ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Proportion And Scale Problems
20 questions · exam conditions
0:00
Proportion And Scale ProblemsQuestion 1 of 20

In a school, the ratio of students to teachers is 22 to 1. The ratio of teachers to administrators is 7 to 2. What is the ratio of students to administrators?

11 to 1
29 to 3
77 to 1
154 to 2
← Back to quizzes

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Proportion And Scale Problems

Practice Proportion And Scale Problems in ACCUPLACER Quantitative Reasoning, Algebra & Statistics 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 Proportion And Scale Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

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

In a school, the ratio of students to teachers is 22 to 1. The ratio of teachers to administrators is 7 to 2. What is the ratio of students to administrators?

  1. 11 to 1
  2. 29 to 3
  3. 77 to 1 (correct answer)
  4. 154 to 2
Explanation: Let S be students, T be teachers, and A be administrators. We are given ST=221\frac{S}{T} = \frac{22}{1} and TA=72\frac{T}{A} = \frac{7}{2}. To find SA\frac{S}{A}, we can multiply the two ratios: ST×TA=221×72\frac{S}{T} \times \frac{T}{A} = \frac{22}{1} \times \frac{7}{2}. The T's cancel out, leaving SA=1542\frac{S}{A} = \frac{154}{2}. This ratio simplifies to 771\frac{77}{1}, or 77 to 1.

Question 2

On a particular day, the exchange rate is 1 U.S. dollar equals 0.92 euros. A currency exchange service charges a 2% commission on the euro amount after conversion. If a person converts 500 U.S. dollars to euros, how many euros do they receive after the commission is deducted?

  1. 450.80 (correct answer)
  2. 451.72
  3. 460.00
  4. 469.20
Explanation: First, convert the U.S. dollars to euros: 500 USD×0.92EURUSD=460 EUR500 \text{ USD} \times 0.92 \frac{\text{EUR}}{\text{USD}} = 460 \text{ EUR}. Next, calculate the 2% commission on this amount: 0.02×460 EUR=9.20 EUR0.02 \times 460 \text{ EUR} = 9.20 \text{ EUR}. Finally, subtract the commission from the converted amount: 460−9.20=450.80 EUR460 - 9.20 = 450.80 \text{ EUR}.

Question 3

A cleaning solution is formed by mixing concentrate and water in a ratio of 3 parts concentrate to 8 parts water. If a total of 550 milliliters of the solution is prepared, how many milliliters of concentrate are used?

  1. 150 (correct answer)
  2. 206
  3. 350
  4. 400
Explanation: The total number of parts in the solution is 3+8=113 + 8 = 11 parts. To find the volume of one part, divide the total volume by the total number of parts: 550 mL11 parts=50 mL per part\frac{550 \text{ mL}}{11 \text{ parts}} = 50 \text{ mL per part}. The amount of concentrate is 3 parts, so multiply the volume of one part by 3: 3 parts×50mLpart=150 mL3 \text{ parts} \times 50 \frac{\text{mL}}{\text{part}} = 150 \text{ mL}.

Question 4

The actual distance between two cities is 210 kilometers. On a certain map, this distance is represented by a line 14 centimeters long. On the same map, what would be the length of a line representing a distance of 144 kilometers?

  1. 9.6 cm (correct answer)
  2. 10.2 cm
  3. 12.0 cm
  4. 15.0 cm
Explanation: First, find the scale of the map in kilometers per centimeter: 210 km14 cm=15 km/cm\frac{210 \text{ km}}{14 \text{ cm}} = 15 \text{ km/cm}. This means each centimeter on the map represents 15 kilometers. To find the map distance for 144 kilometers, divide the actual distance by the scale: 144 km15 km/cm=9.6 cm\frac{144 \text{ km}}{15 \text{ km/cm}} = 9.6 \text{ cm}.

Question 5

To estimate the number of fish in a lake, a biologist tags and releases 60 fish. A week later, a sample of 150 fish is caught, and 8 of them have tags. Based on this sample, what is the best estimate for the total number of fish in the lake?

  1. 900
  2. 1,125 (correct answer)
  3. 1,200
  4. 1,500
Explanation: This is a capture-recapture problem that can be solved with a proportion. The ratio of tagged fish to the total population should be approximately equal to the ratio of tagged fish in the sample to the sample size. Let NN be the total number of fish. tagged in populationtotal population=tagged in samplesample size\frac{\text{tagged in population}}{\text{total population}} = \frac{\text{tagged in sample}}{\text{sample size}}. So, 60N=8150\frac{60}{N} = \frac{8}{150}. Cross-multiply: 8N=60×1508N = 60 \times 150, which is 8N=90008N = 9000. Divide by 8: N=1125N = 1125.

Question 6

The time it takes to complete a construction project is inversely proportional to the number of workers. If it takes 12 workers 20 days to finish the project, how many fewer days would it take if 15 workers were assigned to the project?

  1. 4 (correct answer)
  2. 5
  3. 16
  4. 25
Explanation: For an inverse proportion, the product of the two quantities is constant. The total 'worker-days' needed is 12 workers×20 days=240 worker-days12 \text{ workers} \times 20 \text{ days} = 240 \text{ worker-days}. With 15 workers, the time needed is 240 worker-days15 workers=16 days\frac{240 \text{ worker-days}}{15 \text{ workers}} = 16 \text{ days}. The question asks for how many fewer days it would take. The difference is 20 days−16 days=4 fewer days20 \text{ days} - 16 \text{ days} = 4 \text{ fewer days}.

Question 7

A scale drawing of a triangular park has sides of length 6 cm, 8 cm, and 10 cm. The scale of the drawing is 1:1,500. What is the actual perimeter of the park in meters?

  1. 240
  2. 360 (correct answer)
  3. 2,400
  4. 3,600
Explanation: First, find the perimeter of the park on the drawing: 6+8+10=24 cm6 + 8 + 10 = 24 \text{ cm}. Next, use the scale to find the actual perimeter in centimeters. The scale 1:1,500 means the actual perimeter is 1,500 times the drawing's perimeter: 24 cm×1,500=36,000 cm24 \text{ cm} \times 1,500 = 36,000 \text{ cm}. Finally, convert the perimeter from centimeters to meters. Since there are 100 cm in 1 meter, divide by 100: 36,000 cm100 cm/m=360 meters\frac{36,000 \text{ cm}}{100 \text{ cm/m}} = 360 \text{ meters}.

Question 8

Two gears are connected such that when one rotates, the other rotates in the opposite direction. The first gear has 60 teeth and the second has 24 teeth. For every 5 complete rotations of the second, smaller gear, how many complete rotations does the first, larger gear make?

  1. 2 (correct answer)
  2. 3
  3. 8
  4. 12.5
Explanation: The number of rotations is inversely proportional to the number of teeth. Let R1R_1 and T1T_1 be the rotations and teeth of the first gear, and R2R_2 and T2T_2 be for the second. The relationship is R1T1=R2T2R_1 T_1 = R_2 T_2. We have T1=60T_1 = 60, T2=24T_2 = 24, and R2=5R_2 = 5. Plugging these in gives R1×60=5×24R_1 \times 60 = 5 \times 24, which simplifies to 60R1=12060 R_1 = 120. Divide by 60: R1=2R_1 = 2. The larger gear makes 2 rotations.

Question 9

A 10-liter car radiator is filled with a solution that is 20% antifreeze. The owner needs to increase the concentration to 50% antifreeze. How many liters of the solution must be drained and replaced with pure antifreeze to achieve the desired concentration?

  1. 3.00
  2. 3.75 (correct answer)
  3. 4.25
  4. 5.00
Explanation: Let xx be the number of liters to drain and replace. Initially, there are 10×0.20=210 \times 0.20 = 2 liters of antifreeze. When xx liters of solution are drained, the amount of antifreeze removed is 0.20x0.20x. Then, xx liters of pure (100%) antifreeze are added. The new amount of antifreeze is 2−0.20x+x2 - 0.20x + x. The total volume is still 10 liters. The new concentration should be 50%, so 2−0.20x+x10=0.50\frac{2 - 0.20x + x}{10} = 0.50. Simplify the numerator: 2+0.80x=10×0.502 + 0.80x = 10 \times 0.50, which is 2+0.80x=52 + 0.80x = 5. Subtract 2 from both sides: 0.80x=30.80x = 3. Solve for xx: x=30.80=3.75x = \frac{3}{0.80} = 3.75 liters.

Question 10

A digital artist scales an image that is 800 pixels wide down by a factor of 34\frac{3}{4}. She then takes the resulting image and scales it up by a factor of 52\frac{5}{2}. What is the final width of the image in pixels?

  1. 800
  2. 1,000
  3. 1,250
  4. 1,500 (correct answer)
Explanation: This problem involves applying two successive scale factors. First, scale the image down: 800 pixels×34=600 pixels800 \text{ pixels} \times \frac{3}{4} = 600 \text{ pixels}. Next, take the new width and scale it up: 600 pixels×52=30002=1500 pixels600 \text{ pixels} \times \frac{5}{2} = \frac{3000}{2} = 1500 \text{ pixels}. The final width is 1,500 pixels.

Question 11

A standard medical dosage is 15 milligrams of medicine for every 10 kilograms of a patient's body weight. Following this standard, how many milligrams of medicine should be administered to a patient who weighs 84 kilograms?

  1. 56
  2. 110
  3. 126 (correct answer)
  4. 150
Explanation: Set up a proportion to find the correct dosage. Let xx be the required amount of medicine. 15 mg10 kg=x mg84 kg\frac{15 \text{ mg}}{10 \text{ kg}} = \frac{x \text{ mg}}{84 \text{ kg}}. To solve for xx, cross-multiply: 10x=15×8410x = 15 \times 84, which gives 10x=126010x = 1260. Divide both sides by 10: x=126x = 126. The patient should receive 126 milligrams.

Question 12

A machine produces widgets at a constant rate. In the first 3 hours, it produces 450 widgets. Due to a maintenance issue, its rate decreases to 75% of the original rate. How many additional hours will it take to produce a total of 1,200 widgets?

  1. 5.0 hours
  2. 6.7 hours (correct answer)
  3. 7.5 hours
  4. 8.0 hours
Explanation: Original rate: 450 ÷ 3 = 150 widgets/hour. After maintenance, rate = 0.75 × 150 = 112.5 widgets/hour. Widgets still needed: 1,200 - 450 = 750 widgets. Time needed: 750 ÷ 112.5 = 6.67 ≈ 6.7 hours. Choice A uses the original rate for remaining widgets. Choice C assumes the reduced rate applies to total production time. Choice D miscalculates the rate reduction as 50% instead of 25%.

Question 13

On a map with a scale of 1 inch = 15 miles, two cities are 4.8 inches apart. A third city forms a triangle with the other two cities such that the actual distances form a 3:4:5 ratio triangle. If the longest side of the triangle is the distance between the first two cities, what is the map distance in inches for the shortest side?

  1. 2.88 inches (correct answer)
  2. 3.24 inches
  3. 3.60 inches
  4. 4.32 inches
Explanation: The actual distance between the first two cities is 4.8 × 15 = 72 miles. In a 3:4:5 triangle, if 72 miles is the longest side (ratio 5), then each ratio unit = 72/5 = 14.4 miles. The shortest side (ratio 3) = 3 × 14.4 = 43.2 miles. Converting to map distance: 43.2 ÷ 15 = 2.88 inches. Choice B uses ratio 4. Choice C assumes the given distance is the middle side. Choice D incorrectly applies the scale.

Question 14

A recipe calls for ingredients in the ratio of 4 parts flour to 3 parts sugar to 2 parts butter by weight. If you want to make the recipe but only have 300 grams of butter available, and you need to reduce the sugar by 25% from the original proportion while keeping the flour-to-butter ratio constant, how much flour should you use?

  1. 450 grams
  2. 525 grams
  3. 675 grams
  4. 600 grams (correct answer)
Explanation: When you encounter ratio problems with modifications, break them down step by step rather than trying to solve everything at once. Start with the given information and work systematically through each constraint. The original recipe has a 4:3:2 ratio (flour:sugar:butter). With 300 grams of butter available, you can find the "unit size" by dividing: 300 ÷ 2 = 150 grams per ratio unit. This would normally give you 4 × 150 = 600 grams of flour and 3 × 150 = 450 grams of sugar. However, you're reducing sugar by 25% while keeping the flour-to-butter ratio constant. The flour-to-butter ratio is 4:2 or 2:1, so with 300 grams of butter, you need 600 grams of flour regardless of what happens to the sugar. Looking at the wrong answers: Choice A (450 grams) represents the original amount of sugar, not flour—this suggests confusing the ingredients. Choice B (525 grams) might come from incorrectly applying the 25% reduction to flour instead of sugar, then making calculation errors. Choice C (675 grams) could result from mistakenly increasing the flour amount when reducing sugar, perhaps thinking you need to compensate. The key insight is that reducing sugar by 25% doesn't affect the flour-to-butter ratio, which remains constant at 2:1. Therefore, 600 grams of flour is correct. Strategy tip: In modified ratio problems, identify which relationships stay constant and which change. Handle each constraint separately, then combine your results. Don't let modifications to one ingredient confuse your calculations for others.

Question 15

A water tank can be filled by pipe A alone in 12 hours or by pipe B alone in 18 hours. Both pipes are opened simultaneously to fill the empty tank, but after 4 hours, pipe A is closed and only pipe B continues. How many additional hours will pipe B need to finish filling the tank?

  1. 6.0 hours
  2. 7.2 hours
  3. 9.6 hours
  4. 8.0 hours (correct answer)
Explanation: When you encounter work rate problems, think in terms of "rate of work" - how much of the job each worker (or pipe) completes per unit time. This approach makes complex scenarios much more manageable. Pipe A fills the tank in 12 hours, so its rate is 112\frac{1}{12} tank per hour. Pipe B takes 18 hours, so its rate is 118\frac{1}{18} tank per hour. When both pipes work together, their combined rate is 112+118=336+236=536\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} tank per hour. After 4 hours of working together, they fill 4×536=2036=594 \times \frac{5}{36} = \frac{20}{36} = \frac{5}{9} of the tank. This leaves 1−59=491 - \frac{5}{9} = \frac{4}{9} of the tank remaining. Now only pipe B continues at its rate of 118\frac{1}{18} tank per hour. To fill the remaining 49\frac{4}{9} of the tank: 4/91/18=49×18=8\frac{4/9}{1/18} = \frac{4}{9} \times 18 = 8 hours. Choice A (6.0 hours) likely comes from incorrect calculation of the remaining work. Choice B (7.2 hours) might result from using the combined rate instead of just pipe B's rate for the final phase. Choice C (9.6 hours) could come from miscalculating the initial 4-hour period's work. The key strategy for work rate problems is to always convert to rates first, then multiply by time to find work completed. Remember that when workers combine, you add their individual rates.

Question 16

A concrete mixture requires cement, sand, and gravel in the ratio 2:3:5. A contractor has 18 bags of cement, 24 bags of sand, and 50 bags of gravel. What is the maximum number of complete batches of concrete that can be made, and how many bags of sand will remain unused?

  1. 8 batches, 6 bags of sand unused
  2. 9 batches, 3 bags of sand unused
  3. 8 batches, 0 bags of sand unused (correct answer)
  4. 10 batches, 6 bags of sand unused
Explanation: For ratio 2:3:5, each batch needs 2 bags cement, 3 bags sand, 5 bags gravel. Available batches limited by: cement: 18÷2=9 batches, sand: 24÷3=8 batches, gravel: 50÷5=10 batches. Sand is the limiting factor at 8 batches. For 8 batches: cement used = 8×2=16 (2 unused), sand used = 8×3=24 (0 unused), gravel used = 8×5=40 (10 unused). Choice A miscounts batches. Choice B assumes 9 batches possible. Choice D ignores the limiting factor constraint.

Question 17

A recipe for 8 servings calls for 3 cups of flour and 2.25 cups of sugar. Maria wants to make enough food for 14 people, but she only has 5 cups of flour available. How many cups of sugar should she use to maintain the proper proportions with the flour she has?

  1. 3.75 cups (correct answer)
  2. 3.94 cups
  3. 4.20 cups
  4. 4.50 cups
Explanation: First, find the flour-to-sugar ratio: 3 cups flour to 2.25 cups sugar, which simplifies to 4:3. With 5 cups of flour available, use the proportion 3/2.25 = 5/x to find x = (5 × 2.25)/3 = 11.25/3 = 3.75 cups of sugar. Choice B incorrectly scales for 14 servings then tries to adjust. Choice C scales the sugar for 14 servings without considering flour limitation. Choice D assumes a 1:1 ratio minus the flour shortage.

Question 18

A scale model of a car is constructed using a 1:24 scale. If the length of the actual car is 15 feet, what is the length of the model in inches?

  1. 6.25
  2. 7.50 (correct answer)
  3. 8.00
  4. 9.25
Explanation: First, convert the actual car's length to inches, since the answer is required in inches. There are 12 inches in a foot, so 15 feet×12inchesfoot=180 inches15 \text{ feet} \times 12 \frac{\text{inches}}{\text{foot}} = 180 \text{ inches}. Now, use the scale to find the model's length. The scale 1:24 means the model's length is 124\frac{1}{24} of the actual length. 124×180 inches=7.5 inches\frac{1}{24} \times 180 \text{ inches} = 7.5 \text{ inches}.

Question 19

A blueprint for a rectangular room uses a scale of 1 inch to 4 feet. If the dimensions of the room on the blueprint are 3.5 inches by 5 inches, what is the actual area of the room in square feet?

  1. 70
  2. 84
  3. 140
  4. 280 (correct answer)
Explanation: First, convert the blueprint dimensions to actual dimensions using the scale. The actual width is 3.5 inches×4feetinch=14 feet3.5 \text{ inches} \times 4 \frac{\text{feet}}{\text{inch}} = 14 \text{ feet}. The actual length is 5 inches×4feetinch=20 feet5 \text{ inches} \times 4 \frac{\text{feet}}{\text{inch}} = 20 \text{ feet}. The actual area is the product of the actual dimensions: 14 feet×20 feet=280 square feet14 \text{ feet} \times 20 \text{ feet} = 280 \text{ square feet}.

Question 20

If 5 identical machines can assemble 750 devices in 6 hours, how many hours would it take 3 of these machines to assemble 900 devices?

  1. 8
  2. 10
  3. 12 (correct answer)
  4. 15
Explanation: First, find the rate of one machine. The rate of 5 machines is 750 devices6 hours=125 devices per hour\frac{750 \text{ devices}}{6 \text{ hours}} = 125 \text{ devices per hour}. The rate of one machine is 1255=25 devices per hour\frac{125}{5} = 25 \text{ devices per hour}. Now, find the rate of 3 machines: 3×25=75 devices per hour3 \times 25 = 75 \text{ devices per hour}. Finally, find the time needed to assemble 900 devices at this rate: 900 devices75 devices/hour=12 hours\frac{900 \text{ devices}}{75 \text{ devices/hour}} = 12 \text{ hours}.