SSAT Middle Level Quiz: Proportional Scaling

20 questions · exam conditions

0:00

Proportional ScalingQuestion 1 of 20

A sketch is 9 cm by 6 cm. It is enlarged by a factor of $\tfrac{4}{3}$ . What are new dimensions?

The dimensions are

12

cm by

8

cm.

The dimensions are

13

cm by

9

cm.

The dimensions are

6.75

cm by

4.5

cm.

The dimensions are

36

cm by

24

cm.

← Back to quizzes

SSAT Middle Level Quiz

SSAT Middle Level Quiz: Proportional Scaling

Practice Proportional Scaling in SSAT Middle 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 Scaling, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Middle 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 sketch is 9 cm by 6 cm. It is enlarged by a factor of $\tfrac{4}{3}$ . What are new dimensions?

The dimensions are $12$ cm by $8$ cm. (correct answer)
The dimensions are $13$ cm by $9$ cm.
The dimensions are $6.75$ cm by $4.5$ cm.
The dimensions are $36$ cm by $24$ cm.

Explanation: This question tests middle school proportional reasoning skills, specifically solving scaling problems using ratios. Proportional reasoning involves understanding and applying the constant relationship between quantities, often expressed as ratios or fractions. In this problem, the scenario requires calculating the new dimensions of a sketch enlarged by a factor of 4/3. The correct answer, choice A, accurately reflects the proportional relationship and demonstrates correct application of scaling procedures, as 9 × (4/3) = 12 and 6 × (4/3) = 8 cm. Choice D is incorrect because it represents a common error, such as multiplying by 4 instead of 4/3. To support students, emphasize understanding ratio relationships and practicing with real-world examples, such as adjusting recipes or map reading. Encourage checking calculations for unit consistency and logical accuracy.

Question 2

A photo is 6 in by 4 in. It is enlarged by a factor of 1.25. What are new dimensions?

The dimensions are $7.5$ in by $5$ in. (correct answer)
The dimensions are $6.25$ in by $4.25$ in.
The dimensions are $5$ in by $3$ in.
The dimensions are $30$ in by $20$ in.

Explanation: This question tests middle school proportional reasoning skills, specifically solving scaling problems using ratios. Proportional reasoning involves understanding and applying the constant relationship between quantities, often expressed as ratios or fractions. In this problem, the scenario requires calculating the new dimensions of a photo enlarged by a factor of 1.25. The correct answer, choice A, accurately reflects the proportional relationship and demonstrates correct application of scaling procedures, as 6 × 1.25 = 7.5 and 4 × 1.25 = 5 inches. Choice D is incorrect because it represents a common error, such as multiplying by 5 instead of 1.25. To support students, emphasize understanding ratio relationships and practicing with real-world examples, such as adjusting recipes or map reading. Encourage checking calculations for unit consistency and logical accuracy.

Question 3

A model airplane is built using a scale of 1:48, meaning 1 inch on the model represents 48 inches on the actual airplane. If the wingspan of the actual airplane is 36 feet, what is the wingspan of the model in inches?

6 inches
9 inches (correct answer)
12 inches
18 inches
24 inches

Explanation: Scale problems require you to set up proportions carefully, paying close attention to units. When you see a scale like 1:48, this means 1 unit on the model equals 48 of those same units on the actual object. First, convert the actual wingspan to inches since the answer choices are in inches: 36 feet × 12 inches/foot = 432 inches. Now set up your proportion using the scale 1:48. If 1 inch on the model represents 48 inches on the actual airplane, then:

\frac{1 \text{ inch model}}{48 \text{ inches actual}} = \frac{x \text{ inches model}}{432 \text{ inches actual}}

Cross multiply:

48x = 432

, so

x = 9

inches. Looking at the wrong answers: Choice A (6 inches) might result from incorrectly dividing 36 by 6 instead of properly converting to inches first. Choice C (12 inches) could come from mistakenly thinking the scale means the model is 1/12 the size of the actual plane, confusing the 12 inches per foot conversion. Choice D (18 inches) might result from dividing the actual wingspan in feet (36) by some incorrect scale factor. The correct answer is B (9 inches). Study tip: In scale problems, always make sure your units match throughout your calculation. Convert everything to the same unit before setting up your proportion, and remember that a 1:48 scale means the model is 1/48 the size of the actual object.

Question 4

A recipe that serves 6 people calls for 1.5 cups of flour. Maria wants to make enough for 14 people. How much flour should she use?

3.0 cups of flour
3.5 cups of flour (correct answer)
4.0 cups of flour
4.5 cups of flour
5.0 cups of flour

Explanation: This is a proportion problem where you need to scale a recipe up from serving 6 people to serving 14 people. When you see recipe scaling questions, set up a proportion to find the relationship between the original and new amounts. Start by setting up the proportion:

\frac{\text{original flour}}{\text{original people}} = \frac{\text{new flour}}{\text{new people}}

Substituting the known values:

\frac{1.5 \text{ cups}}{6 \text{ people}} = \frac{x \text{ cups}}{14 \text{ people}}

Cross multiply to solve:

1.5 \times 14 = 6 \times x

, which gives you

21 = 6x

, so

x = 3.5

cups. You can also think of this as finding the flour needed per person first:

1.5 ÷ 6 = 0.25

cups per person. Then multiply by 14 people:

0.25 \times 14 = 3.5

cups. Choice A (3.0 cups) is too small and likely comes from incorrectly doubling the original amount since 14 is roughly twice 6, but this ignores the exact proportional relationship. Choice C (4.0 cups) might result from rounding errors or miscalculating

1.5 \times 14 ÷ 6

. Choice D (4.5 cups) could come from adding instead of using proportional reasoning, perhaps thinking

1.5 + 3 = 4.5

. For proportion problems on the SSAT, always set up your ratios carefully and double-check by working backwards—does 3.5 cups divided by 14 people equal 1.5 cups divided by 6 people? Both equal 0.25 cups per person, confirming your answer.

Question 5

A photograph that is 4 inches by 6 inches is enlarged so that the longer side becomes 15 inches. What is the length of the shorter side in the enlarged photograph?

9 inches
10 inches (correct answer)
11 inches
12 inches
13 inches

Explanation: When you see a photograph enlargement problem, you're dealing with similar figures and proportional relationships. The key insight is that when a photograph is enlarged, both dimensions change by the same scale factor to maintain the same shape. Start by identifying the scale factor. The original longer side is 6 inches, and it becomes 15 inches in the enlargement. So the scale factor is

\frac{15}{6} = 2.5

. This means every dimension of the photograph is multiplied by 2.5. Now apply this scale factor to the shorter side:

4 \times 2.5 = 10

inches. You can verify this makes sense by checking that the ratio of sides remains the same: originally

\frac{6}{4} = 1.5

, and in the enlargement

\frac{15}{10} = 1.5

. Looking at the wrong answers: Choice (A) 9 inches would give a scale factor of

\frac{9}{4} = 2.25

, which doesn't match our scale factor of 2.5. Choice (C) 11 inches represents the trap of adding the same amount (9 inches) to the shorter side as was added to the longer side, but this doesn't preserve proportions. Choice (D) 12 inches might come from incorrectly doubling the original shorter side plus adding 4, but this also breaks the proportional relationship. The correct answer is (B) 10 inches. Strategy tip: In proportion problems, always find the scale factor first by comparing corresponding sides, then apply that same factor to find unknown dimensions. Cross-check by verifying the ratios remain equal.

Question 6

A car travels 280 miles on 8 gallons of gas. At this rate, how many gallons of gas are needed to travel 525 miles?

12 gallons
13 gallons
15 gallons (correct answer)
17 gallons
20 gallons

Explanation: This is a rate problem that tests your ability to find unit rates and use proportional reasoning. When you see a question giving you one rate and asking for a related quantity, think about finding the rate per unit first. Start by finding the car's gas mileage (miles per gallon). Divide 280 miles by 8 gallons:

280 ÷ 8 = 35

miles per gallon. Now you know the car travels 35 miles on each gallon of gas. To find how many gallons are needed for 525 miles, divide the total distance by the miles per gallon:

525 ÷ 35 = 15

gallons. You can verify this with a proportion:

\frac{280 \text{ miles}}{8 \text{ gallons}} = \frac{525 \text{ miles}}{x \text{ gallons}}

. Cross-multiplying gives

280x = 525 \times 8 = 4200

, so

x = 15

gallons. Looking at the wrong answers: Choice A (12 gallons) is too low and might result from incorrectly calculating the unit rate or making an arithmetic error. Choice B (13 gallons) is also too low and could come from rounding errors during calculation. Choice D (17 gallons) is too high and might result from using an incorrect proportion setup or calculation mistake. The correct answer is C) 15 gallons. Strategy tip: For rate problems, always find the unit rate first (like miles per gallon or cost per item). This makes the rest of the problem straightforward division or multiplication. Double-check by seeing if your answer makes sense—525 miles is less than double 280 miles, so the gas needed should be less than double 8 gallons.

Question 7

A machine produces 450 widgets in 3 hours. At the same rate, how long will it take to produce 2,250 widgets?

12 hours
15 hours (correct answer)
18 hours
20 hours
24 hours

Explanation: This is a rate problem that asks you to find how long it takes to complete a larger task when you know the rate for a smaller one. The key is to first calculate the machine's rate of production, then use that rate to find the time needed for the bigger job. Start by finding the machine's rate: if it produces 450 widgets in 3 hours, then it produces

450 ÷ 3 = 150

widgets per hour. Now you can set up the calculation for 2,250 widgets:

2,250 ÷ 150 = 15

hours. So the correct answer is B) 15 hours. Let's see why the other answers are wrong. A) 12 hours would mean the machine produces

2,250 ÷ 12 = 187.5

widgets per hour, which is faster than the given rate of 150 per hour. C) 18 hours gives a rate of

2,250 ÷ 18 = 125

widgets per hour, which is too slow. D) 20 hours yields

2,250 ÷ 20 = 112.5

widgets per hour, also too slow compared to the actual rate. A common mistake is trying to set up a proportion incorrectly or forgetting to calculate the rate first. Always break rate problems into two steps: find the unit rate (widgets per hour, miles per gallon, etc.), then multiply or divide to get your final answer. Double-check by working backwards—does your answer give you the original rate when you plug it back in?

Question 8

A statue is 18 feet tall and casts a shadow that is 24 feet long. At the same time, a nearby tree casts a shadow that is 40 feet long. How tall is the tree?

28 feet tall
30 feet tall (correct answer)
32 feet tall
34 feet tall
36 feet tall

Explanation: When you encounter problems involving shadows and heights, you're dealing with similar triangles and proportional relationships. Objects and their shadows form right triangles, and when the sun's angle is the same for all objects, these triangles are similar. Set up a proportion using the relationship:

\frac{\text{height of object 1}}{\text{shadow of object 1}} = \frac{\text{height of object 2}}{\text{shadow of object 2}}

For this problem:

\frac{18 \text{ feet}}{24 \text{ feet}} = \frac{\text{tree height}}{40 \text{ feet}}

Cross multiply:

18 \times 40 = 24 \times \text{tree height}

This gives you:

720 = 24 \times \text{tree height}

Dividing both sides by 24:

\text{tree height} = 30 \text{ feet}

So answer B) 30 feet tall is correct. Let's examine why the other answers are wrong. Answer A) 28 feet might result from incorrectly subtracting the difference between shadow lengths (40 - 24 = 16) from a rough estimate. Answer C) 32 feet could come from mistakenly adding the statue's height to some calculation error. Answer D) 34 feet might arise from incorrectly adding the statue's height (18) to the difference in shadow lengths (40 - 24 = 16). Remember this key strategy: shadow problems always use proportions. Set up your ratio consistently—keep heights in numerators and shadows in denominators (or vice versa), then cross multiply to solve. Double-check by verifying that taller objects cast longer shadows, which makes sense with your answer.

Question 9

A swimming pool can be filled by a hose in 12 hours. At the same rate, how long would it take to fill a pool that is 2.5 times larger?

28 hours
30 hours (correct answer)
32 hours
35 hours
40 hours

Explanation: When you encounter rate problems involving proportional scaling, the key insight is that if the task gets larger, the time required increases proportionally when the rate stays constant. Since the hose fills at the same rate in both scenarios, you can set up a direct proportion. If the original pool takes 12 hours to fill, and the new pool is 2.5 times larger, then the time required will also be 2.5 times longer:

12 \text{ hours} \times 2.5 = 30 \text{ hours}

Think of it this way: if you're pouring water at a steady rate into a container that's 2.5 times bigger, it logically takes 2.5 times longer to fill completely. Looking at the wrong answers: Choice A (28 hours) might tempt you if you mistakenly calculated

12 + (12 \times 1.33)

or made an arithmetic error. Choice C (32 hours) could result from incorrectly adding 20 to the original 12 hours rather than multiplying. Choice D (35 hours) might come from confusing this with a different type of rate problem or miscalculating the proportion entirely. The correct answer is B (30 hours). Strategy tip: In direct proportion problems, remember that "times larger" means you multiply both quantities by the same factor. When the job gets bigger but the rate stays the same, time increases proportionally. Always double-check by asking: "Does my answer make logical sense?" A pool 2.5 times larger should take 2.5 times longer to fill.

Question 10

On a map with a scale of 1 inch = 20 miles, two towns are 3.5 inches apart. If the scale were changed to 1 inch = 25 miles, how far apart would the same two towns appear on the new map?

2.6 inches apart
2.8 inches apart (correct answer)
3.0 inches apart
3.2 inches apart
3.4 inches apart

Explanation: Map scale problems test your ability to work with proportional relationships. When you encounter these questions, remember that the actual distance between locations never changes—only how that distance appears on different maps. First, find the real distance between the towns using the original scale. If 1 inch = 20 miles, then 3.5 inches represents

3.5 \times 20 = 70

miles. This is the actual distance that won't change. Now determine how this 70-mile distance would appear on the new map where 1 inch = 25 miles. Set up the proportion: if 25 miles = 1 inch, then 70 miles =

\frac{70}{25} = 2.8

inches. Let's examine why the other answers are incorrect. Choice A (2.6 inches) results from incorrectly multiplying 3.5 by

\frac{20}{25}

and rounding down, missing the precise calculation. Choice C (3.0 inches) might come from rough mental math that doesn't account for the scale change properly. Choice D (3.2 inches) could result from accidentally using an inverse relationship or calculation error. The correct answer is B) 2.8 inches apart. Strategy tip: Always work through map scale problems in two steps: first convert the map distance to real-world distance using the original scale, then convert that real distance to the new map distance using the new scale. This systematic approach prevents the common mistake of trying to convert directly between map measurements.

Question 11

A rectangular garden has dimensions of 12 feet by 18 feet. If the garden is scaled up so that the shorter side becomes 20 feet, what will be the area of the new garden?

520 square feet
540 square feet
560 square feet
580 square feet
600 square feet (correct answer)

Explanation: This problem tests your understanding of similar rectangles and proportional scaling. When a rectangle is "scaled up" while maintaining its shape, all dimensions must change by the same ratio to preserve the rectangular proportions. The original garden measures 12 feet by 18 feet, where 12 feet is the shorter side. When this shorter side becomes 20 feet, you need to find the scaling factor:

\frac{20}{12} = \frac{5}{3}

Since the rectangle maintains its shape, the longer side must also be multiplied by this same factor:

18 \times \frac{5}{3} = 30

feet. The new garden dimensions are 20 feet by 30 feet, giving an area of

20 \times 30 = 600

square feet. Since this isn't among choices A-D, the correct answer is E. Let's examine why the given choices are incorrect. Choice A (520) would result from incorrectly using a scaling factor of about 1.44, while choice B (540) might come from adding 240 to the original area of 216 square feet. Choice C (560) could result from miscalculating the new longer side as 28 feet instead of 30 feet. Choice D (580) doesn't correspond to any logical scaling error but represents another trap answer. Remember: when dealing with similar figures, all corresponding dimensions must change by the same ratio. Calculate the scaling factor first, then apply it consistently to all dimensions before finding the area. Watch out for answer choices that don't include the correct result—sometimes "none of the above" is the right choice.

Question 12

A water tank can be emptied by one drain in 15 hours. If a second identical drain is added, how long will it take to empty the same tank using both drains?

6.5 hours
7.0 hours
7.5 hours (correct answer)
8.0 hours
8.5 hours

Explanation: When you encounter work rate problems involving multiple workers (or in this case, drains) working together, think in terms of rates per unit time. The key insight is that rates add when working simultaneously. First, determine the rate of one drain. If one drain empties the tank in 15 hours, its rate is

\frac{1}{15}

of the tank per hour. When you add a second identical drain, you now have two drains each working at

\frac{1}{15}

tank per hour. The combined rate is:

\frac{1}{15} + \frac{1}{15} = \frac{2}{15}

of the tank per hour. To find the time needed to empty one complete tank at this rate, divide 1 by the combined rate:

1 \div \frac{2}{15} = 1 \times \frac{15}{2} = 7.5

hours. Looking at the wrong answers: (A) 6.5 hours represents working too fast—you might get this by incorrectly assuming the drains work more efficiently together than simple addition suggests. (B) 7.0 hours could result from rounding errors or miscalculating the fraction arithmetic. (D) 8.0 hours is too slow and might come from incorrectly subtracting rates instead of adding them, or confusing this with a different type of rate problem. The correct answer is (C) 7.5 hours. Strategy tip: For combined work rate problems, always remember that individual rates add together. Set up the problem as: (rate of worker 1) + (rate of worker 2) = combined rate, then find time by taking the reciprocal.

Question 13

A survey shows that 3 out of every 8 students prefer pizza for lunch. If there are 280 students in the school, how many students prefer pizza?

95 students
105 students (correct answer)
115 students
125 students
135 students

Explanation: This is a classic proportion problem where you need to scale up a ratio to match a larger group. When you see "X out of every Y" language, you're dealing with a part-to-whole relationship that you can apply to any size group. Since 3 out of every 8 students prefer pizza, you can set up a proportion:

\frac{3}{8} = \frac{x}{280}

, where x is the number of pizza-preferring students out of 280 total students. Cross-multiplying gives you

3 \times 280 = 8x

, so

840 = 8x

, which means

x = 105

. Alternatively, you can think of this as finding

\frac{3}{8}

of 280:

\frac{3}{8} \times 280 = \frac{840}{8} = 105

students. Looking at the wrong answers: Choice A (95) is too small and might come from miscalculating the fraction or making an arithmetic error in the division. Choice C (115) could result from adding instead of multiplying somewhere in your calculation, or from rounding errors. Choice D (125) is significantly off and might come from confusing the setup—perhaps trying to find what fraction 280 is of some other number, or making a major computational mistake. Study tip: For proportion problems, always double-check your answer by seeing if it makes sense with the original ratio. Here, 105 out of 280 should equal 3 out of 8. Dividing both by 35 gives you exactly 3 out of 8, confirming your answer.

Question 14

A copy machine can make 180 copies in 4 minutes. At this rate, how many copies can it make in 1 hour and 15 minutes?

3,275 copies
3,375 copies (correct answer)
3,475 copies
3,575 copies
3,675 copies

Explanation: This is a rate problem that tests your ability to work with proportional relationships and unit conversions. When you see questions asking "at this rate," you need to find the unit rate first, then scale it up. Start by finding the machine's rate per minute. If it makes 180 copies in 4 minutes, then it makes

180 ÷ 4 = 45

copies per minute. Next, convert the target time to minutes. One hour and 15 minutes equals

60 + 15 = 75

minutes. Finally, multiply the rate by the time:

45 \text{ copies/minute} × 75 \text{ minutes} = 3,375 \text{ copies}

. This confirms answer choice B. Looking at the wrong answers: Choice A (3,275) likely comes from a calculation error, possibly using 73 minutes instead of 75, or making an arithmetic mistake in the multiplication. Choice C (3,475) might result from accidentally adding 100 to the correct answer or using 77 minutes instead of 75. Choice D (3,575) could come from using 79 minutes or another computational error in the final multiplication step. These answer choices are strategically close to each other, which is common on the SSAT—they're testing whether you can execute the calculation accurately, not just set up the problem correctly. Strategy tip: Always double-check your time conversions in rate problems. Convert everything to the same units (usually minutes for these problems), find the unit rate, then multiply. Writing out each step prevents careless errors that lead to trap answers.

Question 15

A painter can paint 225 square feet of wall in 3 hours. At this rate, how long will it take to paint a room with 525 square feet of wall space?

6 hours
7 hours (correct answer)
8 hours
9 hours
10 hours

Explanation: This is a rate problem that asks you to find how long a task will take given a constant work rate. When you see questions about painters, workers, or machines completing tasks at steady rates, set up a proportion or calculate the rate per unit. First, find the painter's rate:

\frac{225 \text{ square feet}}{3 \text{ hours}} = 75 \text{ square feet per hour}

Now calculate how long it takes to paint 525 square feet:

\frac{525 \text{ square feet}}{75 \text{ square feet per hour}} = 7 \text{ hours}

You can also solve this using proportions:

\frac{225 \text{ sq ft}}{3 \text{ hours}} = \frac{525 \text{ sq ft}}{x \text{ hours}}

Cross-multiply:

225x = 525 \times 3 = 1575

, so

x = 7

hours. Looking at the wrong answers: Choice A (6 hours) might result from incorrectly calculating the rate as 37.5 square feet per hour instead of 75. Choice C (8 hours) could come from using an approximate rate of 65-70 square feet per hour. Choice D (9 hours) might result from miscalculating the original rate as roughly 58 square feet per hour. Remember that rate problems follow the formula: Rate × Time = Work Done. Always double-check your rate calculation first, then use it consistently. On the SSAT, rate problems often include answer choices that result from common arithmetic errors, so take time to verify your division and multiplication.

Question 16

A gear with 24 teeth rotates and turns another gear with 36 teeth. If the first gear makes 150 rotations, how many rotations does the second gear make?

90 rotations
100 rotations (correct answer)
110 rotations
120 rotations
130 rotations

Explanation: When you encounter gear problems, remember that gears with different numbers of teeth rotate at inversely proportional speeds. The gear with fewer teeth spins faster, while the gear with more teeth spins slower, but they cover the same distance along their circumferences. To solve this, use the fundamental gear relationship: the number of teeth times the number of rotations must be equal for both gears. Set up the equation:

24 \times 150 = 36 \times x

, where x is the unknown rotations of the second gear. Solving:

3600 = 36x

, so

x = 100

rotations. You can also think of this as a ratio: since the second gear has

\frac{36}{24} = 1.5

times more teeth, it rotates

\frac{150}{1.5} = 100

times. Looking at the wrong answers: Choice A (90 rotations) likely comes from incorrectly setting up the proportion or making an arithmetic error. Choice C (110 rotations) might result from adding instead of using the proper inverse relationship. Choice D (120 rotations) could come from incorrectly multiplying

150 \times \frac{24}{30}

(perhaps misreading 36 as 30). The correct answer is B) 100 rotations. Study tip: For gear problems, always remember the inverse relationship - more teeth means fewer rotations. Set up the equation "teeth₁ × rotations₁ = teeth₂ × rotations₂" and you'll solve these problems quickly and accurately every time.

Question 17

A company's profit increases proportionally with the number of items sold. If selling 240 items results in a profit of $1,800, what profit would result from selling 320 items?

$2,200
$2,300
$2,400 (correct answer)
$2,500
$2,600

Explanation: When you see a problem stating that one quantity "increases proportionally" with another, you're dealing with direct proportion. This means the ratio between the two quantities stays constant, so you can use cross-multiplication or find the unit rate to solve. Let's find the profit per item first. If 240 items generate $1,800 profit, then each item contributes $

\frac{1800}{240} = 7.50

in profit. For 320 items, the total profit would be

320 \times 7.50 = 2,400

$. Alternatively, you can set up a proportion:

\frac{240 \text{ items}}{1800 \text{ profit}} = \frac{320 \text{ items}}{x \text{ profit}}

. Cross-multiplying gives

240x = 320 \times 1800

, so

x = \frac{320 \times 1800}{240} = 2,400

. Looking at the wrong answers: Choice A (

2,200) represents an increase that's too small—it suggests the profit per item decreased, which contradicts the proportional relationship. Choice B (

2,300) might result from calculation errors or incorrect rounding. Choice D ($2,500) overshoots the correct answer, possibly from adding the wrong increment to the base profit. The correct answer is C) $2,400. Study tip: For proportion problems, always check if your answer makes sense by comparing ratios. Here, selling

\frac{320}{240} = 1.33

times as many items should yield

\frac{2400}{1800} = 1.33

times the profit. When ratios match, you've got the right answer.

Question 18

A model rocket is built to a scale where 2 inches on the model represents 15 feet on the actual rocket. If the model is 8 inches tall, and the actual rocket's fuel tank is 45 feet long, how long should the fuel tank be on the model?

5.5 inches
6.0 inches (correct answer)
6.5 inches
7.0 inches
7.5 inches

Explanation: Scale problems require you to set up a proportion that maintains the same ratio between model and actual measurements. When you see "2 inches represents 15 feet," you're being given the conversion factor that applies to all parts of the rocket. To find the model fuel tank length, set up a proportion using the given scale ratio. You know that 2 inches on the model equals 15 feet on the actual rocket, so:

\frac{2 \text{ inches (model)}}{15 \text{ feet (actual)}} = \frac{x \text{ inches (model)}}{45 \text{ feet (actual)}}

Cross multiply:

2 \times 45 = 15 \times x

, which gives you

90 = 15x

. Solving for x:

x = 6

inches. Choice A (5.5 inches) results from incorrectly using the model height (8 inches) in your calculation instead of the scale ratio. Choice C (6.5 inches) might come from adding the scale numbers incorrectly or making an arithmetic error in the proportion. Choice D (7.0 inches) could result from confusing which measurements go in the numerator versus denominator of your proportion. The correct answer is B (6.0 inches) because it maintains the exact 2:15 ratio given in the problem. Study tip: In scale problems, always identify the given ratio first, then set up your proportion with the same units in corresponding positions. Double-check by verifying that your answer maintains the original scale relationship—here, 6 inches should represent 45 feet just like 2 inches represents 15 feet.

Question 19

The scale on a blueprint shows that 0.25 inches represents 2 feet of actual length. If a room measures 3.5 inches on the blueprint, what is the actual length of the room in feet?

24 feet
26 feet
28 feet (correct answer)
30 feet
32 feet

Explanation: Scale problems are all about setting up proportions to convert between measurements. When you see a blueprint or map scale, you're working with two different units that have a fixed relationship. Here, the scale tells us that 0.25 inches on the blueprint equals 2 feet in real life. To find the actual length when the blueprint shows 3.5 inches, set up a proportion:

\frac{0.25 \text{ inches}}{2 \text{ feet}} = \frac{3.5 \text{ inches}}{x \text{ feet}}

Cross multiply:

0.25x = 2 \times 3.5 = 7

Solving for x:

x = \frac{7}{0.25} = 28 \text{ feet}

This confirms answer C is correct. Looking at the wrong answers: Choice A (24 feet) likely comes from incorrectly setting up the proportion or making an arithmetic error in the division. Choice B (26 feet) might result from rounding errors or miscalculating somewhere in the cross multiplication. Choice D (30 feet) could come from accidentally using simple multiplication (3.5 × 8 = 28, but somehow getting 30) or other computational mistakes. The key strategy for scale problems is to always write out your proportion clearly, making sure the units match up properly on each side. Double-check by asking yourself: "Does this make sense?" Since 3.5 inches is 14 times larger than 0.25 inches, the actual measurement should be 14 times larger than 2 feet, which gives us 28 feet.

Question 20

A recipe calls for a ratio of 2 cups of flour to 3 cups of sugar. If Sarah uses 8 cups of flour, how many cups of sugar should she use to maintain the same ratio?

10 cups of sugar
11 cups of sugar
12 cups of sugar (correct answer)
13 cups of sugar
14 cups of sugar

Explanation: When you encounter ratio problems, you're looking at proportional relationships where quantities must maintain the same relative amounts. The key is setting up a proportion to find the unknown value. The original recipe maintains a ratio of 2 cups flour to 3 cups sugar. You can express this as the fraction

\frac{2}{3}

. When Sarah scales up to 8 cups of flour, she needs to find how much sugar maintains this same ratio. Set up a proportion:

\frac{2 \text{ flour}}{3 \text{ sugar}} = \frac{8 \text{ flour}}{x \text{ sugar}}

Cross multiply:

2x = 3 \times 8 = 24

Therefore:

x = 12

cups of sugar. You can verify this makes sense: Sarah used 4 times as much flour (8 ÷ 2 = 4), so she should use 4 times as much sugar (3 × 4 = 12). Looking at the wrong answers: Choice A (10 cups) represents a common error where students might add the difference in flour amounts to the original sugar amount rather than scaling proportionally. Choice B (11 cups) doesn't follow any clear mathematical relationship and likely represents a calculation error. Choice D (13 cups) might result from incorrectly setting up the proportion or making arithmetic mistakes during cross multiplication. Strategy tip: Always check your ratio answer by verifying the scaling factor is consistent. If one ingredient increases by a certain multiple, the other should increase by that same multiple. This quick check can catch calculation errors and confirm your proportion was set up correctly.