Ratios & Proportions - PSAT Math

A paint color is made by mixing red, blue, and white paint in the ratio $3:4:5$ . If 48 ounces of paint are made in total, how many ounces of blue paint are used?
Answer: 16 oz 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.
Solve for $n$ : $\dfrac{4n-1}{6n+5} = \dfrac{3}{5}$ .
Answer: 10 When you encounter a rational equation like this one, you're dealing with two fractions set equal to each other. The most efficient approach is to cross-multiply, which eliminates the fractions and gives you a linear equation to solve. Starting with $\frac{4n-1}{6n+5} = \frac{3}{5}$ , cross-multiply by multiplying the numerator of each fraction by the denominator of the other: $5(4n-1) = 3(6n+5)$ . Distribute on both sides: $20n - 5 = 18n + 15$ . Now isolate the variable by subtracting $18n$ from both sides: $2n - 5 = 15$ . Add 5 to both sides: $2n = 20$ . Finally, divide by 2: $n = 10$ . You can verify this by substituting back into the original equation: $\frac{4(10)-1}{6(10)+5} = \frac{39}{65} = \frac{3}{5}$ ✓ Looking at the wrong answers: Choice (A) 8 would give you $\frac{31}{53}$ , which doesn't equal $\frac{3}{5}$ . Choice (B) 9 yields $\frac{35}{59}$ , also incorrect. Choice (D) 12 produces $\frac{47}{77}$ , which similarly doesn't work. These incorrect values likely result from arithmetic errors during the cross-multiplication or algebraic manipulation steps. Strategy tip: Always verify your answer by substituting it back into the original equation. This catches calculation mistakes and confirms your solution works. Also, remember that cross-multiplication is your go-to technique whenever you see two fractions equal to each other.
A map uses the scale $1\text{ inch} : 30\text{ miles}$ . On the map, the distance between two towns is $2.75$ inches. The actual driving distance is $96$ miles. What percent of the driving distance is the straight-line map distance? (Round to the nearest whole percent.)
Answer: 86% 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%.
For a certain substance, mass $m$ (in grams) is directly proportional to volume $v$ (in cubic centimeters). If a volume of $250\text{ cm}^3$ has a mass of $700\text{ g}$ , what is the mass of $180\text{ cm}^3$ of the substance?
Answer: 504 g 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.
At a fruit stand, the ratio of apples to oranges is $5:3$ . If the stand has 24 oranges, how many apples does it have?
Answer: 40 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.
If $\dfrac{p}{q} = \dfrac{7}{9}$ and $q = 18k$ for some positive constant $k$ , which expression gives $p$ in terms of $k$ ?
Answer: $14k$ When you encounter problems involving proportional relationships with substitution, your goal is to systematically substitute known values and solve for the unknown variable. Starting with the given information: $\frac{p}{q} = \frac{7}{9}$ and $q = 18k$ . To find $p$ in terms of $k$ , substitute the expression for $q$ into the proportion. Substituting $q = 18k$ into $\frac{p}{q} = \frac{7}{9}$ : $\frac{p}{18k} = \frac{7}{9}$ Cross-multiply to solve for $p$ : $9p = 7 \times 18k$ $9p = 126k$ $p = \frac{126k}{9} = 14k$ Therefore, $p = 14k$ , which is choice C. Let's examine why the other answers are incorrect. Choice A ( $4k$ ) might result from calculation errors or confusion about the relationship between the fractions. Choice B ( $6k$ ) could come from incorrectly simplifying $\frac{18}{9} = 2$ and then somehow getting $6k$ , missing the multiplication by 7. Choice D ( $18k$ ) represents the value of $q$ , not $p$ - this is a common trap where students confuse which variable they're solving for. Strategy tip: In proportion problems with substitution, always substitute first, then solve systematically. Cross-multiplication is your most reliable tool for solving proportions. Double-check your answer by substituting back into the original equation to verify the proportion holds true.
A recipe uses the ratio of flour to sugar as $7:3$ by weight. A baker has $1.8$ kilograms of sugar and wants to keep the ratio the same. How many kilograms of flour are needed?
Answer: 4.2 kg 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.
The variable $y$ varies directly with $x$ , and $y = 3$ when $x = 4$ . What is $y$ when $x = 10$ ?
Answer: 7.5 When you see that one variable "varies directly" with another, you're dealing with direct variation, which means the variables have a constant ratio. This creates the relationship $y = kx$ , where $k$ is the constant of variation. First, you need to find the constant $k$ using the given information. Since $y = 3$ when $x = 4$ , substitute these values: $3 = k(4)$ , so $k = \frac{3}{4} = 0.75$ . Now you can find $y$ when $x = 10$ : $y = 0.75(10) = 7.5$ . This confirms that choice C is correct. Let's examine why the other answers are wrong. Choice A (5.0) appears to come from incorrectly thinking this is inverse variation, where you might calculate $\frac{3 \times 4}{10} = 1.2$ , then making arithmetic errors. Choice B (6.7) likely results from incorrectly setting up a proportion or making calculation mistakes with the constant. Choice D (8.3) might come from confusion about the relationship or rounding errors in an incorrect approach. The key strategy for direct variation problems is to remember the two-step process: first find the constant of variation using the given pair of values, then use that constant to find the unknown value. Also, remember that in direct variation, as one variable increases, the other increases proportionally—if $x$ increases from 4 to 10 (2.5 times larger), then $y$ should increase from 3 to 7.5 (also 2.5 times larger). This proportional thinking can help you check your answer quickly.
A model car is built at a scale of $1:18$ (model:actual). The actual car is $4.32$ meters long. What is the length of the model car, in centimeters?
Answer: 24 cm 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.
In a survey, 35% of students chose option A. If 280 students took the survey, how many students chose option A?
Answer: 98 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.
At a fundraiser, adult tickets and student tickets were sold in the ratio $7:3$ . An adult ticket costs $12 and a student ticket costs $8. If 200 tickets were sold in total, what was the total revenue from ticket sales?
Answer: $\$2{,}160$ 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: $\$1{,}680 + \$480 = \$2{,}160$ The answer is C. Looking at the wrong answers: A ( $\$1{,}920$ ) likely comes from incorrectly calculating $160 × \$12$ , suggesting you found 160 adult tickets instead of 140. B ( $\$2{,}040$ ) might result from switching the ticket prices or making an arithmetic error in the final addition. D ( $\$2{,}240$ ) could come from calculating $140 × \$16$ , possibly adding the two ticket prices together instead of using them separately. Study tip: When working with ratios, always verify your individual quantities add up to the given total before moving to the next step. This catches proportion errors early and prevents them from compounding in your final calculation.
A class has a ratio of freshmen to sophomores to juniors of $4:5:3$ . There are $72$ students in the class total. How many sophomores are in the class?
Answer: 30 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.
A model car is built at a scale of $1:24$ (model:actual). The model's length is 18.5 cm. What is the actual length of the car, in centimeters?
Answer: 444 cm 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.
On a certain map, $1.5\text{ cm}$ represents $4\text{ km}$ of actual distance. If two towns are $6.75\text{ cm}$ apart on the map, what is their approximate actual distance?
Answer: 18 km Map scale problems test your ability to set up and solve proportions. When you see a question giving you a map scale and asking for actual distances, you're working with equivalent ratios. The key is setting up a proportion where the relationship between map distance and actual distance stays constant. Given that 1.5 cm represents 4 km, you can write this as a ratio: $\frac{1.5 \text{ cm}}{4 \text{ km}}$ . Since the towns are 6.75 cm apart on the map, you need to find the actual distance $x$ that maintains this same ratio: $\frac{1.5 \text{ cm}}{4 \text{ km}} = \frac{6.75 \text{ cm}}{x \text{ km}}$ Cross-multiplying gives you: $1.5x = 4 \times 6.75 = 27$ , so $x = \frac{27}{1.5} = 18$ km. Looking at the wrong answers: (A) 12 km results from incorrectly multiplying 6.75 by 4, then dividing by 2.25 instead of 1.5. (C) 20 km might come from rounding errors or mishandling the decimal calculations. (D) 27 km is what you get if you multiply 6.75 by 4 but forget to divide by 1.5 - essentially treating the scale as if 1 cm = 4 km instead of 1.5 cm = 4 km. The correct answer is (B) 18 km. Study tip: Always set up your proportion carefully, making sure the units match up in the same positions. Double-check by asking: "Does my scale factor make sense?" Here, 6.75 ÷ 1.5 = 4.5, and 4.5 × 4 km = 18 km.
A photo is enlarged so that all lengths are multiplied by the same scale factor. In the original photo, the width is $9\text{ cm}$ and the height is $13.5\text{ cm}$ . In the enlarged photo, the width is $15\text{ cm}$ . What is the height of the enlarged photo?
Answer: 22.5 cm 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.
Triangles $\triangle ABC$ and $\triangle DEF$ are similar. In $\triangle ABC$ , side $AB=9$ cm and $AC=12$ cm. In $\triangle DEF$ , the side corresponding to $AB$ is $DE=15$ cm. What is the length of the side in $\triangle DEF$ that corresponds to $AC$ ?
Answer: 20 cm 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.
A recipe that yields 8 servings calls for 3 cups of flour. Assuming the recipe scales proportionally, how many cups of flour are needed to prepare 14 servings?
Answer: 5.25 cups This is a proportional reasoning problem where you need to scale a recipe up from one serving size to another. When you see "scales proportionally," you're looking at a direct proportion where the ratio of flour to servings stays constant. Set up the proportion using the given information: $\frac{3 \text{ cups}}{8 \text{ servings}} = \frac{x \text{ cups}}{14 \text{ servings}}$ Cross multiply to solve: $3 \times 14 = 8 \times x$ , which gives you $42 = 8x$ . Dividing both sides by 8: $x = \frac{42}{8} = 5.25$ cups. You can verify this makes sense: if 8 servings need 3 cups, then 16 servings (double) would need 6 cups. Since 14 servings is slightly less than 16, you'd expect slightly less than 6 cups of flour. Looking at the wrong answers: Choice A (4.0 cups) likely comes from incorrectly adding a fixed amount rather than scaling proportionally. Choice B (4.5 cups) might result from setting up the proportion incorrectly or making an arithmetic error. Choice D (6.0 cups) is what you'd need for 16 servings, suggesting someone rounded 14 up to 16 instead of calculating precisely. When tackling proportion problems, always set up your ratios consistently (same units in numerator and denominator positions) and double-check that your answer makes logical sense compared to the original quantities. The PSAT often includes answer choices that result from common computational mistakes, so verify your arithmetic carefully.
A paint mixture must maintain a red-to-white ratio of $7:4$ . If a painter has exactly 5 gallons of red paint, how many gallons of white paint are needed to keep the required ratio?
Answer: 2.9 gallons 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.
Suppose $A:B = 2:5$ and $B:C = 3:4$ . What is $A:C$ ?
Answer: 3:10 When you encounter chained ratios like this, you need to find a common term to connect them. Here, B appears in both ratios, so you'll use it as your bridge. Start with the given ratios: $A:B = 2:5$ and $B:C = 3:4$ . To combine these, you need the B values to match. In the first ratio, B corresponds to 5, while in the second ratio, B corresponds to 3. Find the least common multiple of 3 and 5, which is 15. Scale each ratio so that B equals 15. For $A:B = 2:5$ , multiply both terms by 3 to get $A:B = 6:15$ . For $B:C = 3:4$ , multiply both terms by 5 to get $B:C = 15:20$ . Now that B = 15 in both ratios, you can directly read that $A:C = 6:20$ , which simplifies to $A:C = 3:10$ . Choice A (2:15) represents a common error where students incorrectly use the original A value with the scaled C value without proper ratio connection. Choice C (4:9) might result from incorrectly adding or manipulating the ratio terms arithmetically rather than maintaining proportional relationships. Choice D (5:12) could come from using the B values incorrectly or mixing up which terms to scale. Remember this key strategy: when working with chained ratios, always identify the common term, scale both ratios so that common term has the same value, then read off your answer. This systematic approach prevents the algebraic confusion that creates most wrong answers on ratio problems.
Machine A produces 300 widgets in 12 minutes, and Machine B produces 500 widgets in 25 minutes. At these constant rates, how many more widgets per minute does Machine A produce than Machine B?
Answer: 5 When you encounter rate problems like this, you need to find each machine's rate per unit of time, then compare them directly. First, calculate each machine's rate in widgets per minute. Machine A produces 300 widgets in 12 minutes, so its rate is $\frac{300}{12} = 25$ widgets per minute. Machine B produces 500 widgets in 25 minutes, so its rate is $\frac{500}{25} = 20$ widgets per minute. To find how many more widgets per minute Machine A produces than Machine B, subtract: $25 - 20 = 5$ widgets per minute. This confirms answer choice C is correct. Let's examine why the other answers are wrong. Choice A (3) might result from incorrectly calculating one of the rates—perhaps finding Machine B's rate as 22 instead of 20. Choice B (4) could come from a similar calculation error or from rounding mistakes in the division. Choice D (6) might occur if you mistakenly calculated Machine A's rate as 26 widgets per minute instead of 25, possibly from computational errors in the division. The key strategy for rate problems is to always convert to the same time unit first, then perform your comparison. Watch out for the common trap of comparing totals instead of rates—the fact that Machine B produces more widgets overall (500 vs 300) doesn't mean it has a higher rate. Always divide total output by total time to find the true rate per unit of time.

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