Proportional Scaling - ISEE Lower Level Quantitative Reasoning

A small box containing 6 crayons costs $1.50. A large box containing 15 crayons is sold at the same price per crayon. What is the cost of the large box of crayons?
Answer: $3.75 First, find the price of a single crayon from the small box: $\$1.50 \div 6 \text{ crayons} = \$0.25$ per crayon. Then, multiply this unit price by the number of crayons in the large box: $15 \text{ crayons} \times \$0.25/\text{crayon} = \$3.75$ . Alternatively, the scaling factor for the number of crayons is $15 \div 6 = 2.5$ . So the cost is $\$1.50 \times 2.5 = \$3.75$ .
A family of 4 people eats 2 boxes of cereal in 5 days. Assuming the same rate of consumption, how many boxes of cereal would a family of 8 people eat in 5 days?
Answer: 4 boxes The time period of 5 days is the same in both scenarios, so we only need to scale the amount of cereal based on the number of people. The family size has doubled from 4 people to 8 people. Therefore, the amount of cereal they eat will also double. $2 \text{ boxes} \times 2 = 4$ boxes.
A standard photograph is 4 inches wide and 6 inches long. If it is enlarged proportionally so that its width becomes 10 inches, what is its new length?
Answer: 15 inches The ratio of width to length must remain the same for the photo to be enlarged proportionally. The new width is 10 inches, and the original width was 4 inches. The scaling factor is $10 \div 4 = 2.5$ . We must apply the same scaling factor to the original length: $6 \text{ inches} \times 2.5 = 15$ inches.
On a map, $1$ inch equals $10$ miles. A road is 4 inches long on the map. The map uses this scale for all roads. How many miles long is the road?
Answer: 40 miles This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the actual length of a road using a map scale of 1 inch = 10 miles. The correct answer works because multiplying the map length of 4 inches by the scale factor of 10 miles per inch gives 40 miles. A common distractor may fail because it multiplies by a wrong factor or uses division, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately. Encourage double-checking calculations and understanding the relationship between map and actual distances. Practice with different map scenarios to build confidence in identifying and applying the correct scaling methods.
A building model uses a scale of $1:50$ . A wall in the model is 3 inches long. The real building keeps the same proportions. What is the real wall length in inches?
Answer: 150 inches This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the real length of a wall from a 1:50 building model where the model wall is 3 inches long. The correct answer works because multiplying the model measurement of 3 inches by the scale factor of 50 gives 150 inches for the real wall. A common distractor may fail because it divides by the scale instead of multiplying, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately. Encourage double-checking calculations and understanding the relationship between model and real sizes. Practice with different model scenarios to build confidence in identifying and applying the correct scaling methods.
A pancake recipe makes 6 pancakes using 3 cups of milk. You want to make 12 pancakes for breakfast. You scale the recipe to keep the same batter thickness. How many cups of milk are needed?
Answer: 6 cups This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to adjust a pancake recipe from 6 pancakes to 12 pancakes while keeping the batter thickness the same. The correct answer works because the scaling factor is 12/6 = 2, and multiplying the original 3 cups of milk by 2 gives 6 cups. A common distractor may fail because it misapplies the scaling factor or confuses the number of pancakes with servings, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately to each ingredient. Encourage double-checking calculations and understanding the relationship between recipe yield and ingredients. Practice with different recipe scenarios to build confidence in proportional scaling.
A model airplane is built at a scale of $1:20$ . The model's wingspan is 9 inches. The real airplane keeps the same proportions. What is the real wingspan in inches?
Answer: 180 inches This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the real wingspan of an airplane from a 1:20 model where the model's wingspan is 9 inches. The correct answer works because multiplying the model measurement of 9 inches by the scale factor of 20 gives 180 inches for the real wingspan. A common distractor may fail because it confuses the scale ratio or uses addition instead of multiplication, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately. Encourage double-checking calculations and understanding the relationship between model and real sizes. Practice with different model scenarios to build confidence in identifying and applying the correct scaling methods.
A map scale shows $1$ inch = $4$ miles. A river is 5 inches long on the map. The scale is consistent across the map. How long is the river in miles?
Answer: 20 miles This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the actual length of a river using a map scale of 1 inch = 4 miles. The correct answer works because multiplying the map length of 5 inches by the scale factor of 4 miles per inch gives 20 miles. A common distractor may fail because it adds instead of multiplying or misinterprets the scale, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately. Encourage double-checking calculations and understanding the relationship between map and actual distances. Practice with different map scenarios to build confidence in identifying and applying the correct scaling methods.
A car uses 4 gallons of gasoline to travel 100 miles. At this rate, how many miles can the car travel on a full tank of 5 gallons of gasoline?
Answer: 125 miles First, determine the car's fuel efficiency in miles per gallon. The car travels 100 miles on 4 gallons, so it gets $100 \div 4 = 25$ miles per gallon. With 5 gallons of gasoline, the car can travel $5 \text{ gallons} \times 25 \text{ miles/gallon} = 125$ miles.
A model train is built at a scale of $1:25$ . The model engine is 4 inches long. The real engine is the same shape, just larger. What is the real length in inches?
Answer: 100 inches This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the real length of a train engine from a 1:25 model where the model is 4 inches long. The correct answer works because multiplying the model length of 4 inches by the scale factor of 25 gives 100 inches for the real engine. A common distractor may fail because it confuses the scale direction or uses incorrect multiplication, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately. Encourage double-checking calculations and understanding the relationship between model and real sizes. Practice with different model scenarios to build confidence in identifying and applying the correct scaling methods.
On a local map, the scale shows that 2 inches represents 5 miles. The distance on the map between the library and the park is 6 inches. What is the actual distance between the library and the park?
Answer: 15 miles The ratio of map distance to actual distance is 2 inches to 5 miles. The measured map distance is 6 inches. Since 6 inches is 3 times 2 inches ( $6 \div 2 = 3$ ), the actual distance must be 3 times the corresponding 5 miles. Therefore, the actual distance is $3 \times 5 = 15$ miles.
A smoothie recipe serves four and uses two cups of yogurt. You want to make it for 12 servings for a team snack. You keep the taste the same by scaling every ingredient by the same factor. The yogurt amount must match the new number of servings. How many cups of yogurt are needed for 12 servings?
Answer: 6 cups This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to a recipe that serves 4 people and needs to serve 12 people, requiring you to find the scale factor (12 ÷ 4 = 3) and apply it to the yogurt amount. The correct answer works because it correctly applies the scaling factor of 3 to the original 2 cups of yogurt: 2 × 3 = 6 cups. A common distractor like 8 cups may fail because it adds the difference in servings (8) to the original amount instead of using proportional scaling. To help students: Teach them to identify the scale factor by dividing the new quantity by the original quantity. Encourage them to check their work by verifying that all ingredients maintain the same ratio to preserve the recipe's taste.
It takes 2 painters 6 hours to paint a room. Assuming they both work at the same constant rate, how long would it take 4 painters to paint the same room?
Answer: 3 hours This is an inverse proportion problem. If you double the number of workers, you halve the time it takes. The total amount of work is $2 \text{ painters} \times 6 \text{ hours} = 12$ "painter-hours." To find the time for 4 painters, divide the total work by the new number of painters: $12 \text{ painter-hours} \div 4 \text{ painters} = 3$ hours.
A map scale says 1 inch = 6 miles. Two towns are 5 inches apart on the map. You use the same scale for the whole map. Multiply to scale up from map inches to real miles. How many miles apart are the towns?
Answer: 30 miles This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to convert map distances to real distances using the given scale of 1 inch = 6 miles. The correct answer works because it correctly applies the scaling factor to achieve the desired proportion: 5 inches × 6 miles/inch = 30 miles. A common distractor like 11 miles may fail because it adds the scale factor to the map distance instead of multiplying, misunderstanding how map scales work. To help students: Teach them to set up the proportion clearly (1 inch : 6 miles = 5 inches : x miles) or use direct multiplication. Encourage them to label their units throughout the calculation to avoid confusion between map inches and real miles.
Sarah can read 15 pages of her book in 20 minutes. If she continues to read at this constant speed, how many pages can she read in one full hour?
Answer: 45 pages This is a rate problem that asks you to find how much work gets done over a different time period. When you see questions about constant rates or speeds, you need to set up a proportion or find the unit rate. First, let's find Sarah's reading rate per minute. If she reads 15 pages in 20 minutes, her rate is $\frac{15 \text{ pages}}{20 \text{ minutes}} = \frac{3}{4} = 0.75$ pages per minute. Now you can find how many pages she reads in one hour (60 minutes): $0.75 \text{ pages/minute} \times 60 \text{ minutes} = 45 \text{ pages}$ . This confirms answer choice D is correct. Let's examine why the other answers are wrong. Choice A (30 pages) represents a common error where students might double the original 15 pages, incorrectly thinking that 60 minutes is twice 20 minutes (it's actually three times). Choice B (60 pages) likely comes from students who mistakenly think she reads one page per minute, perhaps by confusing the 15 pages and 20 minutes. Choice C (55 pages) doesn't follow any clear mathematical relationship to the given information and may be included as a distractor. When solving rate problems, always identify what you're given, what you need to find, and set up the relationship clearly. Converting to a unit rate (pages per minute, miles per hour, etc.) often makes these problems much easier to solve and helps you avoid calculation errors.
A model airplane uses a scale of 1:20. The model is 9 inches long. The real airplane is 20 times the model length. Use the scale to enlarge the model measurement. What is the actual length in inches?
Answer: 180 inches This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the actual airplane length using a 1:20 scale, where the model is 9 inches long. The correct answer works because it correctly applies the scaling factor of 20 to the model length: 9 inches × 20 = 180 inches. A common distractor like 29 inches may fail because it adds the scale factor to the model length instead of multiplying, showing confusion about how scale ratios work. To help students: Teach them that in a 1:20 scale, the real object is 20 times larger than the model. Encourage them to write out the multiplication clearly and understand that scale ratios represent a multiplicative relationship, not an additive one.
A map uses the scale $1$ inch = $3$ miles. A park is 6 inches from a school on the map. The scale is the same for all distances. How far is the park from the school?
Answer: 18 miles This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the actual distance between a park and a school using a map scale of 1 inch = 3 miles. The correct answer works because multiplying the map distance of 6 inches by the scale factor of 3 miles per inch gives 18 miles. A common distractor may fail because it uses addition or misreads the scale, leading to incorrect results. To help students: Teach them to identify the scale factor clearly and apply it accurately. Encourage double-checking calculations and understanding the relationship between map and actual distances. Practice with different map scenarios to build confidence in identifying and applying the correct scaling methods.
A special type of wire is sold for $0.15 per inch. How much would it cost to buy 2 feet of this wire? (1 foot = 12 inches)
Answer: $3.60 First, convert the length from feet to inches, since the price is given per inch. There are 12 inches in a foot, so 2 feet is equal to $2 \times 12 = 24$ inches. Next, calculate the total cost by multiplying the length in inches by the cost per inch: $24 \text{ inches} \times \$0.15/\text{inch} = \$3.60$ .
To make a special shade of orange, an artist mixes 4 parts of red paint with 3 parts of yellow paint. If the artist needs to make a total of 21 ounces of this orange paint, how many ounces of red paint are required?
Answer: 12 ounces First, find the total number of parts in one batch of the mixture: 4 parts red + 3 parts yellow = 7 total parts. The artist needs a total of 21 ounces. The scaling factor is the desired amount divided by the parts in one batch: $21 \text{ ounces} \div 7 \text{ parts} = 3$ ounces per part. Since the recipe calls for 4 parts of red paint, the artist needs $4 \text{ parts} \times 3 \text{ ounces/part} = 12$ ounces of red paint.
A building model uses a 1:50 scale. The model height is 3 inches. The real building is 50 times taller than the model. Scale up the height using multiplication. What is the actual height in inches?
Answer: 150 inches This question tests ISEE Lower Level quantitative reasoning skills: solving proportional situations using scaling. Proportional scaling involves multiplying or dividing quantities to maintain a consistent ratio or proportion. In this problem, you apply the concept of scaling to find the actual building height from a 1:50 scale model that is 3 inches tall. The correct answer works because it correctly applies the scaling factor of 50 to the model height: 3 inches × 50 = 150 inches. A common distractor like 53 inches may fail because it adds the scale factor to the model height (3 + 50) instead of multiplying, showing confusion about how scale ratios work. To help students: Teach them that in a 1:50 scale, every measurement on the real building is 50 times larger than on the model. Encourage them to set up the problem clearly and remember that scaling involves multiplication, making the real building much taller than just adding numbers together would suggest.

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