectly calculating the unit rate as 0.28perpencil,possiblyfromadivisionerror.ChoiceC(0.28 per pencil, possibly from a division error. Choice C (3.60) comes from using $0.24 per pencil as the unit rate, which could happen if you mistakenly divided 1.20by5insteadof4.ChoiceD(1.20 by 5 instead of 4. Choice D (5.00) might occur if you rounded the unit rate to $0.33 and then multiplied, or made an arithmetic error in your calculations. When solving unit rate problems, always double-check your unit rate calculation first since any error there will carry through to your final answer. Setting up a proportion is also a reliable backup method to verify your work.

Question 20

Two similar rectangles have areas in the ratio 4:9. If the length of the smaller rectangle is 8 inches, what is the length of the larger rectangle?

  1. 12 inches (correct answer)
  2. 18 inches
  3. 10 inches
  4. 15 inches
Explanation: When you encounter problems involving similar figures, remember that their corresponding linear dimensions are proportional, but their areas relate to the square of that proportion. Since these rectangles are similar with areas in the ratio 4:9, you need to find the ratio of their corresponding linear dimensions. If the area ratio is 4:9, then the linear ratio is 4:9=2:3\sqrt{4}:\sqrt{9} = 2:3. This means every linear measurement of the larger rectangle is 32\frac{3}{2} times the corresponding measurement of the smaller rectangle. Given that the smaller rectangle has a length of 8 inches, the larger rectangle's length is 8×32=128 \times \frac{3}{2} = 12 inches. Looking at the wrong answers: Choice B (18 inches) incorrectly uses the area ratio directly as the linear ratio, calculating 8×94=188 \times \frac{9}{4} = 18. Choice C (10 inches) might come from adding half of the smaller length (8+2=108 + 2 = 10), which has no mathematical basis for similar figures. Choice D (15 inches) could result from various calculation errors, perhaps mixing up ratios or making arithmetic mistakes. The correct answer is A) 12 inches. Study tip: For similar figures, always remember that if the area ratio is a:ba:b, then the linear ratio is a:b\sqrt{a}:\sqrt{b}. This square root relationship is crucial because area involves two dimensions while length involves just one. Practice identifying whether a problem asks for linear or area measurements to avoid the common trap of using ratios incorrectly.