PSAT Math Quiz: Ratios And Proportions

What this quiz covers

This quiz focuses on Ratios And Proportions, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT Math.

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.

Question 1

1. 120 m$^2$ (correct answer)
2. 48 m$^2$
3. 30 m$^2$
4. 80 m$^2$

Explanation: The question asks for the actual area of a garden in square meters, given a scale drawing of 6.4 cm by 3.0 cm with a scale of 1 cm to 2.5 m. Set up the proportion for each dimension: actual length = drawing length × scale factor, and similarly for width. Convert the dimensions: actual length = 6.4 × 2.5 = 16 m, actual width = 3.0 × 2.5 = 7.5 m. Then, calculate the area: 16 × 7.5 = 120 m². A key error might be forgetting to square the scale factor for area or mixing up units between cm and m. Emphasize careful unit conversion when setting up proportions in scale problems. For test-taking, quickly estimate the area to check if the answer makes sense with the scale.

Question 2

1. $\$1{,}920$
2. $\$2{,}040$
3. $\$2{,}160$ (correct answer)
4. $\$2{,}240$

Explanation: When you encounter ratio problems involving total amounts, you need to find the actual quantities from the given ratio, then calculate based on those specific numbers. Given the ratio $7:3$ for adult to student tickets, this means for every 7 adult tickets sold, 3 student tickets were sold. To find the actual numbers from 200 total tickets, think of this as $7x + 3x = 200$, where $x$ is the multiplier. This gives us $10x = 200$, so $x = 20$. Therefore: Adult tickets = $7 \times 20 = 140$ and Student tickets = $3 \times 20 = 60$. Now calculate revenue: Adult revenue = $140 \times \$12 = \$1{,}680$ and Student revenue = $60 \times \$8 = \$480$. Total revenue = $\$1{,}680 + \$480 = \$2{,}160$, which is answer choice C. Looking at the wrong answers: A) $\$1{,}920$ likely comes from incorrectly calculating one of the ticket quantities or prices. B) $\$2{,}040$ might result from mixing up the ratio (using $3:7$ instead of $7:3$) or making an arithmetic error. D) $\$2{,}240$ could come from assuming all 200 tickets were sold at some average price without properly working through the ratio. Strategy tip: In ratio problems, always convert the ratio to actual quantities first using the total given, then perform your calculations. Double-check that your individual quantities add up to the stated total before proceeding with further computations.

Question 3

1. 34.5 mL
2. 45 mL (correct answer)
3. 51 mL
4. 85 mL

Explanation: The question asks how many milliliters of acid should be added to 255 mL of water to maintain an acid-to-water ratio of 3:17. Set up the proportion where water is 17 parts corresponding to 255 mL, so find the value of one part: 255 ÷ 17 = 15 mL per part. Then, for acid, multiply by its parts: 3 × 15 = 45 mL. This keeps the ratio exact with total mixture of 20 parts. Common errors include reversing the ratio or forgetting to find the per-part value before scaling. Always verify units are consistent, like mL for both components. A useful strategy is to check if the final ratio matches the original after adding the amounts.

Question 4

1. 432 g
2. 504 g (correct answer)
3. 520 g
4. 540 g

Explanation: When you see "directly proportional" in a math problem, you're dealing with a constant ratio relationship. This means as one quantity increases, the other increases at a steady rate, and you can write the relationship as $m = kv$ where $k$ is the constant of proportionality. First, find the constant using the given information. With $m = 700$ grams and $v = 250$ cubic centimeters: $700 = k \cdot 250$ $k = \frac{700}{250} = 2.8$ Now you can find the mass for any volume using $m = 2.8v$. For $180$ cubic centimeters: $m = 2.8 \times 180 = 504$ grams This confirms answer choice B is correct. Looking at the wrong answers: Choice A (432 g) might result from incorrectly setting up a proportion or making an arithmetic error in the calculation. Choice C (520 g) could come from mistakenly using $k = 2.6$ instead of $2.8$, perhaps from a computational slip. Choice D (540 g) might result from using an incorrect constant like $k = 3$. The key strategy for direct proportion problems is to always find the constant first, then apply it to the new situation. You can also solve these using equivalent ratios: $\frac{700}{250} = \frac{m}{180}$, which gives the same result. Remember that "directly proportional" means the ratio between the two quantities stays constant—this is actually the definition of density in science contexts.

Question 5

1. 3:4
2. 4:3 (correct answer)
3. 5:4
4. 7:6

Explanation: When you encounter mixture problems involving percentages, you're dealing with weighted averages. The key is setting up equations that account for both the total volume and the total amount of the substance (acid, in this case). Let's say you need $x$ liters of Solution A and $y$ liters of Solution B. You can set up two equations: First, the volumes must add to 15 liters: $x + y = 15$. Second, the total acid content must equal 40% of 15 liters: $0.25x + 0.60y = 0.40(15) = 6$. From the first equation, $x = 15 - y$. Substituting into the second equation: $0.25(15 - y) + 0.60y = 6$. Simplifying: $3.75 - 0.25y + 0.60y = 6$, which gives $0.35y = 2.25$, so $y = 6.43$ liters. Therefore, $x = 15 - 6.43 = 8.57$ liters. The ratio of Solution A to Solution B is approximately $8.57:6.43$, which simplifies to $4:3$ when you divide both parts by their greatest common factor. Choice A (3:4) reverses this ratio—a common trap when students mix up which solution is which. Choice C (5:4) would result from calculation errors in the algebraic manipulation. Choice D (7:6) represents a ratio that's too close to 1:1, which would happen if you incorrectly assumed the concentrations were closer together than they actually are. Remember: In mixture problems, always define your variables clearly, set up both a volume equation and a concentration equation, then solve systematically. Double-check by verifying your answer produces the target concentration.

Question 6

1. 4.2 kg (correct answer)
2. 3.0 kg
3. 2.7 kg
4. 1.2 kg

Explanation: The question asks how many kilograms of flour are needed to match 1.8 kilograms of sugar in a 7:3 flour-to-sugar ratio. Set up the proportion where flour/sugar = 7/3, so flour = (7/3) * sugar = (7/3) * 1.8. Calculate 7 * 1.8 = 12.6, then divide by 3 to get 4.2 kilograms. This maintains the ratio by scaling the flour accordingly. A common mistake is inverting the ratio, like using 3/7, which would give an incorrect smaller amount. Always note the units, here kilograms for both ingredients. For tests, verify by checking if the total mixture follows the original ratio when combined.

Question 7

1. 24
2. 30 (correct answer)
3. 32
4. 40

Explanation: We need to find the number of sophomores when the ratio of freshmen:sophomores:juniors is 4:5:3 and there are 72 total members. First, find the total parts in the ratio: 4 + 5 + 3 = 12 parts. Each part represents 72 ÷ 12 = 6 students. Since sophomores make up 5 parts of the ratio, there are 5 × 6 = 30 sophomores. A common mistake is dividing 72 by just one part of the ratio instead of the sum of all parts. Always add all ratio parts first to find what one part represents.

Question 8

1. 24
2. 30 (correct answer)
3. 36
4. 40

Explanation: The question asks how many sophomores are in a class of 72 students with a freshmen-to-sophomores-to-juniors ratio of 4:5:3. Set up the proportion with total parts: 4 + 5 + 3 = 12 parts for 72 students, so one part = 72 ÷ 12 = 6 students. For sophomores, multiply by their parts: 5 × 6 = 30. This distributes the total correctly according to the ratio. A common error is forgetting to sum the parts or assigning the wrong part to a group. Ensure units (students) are consistent across the proportion. For strategy, after calculating, add up all groups to verify they total 72.

Question 9

1. 84
2. 98 (correct answer)
3. 112
4. 182

Explanation: This percentage problem asks how many students chose option A if 35% of 280 students made this choice. To find 35% of 280, multiply: 0.35 × 280 = 98 students. You can also think of this as (35/100) × 280 = 9800/100 = 98. A common mistake is forgetting to convert the percentage to a decimal or fraction before multiplying. When working with percentages, always convert to decimal form (divide by 100) before calculating.

Question 10

1. 78%
2. 86% (correct answer)
3. 94%
4. 116%

Explanation: The question asks what percent the straight-line map distance is of the actual driving distance, given a scale of 1 inch to 30 miles, map distance 2.75 inches, and driving distance 96 miles. Set up the proportion for actual straight-line distance: 2.75 × 30 = 82.5 miles. Then, find the percentage: (82.5 / 96) × 100 ≈ 85.9375%, which rounds to 86%. Calculate carefully and round to the nearest whole percent as instructed. Errors can occur in forgetting to multiply by the scale or misrounding. Ensure units match when comparing distances. A strategy is to estimate: 2.75 × 30 is about 82.5, and 82.5/96 is roughly 86%.

Question 11

1. 18 cm
2. 20 cm
3. 22.5 cm (correct answer)
4. 24 cm

Explanation: The question asks for the height of an enlarged photo where the original width is 9 cm and height is 13.5 cm, and the enlarged width is 15 cm, with all lengths scaled by the same factor. Set up the proportion for the scale factor: enlarged width / original width = 15 / 9 = 5/3. Apply this factor to the height: enlarged height = 13.5 × (5/3) = 22.5 cm. Simplify fractions before multiplying to avoid calculation errors. A frequent mistake is using the wrong dimension for the scale factor or not simplifying ratios. Pay close attention to which measurements are being compared in proportion setups. As a strategy, cross-check by verifying the ratio of widths equals the ratio of heights.

Question 12

1. 8 cm
2. 18 cm
3. 20 cm (correct answer)
4. 27 cm

Explanation: The question asks for the length of the side in triangle DEF corresponding to AC in similar triangle ABC, where AB=9 cm, AC=12 cm, and DE=15 cm corresponds to AB. Set up the proportion using the scale factor from ABC to DEF as DE/AB = 15/9 = 5/3. Apply this to AC: corresponding side = (5/3) * 12 = 20 cm. This ensures similarity by scaling all sides equally. A key error is mismatching corresponding sides, leading to wrong proportions. Confirm units are consistent, here all in centimeters. As a strategy, list corresponding sides clearly before setting up ratios.

Question 13

1. 12 oz
2. 15 oz
3. 16 oz (correct answer)
4. 20 oz

Explanation: The paint mixture uses red:blue:white in ratio 3:4:5, totaling 48 ounces. First, find total parts: 3 + 4 + 5 = 12 parts. Each part represents 48 ÷ 12 = 4 ounces. Since blue paint is 4 parts of the ratio, the amount of blue paint is 4 × 4 = 16 ounces. A common error is using the ratio number directly as the amount instead of calculating what each part represents. Always find the value of one part first, then multiply by the specific ratio number.

Question 14

1. $\$1{,}920$
2. $\$2{,}040$
3. $\$2{,}160$ (correct answer)
4. $\$2{,}240$

Explanation: This problem combines ratios with revenue calculation, testing your ability to work with proportional relationships and apply them to real-world scenarios. Start by using the ratio $7:3$ to find how many of each ticket type were sold. Since the ratio represents 7 parts adult tickets to 3 parts student tickets, that's $7 + 3 = 10$ total parts. With 200 tickets total, each part equals $200 ÷ 10 = 20$ tickets. Therefore: Adult tickets = $7 × 20 = 140$ and Student tickets = $3 × 20 = 60$ Now calculate the revenue from each ticket type:

• Adult revenue: $140 × \$12 = \$1{,}680$
• Student revenue: $60 × \$8 = \$480$
• Total revenue: $\$1{,}680 + \$480 = \$2{,}160$

Question 15

1. 15
2. 16
3. 20 (correct answer)
4. 24

Explanation: The question asks how many sophomores are in a club with a 4:5:3 ratio of freshmen to sophomores to juniors and 48 total students. Set up the proportion by finding total parts: 4 + 5 + 3 = 12 parts, where sophomores are 5 parts. Sophomores = (5/12) * 48 = 20. Scale by dividing 48 by 12 to get 4 per part, then 5 * 4 = 20. Avoid the error of dividing total by one ratio only, which ignores the full proportion. Units are counts of students, so no conversion needed. For tests, check if the sum of all groups equals the total after calculation.

Question 16

1. 44.4 cm
2. 77.0 cm
3. 222 cm
4. 444 cm (correct answer)

Explanation: With a scale of 1:24 (model:actual), we need to find the actual car length when the model is 18.5 cm. The scale means 1 unit on the model represents 24 units on the actual car. Therefore, actual length = model length × 24 = 18.5 × 24 = 444 cm. Students often confuse scale ratios and divide instead of multiply, or use the ratio backwards. Remember: when the scale is model:actual and you have the model size, multiply by the second number to get actual size.

Question 17

1. 3.0 L
2. 4.8 L (correct answer)
3. 5.0 L
4. 6.0 L

Explanation: The question asks how many liters of white paint are needed for 9 liters of gray paint mixed in the ratio of blue to white to black as 5:8:2. Set up the proportion by finding the total parts in the ratio, which is 5 + 8 + 2 = 15 parts, where white paint represents 8 parts. The amount of white paint is then (8/15) of the total 9 liters, so calculate (8/15) * 9 = 72/15 = 4.8 liters. This setup ensures the ratio remains consistent by scaling all components equally. A key error to avoid is forgetting to find the total parts, which might lead to incorrectly dividing by just one ratio value. Always double-check units to confirm they match, here all in liters. As a test-taking strategy, verify the answer by checking if all parts sum to the total volume when scaled.

Question 18

1. 8 cm
2. 18 cm
3. 24 cm (correct answer)
4. 240 cm

Explanation: The question asks for the length of a model car in centimeters, scaled 1:18 from an actual 4.32 meters long. Set up the proportion: model = actual / 18 = 4.32 / 18 = 0.24 meters. Convert to centimeters: 0.24 * 100 = 24 cm. Alternatively, convert actual to cm first: 4.32 * 100 = 432 cm, then 432 / 18 = 24 cm. Avoid errors like multiplying instead of dividing by the scale factor, which would inflate the size. Carefully handle unit conversions from meters to centimeters. For tests, confirm the scale direction (model to actual) before calculating.

Question 19

1. 32
2. 36
3. 40 (correct answer)
4. 42

Explanation: When you encounter ratio problems, you're working with proportional relationships between quantities. The key is understanding that ratios tell you the relative amounts, not the actual amounts, until you're given one concrete value. The ratio $5:3$ means that for every 5 apples, there are 3 oranges. Since you know there are 24 oranges, you can set up a proportion to find the number of apples. Think of this as: if 3 parts equal 24 oranges, then 5 parts equal how many apples? First, find what one "part" represents: $24 \div 3 = 8$. So each part in the ratio equals 8 fruits. Since apples represent 5 parts, multiply: $5 \times 8 = 40$ apples. You can verify this by checking that $\frac{40}{24} = \frac{5}{3}$, which simplifies correctly. Looking at the wrong answers: A) 32 represents a calculation error where someone might have multiplied $4 \times 8$ instead of $5 \times 8$. B) 36 could result from incorrectly thinking the ratio is $3:2$ instead of $5:3$, then calculating $\frac{3}{2} \times 24 = 36$. D) 42 might come from adding the ratio parts incorrectly or making an arithmetic mistake in the proportion. The correct answer is C) 40. Strategy tip: For ratio problems, always identify what one "part" equals by dividing the known quantity by its ratio number, then multiply by the unknown quantity's ratio number. Double-check by verifying the final ratio matches the original.

Question 20

1. 1.6 gallons
2. 2.1 gallons
3. 2.9 gallons (correct answer)
4. 3.6 gallons

Explanation: When you encounter ratio problems, you're working with proportional relationships where two quantities must maintain a constant relationship to each other. The key is setting up a proportion that relates the given information to what you're trying to find. Here, the red-to-white ratio must be $7:4$, which means for every 7 parts red paint, you need 4 parts white paint. Since you have exactly 5 gallons of red paint, you can set up the proportion: $\frac{7}{4} = \frac{5}{x}$, where $x$ is the gallons of white paint needed. Cross-multiplying: $7x = 4 \times 5 = 20$, so $x = \frac{20}{7} = 2.857...$ gallons. Rounding to one decimal place gives approximately 2.9 gallons. Looking at the wrong answers: Choice A (1.6 gallons) likely comes from incorrectly setting up the proportion as $\frac{4}{7} = \frac{5}{x}$ and making calculation errors. Choice B (2.1 gallons) might result from using an incorrect ratio or arithmetic mistakes. Choice D (3.6 gallons) could come from confusing the setup entirely, perhaps thinking you need more white than the ratio actually requires. The correct answer is C) 2.9 gallons. Strategy tip: Always double-check ratio problems by substituting your answer back into the original ratio. Here, $\frac{5}{2.9} \approx \frac{7}{4} = 1.75$, confirming your answer maintains the required proportion. Write out your proportion clearly before solving to avoid setup errors.