HSPT Quiz: Solve Ratio Problems

What this quiz covers

This quiz focuses on Solve Ratio Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT.

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. 72 (correct answer)
2. 80
3. 82
4. 90

Explanation: This problem requires finding a linear relationship, then applying a change. Let F be the fixed fee and C be the per-guest charge. We have a system of two equations: 1) $F + 40C = 2100$ and 2) $F + 60C = 2900$. Subtracting equation (1) from (2) gives: $(F + 60C) - (F + 40C) = 2900 - 2100$, which simplifies to $20C = 800$, so (C = 40\). Substitute C back into equation (1): F + 40(40) = 2100$, so$F + 1600 = 2100, which gives \(F = 500). Now, the per-guest charge is increased by 25%. The new charge, C', is (C' = 40 \times (1 + 0.25) = 40 \times 1.25 = 50\). The fixed fee F remains $500. The budget is $4100. Let G be the number of guests. The new cost equation is 500 + 50G = 4100$. Solving for G:$50G = 4100 - 500$, so$50G = 3600$.$G = \frac{3600}{50} = 72$. The maximum number of guests is 72.

Question 2

1. 42:31
2. 7:5
3. 6:7 (correct answer)
4. 35:31

Explanation: Let the son's share of the inheritance be 7k and the daughter's share be 5k for some constant k. The formula for the total amount after simple interest is A = P(1 + rt), where P is the principal, r is the annual rate, and t is the time in years. For the son: P = 7k, r = 10% = 0.10, t = 2 years. The son's total amount is $A_{son} = 7k(1 + 0.10 \times 2) = 7k(1 + 0.20) = 7k(1.2) = 8.4k$. For the daughter: P = 5k, r = 12% = 0.12, t = 2 years. The daughter's total amount is $A_{daughter} = 5k(1 + 0.12 \times 2) = 5k(1 + 0.24) = 5k(1.24) = 6.2k$. The ratio of the son's total amount to the daughter's total amount is $8.4k : 6.2k$. The k cancels out, leaving the ratio 8.4 : 6.2. To express this as a ratio of integers, multiply both parts by 10 to get 84 : 62. This ratio can be simplified by dividing both parts by their greatest common divisor, 2. The final simplified ratio is 42:31.

Question 3

1. 2 hours
2. 2.5 hours
3. 3 hours (correct answer)
4. 3.2 hours

Explanation: Let the initial height of both candles be H. The rate of burning for the first candle (C1) is H/6 per hour. The rate of burning for the second candle (C2) is H/4 per hour. Let 't' be the time in hours after they are lit. The height of C1 at time 't' is $H_1(t) = H - \frac{H}{6}t = H(1 - \frac{t}{6})$. The height of C2 at time 't' is $H_2(t) = H - \frac{H}{4}t = H(1 - \frac{t}{4})$. We want to find the time 't' when the ratio of their heights is 2:1, i.e., $\frac{H_1(t)}{H_2(t)} = \frac{2}{1}$. Substituting the expressions for the heights: $\frac{H(1 - t/6)}{H(1 - t/4)} = 2$. The H terms cancel out: $1 - \frac{t}{6} = 2(1 - \frac{t}{4})$. Distribute the 2 on the right side: $1 - \frac{t}{6} = 2 - \frac{2t}{4} = 2 - \frac{t}{2}$. Now, solve for t. It's helpful to clear the denominators by multiplying the entire equation by the least common multiple of 6 and 2, which is 6. $6(1) - 6(\frac{t}{6}) = 6(2) - 6(\frac{t}{2})$. This gives $6 - t = 12 - 3t$. Rearranging the terms to solve for t: $3t - t = 12 - 6$, which simplifies to $2t = 6$. Therefore, $t = 3$ hours.

Question 4

1. $18$ cm
2. $12$ cm (correct answer)
3. $15$ cm
4. $20$ cm

Explanation: When you encounter gear problems involving inverse proportionality, remember that as one quantity increases, the other decreases at a predictable rate. Here, rotation speed and radius have an inverse relationship, meaning $\text{speed} \times \text{radius} = \text{constant}$ for all connected gears. First, find the constant using gear A's information: $150 \text{ rpm} \times 6 \text{ cm} = 900$. You can verify this works for gear B: if its radius is 9 cm, then its speed should be $900 ÷ 9 = 100$ rpm, which makes sense since larger gears rotate slower. For gear C rotating at 75 rpm, use the same constant: $75 \times r_C = 900$, so $r_C = 900 ÷ 75 = 12$ cm. Looking at the wrong answers: Choice A (18 cm) would give a speed of $900 ÷ 18 = 50$ rpm, not 75 rpm. Choice C (15 cm) would result in $900 ÷ 15 = 60$ rpm. Choice D (20 cm) would produce $900 ÷ 20 = 45$ rpm. These represent common calculation errors or misunderstanding the inverse relationship. The answer is B) 12 cm. Study tip: For inverse proportion problems, always establish the constant first by multiplying the two given quantities. Then use this constant to find missing values. Watch for the key phrase "inversely proportional" – it signals that one value increases as the other decreases, and their product remains constant.

Question 5

1. $48$ rotations in the same direction as A (correct answer)
2. $48$ rotations in the opposite direction to A
3. $64$ rotations in the same direction as A
4. $64$ rotations in the opposite direction to A

Explanation: When you encounter gear system problems, remember two key principles: gears with more teeth rotate slower, and adjacent gears always rotate in opposite directions. First, let's find how many rotations gear C makes. In any gear system, the product of teeth and rotations remains constant. So if gear A (12 teeth) makes 80 rotations, we can write: $12 \times 80 = 20 \times R_C$, where $R_C$ is gear C's rotations. Solving: $R_C = \frac{12 \times 80}{20} = \frac{960}{20} = 48$ rotations. Next, we need to determine the direction. Since adjacent gears rotate in opposite directions, we must trace through the connections. Gear A connects to gear B (opposite direction), then gear B connects to gear C (opposite direction again). Two direction changes means gear C rotates in the same direction as gear A. Looking at the wrong answers: Choice B gives the correct number of rotations (48) but incorrectly states the direction as opposite to A. This ignores that there are two gear connections, not one. Choices C and D both incorrectly calculate 64 rotations, likely from using gear B's tooth count (16) in the calculation instead of gear C's count (20). Choice D compounds this error with the wrong direction as well. For gear problems, always set up the relationship that teeth times rotations equals a constant, and carefully count the number of gear connections to determine final direction. Each connection reverses the direction once.

Question 6

1. $72$ gallons of red and yellow paint combined (correct answer)
2. $60$ gallons of red and yellow paint combined
3. $54$ gallons of red and yellow paint combined
4. $48$ gallons of red and yellow paint combined

Explanation: When you encounter ratio problems, your key strategy is to find the scaling factor that connects the given information to the ratio. Here, the paint mixture uses red, blue, and yellow in the ratio $4:3:2$, and you have $36$ gallons of blue paint to use completely. Since blue paint corresponds to the "$3$" in the ratio $4:3:2$, you can set up the equation: $3x = 36$, where $x$ is your scaling factor. Solving gives you $x = 12$. Now you can find the actual amounts needed:

• Red paint: $4x = 4(12) = 48$ gallons
• Yellow paint: $2x = 2(12) = 24$ gallons
• Combined red and yellow: $48 + 24 = 72$ gallons

Question 7

1. Team B has better percentage: $71.4\%$ vs Team A's $70\%$, difference of $1.4\%$ (correct answer)
2. Team A has better percentage: $70\%$ vs Team B's $71.4\%$, difference of $1.4\%$
3. Team B has better percentage: $71.4\%$ vs Team A's $70\%$, difference of $2.4\%$
4. Team A has better percentage: $70\%$ vs Team B's $68.6\%$, difference of $1.4\%$

Explanation: When you encounter ratio problems involving percentages, you need to convert the ratios into actual numbers of wins and losses, then calculate the winning percentage for each team. For Team A with a $7:3$ win-to-loss ratio over $40$ games: The ratio parts total $7+3=10$. So Team A wins $\frac{7}{10} \times 40 = 28$ games. Their winning percentage is $\frac{28}{40} = 0.70 = 70\%$. For Team B with a $5:2$ win-to-loss ratio over $35$ games: The ratio parts total $5+2=7$. So Team B wins $\frac{5}{7} \times 35 = 25$ games. Their winning percentage is $\frac{25}{35} = \frac{5}{7} ≈ 0.714 = 71.4\%$. Team B has the better winning percentage at $71.4\%$ versus Team A's $70\%$, with a difference of $71.4\% - 70\% = 1.4\%$. Choice A correctly identifies this relationship. Choice B makes a statement error—it claims Team A has the better percentage but then shows Team B's percentage ($71.4\%$) is actually higher than Team A's ($70\%$). Choice C has the right percentages and comparison but calculates the wrong difference ($2.4\%$ instead of $1.4\%$). Choice D incorrectly calculates Team B's percentage as $68.6\%$, likely from a ratio misinterpretation. Study tip: Always convert ratios to actual numbers first, then calculate percentages. Double-check that your final comparison statement matches your calculated values—many wrong answers contain contradictory statements.

Question 8

1. Medium gear: $120$ revolutions, large gear: $80$ revolutions; total: $380$ revolutions (correct answer)
2. Medium gear: $270$ revolutions, large gear: $405$ revolutions; total: $855$ revolutions
3. Medium gear: $120$ revolutions, large gear: $80$ revolutions; total: $280$ revolutions
4. Medium gear: $270$ revolutions, large gear: $405$ revolutions; total: $675$ revolutions

Explanation: Gear problems test your understanding of inverse relationships: when gears are connected, smaller gears turn faster than larger gears. The key insight is that all gears must move the same linear distance along their circumferences in the same time period. Since the gear teeth are in the ratio $4:6:9$, the number of teeth determines each gear's circumference. When gears mesh, they cover equal arc lengths, so: (revolutions of gear 1) × (teeth of gear 1) = (revolutions of gear 2) × (teeth of gear 2). The smallest gear (4 teeth) makes 180 revolutions. For the medium gear (6 teeth): $180 \times 4 = \text{revolutions} \times 6$, so $720 = 6 \times \text{revolutions}$, giving us $120$ revolutions. For the large gear (9 teeth): $180 \times 4 = \text{revolutions} \times 9$, so $720 = 9 \times \text{revolutions}$, giving us $80$ revolutions. Total: $180 + 120 + 80 = 380$ revolutions. Choice A correctly shows these calculations. Choice B incorrectly assumes the gears turn in the same direction as their size ratio (bigger gears turning more), which violates the physics of gear systems. Choice C uses the correct individual gear calculations but miscalculates the total as $280$ instead of $380$. Choice D combines the wrong individual calculations from choice B with an incorrect total. Remember: in gear systems, revolutions are inversely proportional to the number of teeth. Always multiply revolutions by teeth to find the constant, then divide by the other gear's teeth.

Question 9

1. $375$ mL (correct answer)
2. $750$ mL
3. $300$ mL
4. $450$ mL

Explanation: When you encounter ratio problems with measurement constraints, you need to find the smallest actual volumes that maintain the given ratio while meeting the physical limitations. The ratio $3:7:5$ means for every 3 parts of solution A, you need 7 parts of solution B and 5 parts of solution C. To find actual volumes, multiply each ratio part by the same factor $k$: $3k : 7k : 5k$. Since containers only measure in 25 mL increments, each volume ($3k$, $7k$, and $5k$) must be divisible by 25. This means $k$ itself must make each product divisible by 25. For $3k$ to be divisible by 25: since $\gcd(3,25) = 1$, we need $k$ divisible by 25. For $7k$ to be divisible by 25: since $\gcd(7,25) = 1$, we need $k$ divisible by 25. For $5k$ to be divisible by 25: since $5 \times 5 = 25$, we need $k$ divisible by 5. The smallest $k$ satisfying all conditions is $k = 25$. This gives volumes: $3(25) = 75$ mL, $7(25) = 175$ mL, and $5(25) = 125$ mL. Total volume: $75 + 175 + 125 = 375$ mL. Answer A (375 mL) is correct. Answer B (750 mL) uses $k = 50$, which works but isn't minimal. Answer C (300 mL) doesn't maintain the ratio with 25 mL increments. Answer D (450 mL) also fails to satisfy both the ratio and measurement constraints. Remember: when ratios meet measurement constraints, find the least common multiple that makes each ratio component measurable.

Question 10

1. $2.1$ liters (correct answer)
2. $1.8$ liters
3. $2.4$ liters
4. $1.5$ liters

Explanation: When you encounter ratio problems with constraints, you need to identify which component limits your total mixture. The ratio $7:4:3$ tells you that for every 7 parts red, you need 4 parts blue and 3 parts yellow, totaling 14 parts. First, determine how much mixture you could make if blue paint is your limiting factor. With 0.6 liters of blue paint representing 4 parts of the ratio, each part equals $0.6 ÷ 4 = 0.15$ liters. Therefore, the maximum total mixture is $14 × 0.15 = 2.1$ liters, which is answer A. Let's see why the other answers miss the mark. Answer B (1.8 liters) incorrectly assumes each part is $1.8 ÷ 14 = 0.129$ liters, which would only require $4 × 0.129 = 0.514$ liters of blue paint—but you have 0.6 liters available, so this underuses your constraint. Answer C (2.4 liters) makes the error of assuming you can use the full 2.8 liters mentioned in the problem, ignoring that the blue paint limitation prevents this. Answer D (1.5 liters) seems to assume each part equals $1.5 ÷ 14 = 0.107$ liters, which would need only 0.429 liters of blue paint, again underutilizing what's available. Remember this key strategy: in constrained ratio problems, always identify your limiting component first, then work backward to find the maximum total. The component with the smallest available amount relative to its ratio requirement determines your ceiling.

Question 11

1. $1650$ widgets (correct answer)
2. $1575$ widgets
3. $1725$ widgets
4. $1800$ widgets

Explanation: When you encounter problems involving proportional rates and changing conditions, break them down into clear steps: find the individual rates, then apply any efficiency changes. First, let's determine each production line's rate during the normal 8-hour shift. The lines operate in a $3:4:5$ ratio, totaling $3+4+5=12$ parts. With 2400 total widgets, each part represents $2400÷12=200$ widgets. So the hourly rates are:

• Line 1: $3×200÷8=75$ widgets/hour
• Line 2: $4×200÷8=100$ widgets/hour
• Line 3: $5×200÷8=125$ widgets/hour

• Line 1: $75×6=450$ widgets
• Line 2: $100×0.75×6=450$ widgets
• Line 3: $125×6=750$ widgets
• Total: $450+450+750=1650$ widgets

Question 12

1. New width is $15$ inches; perimeter ratio is $5:2$ (correct answer)
2. New width is $15$ inches; perimeter ratio is $25:4$
3. New width is $37.5$ inches; perimeter ratio is $5:2$
4. New width is $37.5$ inches; perimeter ratio is $25:4$

Explanation: When you see questions about enlarging or scaling geometric figures, remember that area scales with the square of the linear scale factor, while perimeter scales directly with the linear scale factor. Since the area ratio is $25:4$, you need to find the linear scale factor first. Because area scales as the square of linear dimensions, if the linear scale factor is $k$, then $k^2 = \frac{25}{4}$. Taking the square root: $k = \frac{5}{2} = 2.5$. For the new width: multiply the original width by the scale factor. The original width was 6 inches, so the new width is $6 \times 2.5 = 15$ inches. For the perimeter ratio: since perimeter is a linear measurement, it scales directly with the linear scale factor. Therefore, the perimeter ratio is $5:2$. Looking at the wrong answers: Choice B gives the correct width but incorrectly uses the area ratio ($25:4$) for the perimeter ratio. This is a common mistake—forgetting that perimeter scales linearly, not quadratically. Choice C calculates the new width as $6 \times \frac{25}{4} = 37.5$ inches, incorrectly applying the area ratio directly to a linear dimension. Choice D makes both errors: using the area ratio for both the width calculation and the perimeter ratio. Study tip: Always identify whether you're working with linear measurements (length, width, perimeter) or area measurements. Linear measurements scale with the scale factor $k$, while areas scale with $k^2$. When given an area ratio, take its square root to find the linear scale factor.

Question 13

1. $32$ cm, because the ratio of perimeters equals the ratio of areas
2. $32$ cm, because the ratio of perimeters is $3:4$ (correct answer)
3. $42.67$ cm, because the ratio of perimeters equals the ratio of areas
4. $42.67$ cm, because the ratio of perimeters is $3:4$

Explanation: When working with similar triangles, you need to understand the relationship between different types of measurements. Similar triangles have the same shape but different sizes, which creates predictable ratios between corresponding parts. The key insight is that while areas have a squared relationship to linear dimensions, perimeters have a direct linear relationship. If two similar triangles have corresponding sides in the ratio $a:b$, then their perimeters are also in the ratio $a:b$, but their areas are in the ratio $a^2:b^2$. Since the area ratio is $9:16$, you need to find the ratio of corresponding linear dimensions. Taking the square root of both parts: $\sqrt{9}:\sqrt{16} = 3:4$. This $3:4$ ratio applies to all linear measurements, including perimeters. If the smaller triangle's perimeter is $24$ cm and the perimeter ratio is $3:4$, then: $\frac{24}{x} = \frac{3}{4}$, where $x$ is the larger triangle's perimeter. Cross-multiplying: $3x = 96$, so $x = 32$ cm. Answer A gives the correct numerical result but incorrectly states that perimeter ratios equal area ratios. Answer C incorrectly uses the area ratio $9:16$ directly for perimeters, calculating $24 \times \frac{16}{9} = 42.67$ cm, and compounds this by claiming perimeters equal area ratios. Answer D uses the same incorrect calculation as C but correctly identifies the $3:4$ ratio relationship. Remember: for similar figures, linear measurements (like perimeter) relate by taking the square root of the area ratio. Always convert area ratios to linear ratios first.

Question 14

1. Order $50$ cubic yards of sand; can make $133.33$ cubic yards of concrete (correct answer)
2. Order $50$ cubic yards of sand; can make $130$ cubic yards of concrete
3. Order $40$ cubic yards of sand; can make $106.67$ cubic yards of concrete
4. Order $40$ cubic yards of sand; can make $120$ cubic yards of concrete

Explanation: When you encounter ratio problems with real-world constraints, you need to work backwards from the limiting factor rather than the total amount needed. The concrete ratio is $1:3:4$ (cement:sand:gravel), meaning for every 8 total parts, 3 parts are sand. To make exactly 120 cubic yards, you'd need $120 \times \frac{3}{8} = 45$ cubic yards of sand. However, sand only comes in 10-cubic-yard increments, so you must round up to 50 cubic yards to have enough material. With 50 cubic yards of sand, you can determine the maximum concrete possible. Since sand represents 3 parts out of 8 total parts, you can make $50 \times \frac{8}{3} = 133.33$ cubic yards of concrete. This confirms answer A is correct. Answer B incorrectly calculates the total concrete as 130 cubic yards, likely from a rounding error in the ratio calculation. Answer C suggests ordering only 40 cubic yards of sand, but this falls short of the 45 cubic yards needed for 120 cubic yards of concrete. Answer D makes the impossible claim that 40 cubic yards of sand can produce 120 cubic yards of concrete, which violates the given ratio entirely. Strategy tip: In ratio problems with quantity constraints, always identify which component has the restriction, then work backwards to find the maximum possible output. Don't assume you can make exactly what's "needed" when real-world limitations apply.

Question 15

1. 144 sq cm
2. 204 sq cm
3. 340 sq cm
4. 400 sq cm (correct answer)

Explanation: The ratio of the perimeters of two similar polygons is equal to the ratio of their corresponding sides. Let this ratio be $s_1:s_2 = 3:5$. The ratio of the areas of two similar polygons is the square of the ratio of their corresponding sides. So, the ratio of the areas is $A_1:A_2 = (s_1)^2:(s_2)^2 = 3^2:5^2 = 9:25$. Let the areas be 9x and 25x. The sum of the areas is given as 544 sq cm. So, $9x + 25x = 544$. This simplifies to $34x = 544$. Solving for x, we get $x = \frac{544}{34} = 16$. The area of the larger polygon is 25x. Therefore, the area is $25 \times 16 = 400$ square centimeters.

Question 16

1. 3.75
2. 4.5
3. 5.25 (correct answer)
4. 6.0

Explanation: When you see a ratio problem, you're working with proportional relationships where the quantities maintain a constant relationship to each other. Here, milk and water must always be in the ratio 2:7, meaning for every 2 parts milk, you need 7 parts water. To solve this, set up a proportion. If the ratio is 2:7 (milk to water), then $\frac{2}{7} = \frac{1.5}{x}$, where x is the cups of water needed. Cross multiply: $2x = 7 \times 1.5 = 10.5$. Therefore, $x = 5.25$ cups of water. You can verify this: $\frac{1.5}{5.25} = \frac{2}{7}$ when you simplify both fractions to $\frac{2}{7}$. Looking at the wrong answers: Choice A (3.75) comes from incorrectly setting up the proportion as $\frac{2}{5} = \frac{1.5}{x}$ - this treats the ratio as 2:5 instead of 2:7. Choice B (4.5) results from multiplying 1.5 by 3, which would be correct if the ratio were 1:3 instead of 2:7. Choice D (6.0) comes from simply multiplying 1.5 by 4, showing a misunderstanding of how ratios work. Remember: in ratio problems, always set up your proportion carefully with the correct corresponding parts. Double-check by verifying that your answer maintains the original ratio when you divide the quantities you found.

Question 17

1. 48 (correct answer)
2. 60
3. 72
4. 90

Explanation: When you encounter work rate problems involving multiple workers with different speeds, you need to break down the total work based on their individual rates and then distribute accordingly. Start by finding the combined rate: 120 pages in 8 minutes means they print $\frac{120}{8} = 15$ pages per minute together. The key insight is interpreting the ratio. "Printer A prints at a rate 3:2 compared with Printer B" means for every 3 pages A prints, B prints 2 pages. This creates a total ratio of $3 + 2 = 5$ parts. Since they work together at 15 pages per minute, you can find each printer's individual rate:

• Printer A gets $\frac{3}{5}$ of the work: $\frac{3}{5} \times 15 = 9$ pages per minute
• Printer B gets $\frac{2}{5}$ of the work: $\frac{2}{5} \times 15 = 6$ pages per minute

Question 18

1. 30
2. 36
3. 42 (correct answer)
4. 48

Explanation: Map scale problems test your ability to use proportional reasoning to convert between map measurements and real-world distances. When you see a scale like "1 inch represents 12 miles," you're looking at a ratio that stays constant across the entire map. To find the actual distance, set up a proportion using the given scale. Since 1 inch on the map equals 12 miles in reality, you can write this as $\frac{1 \text{ inch}}{12 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}$. Cross-multiplying gives you $1 \times x = 12 \times 3.5$, so $x = 42$ miles. Alternatively, you can think of this as a simple multiplication: if each inch represents 12 miles, then 3.5 inches represents $3.5 \times 12 = 42$ miles. Looking at the wrong answers: Choice A (30) might result from incorrectly calculating $3.5 \times 8$ or making an arithmetic error. Choice B (36) could come from multiplying $3.5 \times 10$ instead of 12, possibly misremembering the scale. Choice D (48) might occur if you mistakenly thought the scale was 1 inch = 13.7 miles, or made an error in your multiplication. Study tip: For map scale problems, always identify the scale ratio first, then multiply the map distance by the scale factor. Double-check your arithmetic since these problems often include answer choices that result from common calculation mistakes. Practice recognizing that "1 inch represents X miles" means you multiply map inches by X to get real miles.

Question 19

1. 0.45
2. 0.63 (correct answer)
3. 0.90
4. 1.05

Explanation: When you encounter ratio problems with a given total, you need to find what fraction each part represents of the whole. The ratio 7 : 3 : 5 tells you the relative weights of almonds, cashews, and peanuts respectively. First, find the total ratio parts: $7 + 3 + 5 = 15$ parts total. This means cashews represent $\frac{3}{15} = \frac{1}{5}$ of the total weight. To find the weight of cashews, multiply this fraction by the total weight: $\frac{1}{5} \times 3.15 = 0.63$ kg. This confirms answer choice B is correct. Let's examine why the other answers are wrong: A) 0.45 kg would result if you mistakenly calculated $\frac{3}{20} \times 3.15$, perhaps by incorrectly adding the ratio parts or making an arithmetic error. C) 0.90 kg represents a calculation error where you might have found $\frac{3}{10.5} \times 3.15$ or confused the setup entirely. D) 1.05 kg would result from calculating $\frac{1}{3} \times 3.15$, which treats the ratio as if cashews were one-third of the total instead of 3 out of 15 parts. Remember this key strategy for ratio problems: always add up all the ratio parts first to find your denominator, then use the specific part you want as your numerator. The fraction $\frac{\text{part you want}}{\text{sum of all parts}}$ times the total quantity gives you your answer. Double-check by verifying that all parts add up to the given total.

Question 20

1. 390
2. 450 (correct answer)
3. 480
4. 525

Explanation: When you encounter ratio problems, you're working with proportional relationships between different groups. The key insight is that ratios tell you the relative sizes of groups, and you can use any known quantity to find the others. Given the ratio 6:5:4 for freshmen:sophomores:juniors, this means that for every 5 sophomores, there are 6 freshmen and 4 juniors. Since you know there are 150 sophomores, you can find the ratio multiplier. If the ratio calls for 5 sophomores but you actually have 150, then $150 ÷ 5 = 30$. This means each part of the ratio represents 30 students. Therefore: freshmen = $6 × 30 = 180$, sophomores = $5 × 30 = 150$ (which confirms our calculation), and juniors = $4 × 30 = 120$. The total is $180 + 150 + 120 = 450$ students. Looking at the wrong answers: (A) 390 likely comes from miscalculating the ratio multiplier or making an arithmetic error in the final addition. (C) 480 might result from incorrectly treating 150 as corresponding to a different part of the ratio, perhaps thinking it represents the "6" portion instead of the "5" portion. (D) 525 could come from adding 150 to each ratio number (6+5+4=15, then 15×150÷something) or another fundamental misunderstanding of how ratios work. Remember: in ratio problems, always identify which quantity corresponds to which part of the ratio, then find the multiplier to scale up the entire ratio proportionally.