SSAT Middle Level Quiz: Unit Rate Comparisons
20 questions · exam conditions
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Unit Rate ComparisonsQuestion 1 of 20

A factory produces toy cars and toy trucks in the ratio 5:3. If the factory produces 240 toy cars in one day, how many total toys does it produce that day?

384 toys
400 toys
420 toys
456 toys
480 toys
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SSAT Middle Level Quiz

SSAT Middle Level Quiz: Unit Rate Comparisons

Practice Unit Rate Comparisons 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 Unit Rate Comparisons, 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 factory produces toy cars and toy trucks in the ratio 5:3. If the factory produces 240 toy cars in one day, how many total toys does it produce that day?

  1. 384 toys (correct answer)
  2. 400 toys
  3. 420 toys
  4. 456 toys
  5. 480 toys
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 absolute amounts. Given that toy cars and trucks are produced in a 5:3 ratio, this means for every 5 cars produced, 3 trucks are made. Since you know 240 cars are produced, you can find the number of trucks by setting up a proportion. If 5 parts represent 240 cars, then each part equals 240÷5=48240 ÷ 5 = 48 toys. Since trucks represent 3 parts, the factory produces 3×48=1443 × 48 = 144 trucks. The total production is 240+144=384240 + 144 = 384 toys. Looking at the wrong answers: B) 400 toys likely comes from incorrectly assuming the ratio means 240 cars plus 160 trucks (perhaps by thinking 3/5 of 240 is added to 240). C) 420 toys might result from mistakenly calculating 240 + 180, possibly by confusing the ratio setup. D) 456 toys could come from incorrectly multiplying 240 by some factor derived from misunderstanding the 5:3 relationship. The correct answer is A) 384 toys because it properly accounts for both the 240 cars and the proportionally calculated 144 trucks. Strategy tip: In ratio problems, always identify what quantity you know, determine what one "part" of the ratio represents, then calculate the unknown quantities before finding totals. Set up your ratios clearly: if A:B = 5:3 and A = 240, then one part = 240÷5 = 48.

Question 2

A factory produces 840 widgets in 6 hours, while another factory produces 1,050 widgets in 7 hours. At these rates, how many more widgets will the second factory produce than the first factory in 42 hours?

  1. 210 widgets
  2. 420 widgets (correct answer)
  3. 630 widgets
  4. 840 widgets
  5. 1,260 widgets
Explanation: When you encounter rate problems involving multiple factories or workers, you need to find each unit's individual rate, then scale up to the target time period. First, calculate each factory's hourly production rate. The first factory produces 840 widgets in 6 hours, so its rate is 840÷6=140840 ÷ 6 = 140 widgets per hour. The second factory produces 1,050 widgets in 7 hours, so its rate is 1,050÷7=1501,050 ÷ 7 = 150 widgets per hour. Next, find how many widgets each factory produces in 42 hours. The first factory will produce 140×42=5,880140 × 42 = 5,880 widgets. The second factory will produce 150×42=6,300150 × 42 = 6,300 widgets. The question asks how many MORE widgets the second factory produces, so subtract: 6,300−5,880=4206,300 - 5,880 = 420 widgets. This confirms answer B is correct. Looking at the wrong answers: A) 210 widgets is exactly half the correct answer, likely resulting from a calculation error or confusing the time period. C) 630 widgets might come from incorrectly using the original 6-hour and 7-hour production differences without proper scaling. D) 840 widgets equals the first factory's 6-hour production, suggesting someone confused the given data with the final answer. Study tip: In rate problems, always convert to a common unit rate first (like "per hour"), then scale up to your target time. Double-check that you're answering what the question actually asks—here it's the difference, not the total production of either factory.

Question 3

A recipe calls for 2 cups of flour for every 3 cups of milk. If Maria has 10 cups of flour, what is the maximum number of complete batches of the recipe she can make, and how much milk will she need?

  1. 4 batches, 12 cups of milk
  2. 5 batches, 15 cups of milk (correct answer)
  3. 5 batches, 22.5 cups of milk
  4. 6 batches, 15 cups of milk
  5. 6 batches, 18 cups of milk
Explanation: This is a ratio and proportion problem that tests your ability to work with scaling recipes. When you see questions about recipes or mixing ingredients, focus on the relationship between ingredients and how that ratio stays constant across multiple batches. The recipe uses a 2:3 ratio of flour to milk (2 cups flour for every 3 cups milk). To find how many complete batches Maria can make with 10 cups of flour, divide the flour she has by the flour needed per batch: 10÷2=510 \div 2 = 5 batches. Since each batch requires 3 cups of milk, the total milk needed is 5×3=155 \times 3 = 15 cups. Looking at the wrong answers: Choice A suggests only 4 batches with 12 cups of milk. This represents someone who miscalculated the division or didn't fully use all available flour. Choice C gives the right number of batches (5) but calculates 22.5 cups of milk, which seems to come from incorrectly multiplying or confusing the ratio. Choice D claims 6 batches are possible with only 15 cups of milk. Six batches would require 6×2=126 \times 2 = 12 cups of flour, but would also need 6×3=186 \times 3 = 18 cups of milk, making this answer internally inconsistent. Remember that in ratio problems, once you determine how many complete units you can make, multiply that number by each ingredient requirement separately. Don't try to scale ingredients independently—always work from the number of complete batches first.

Question 4

A printing machine produces 450 flyers in 30 minutes. At this rate, how many complete hours are needed to produce at least 2,700 flyers?

  1. 2 hours
  2. 3 hours (correct answer)
  3. 4 hours
  4. 5 hours
  5. 6 hours
Explanation: This is a rate problem that requires you to find how long it takes to produce a specific quantity at a given rate, then round up to complete hours. First, calculate the machine's rate per hour. If 450 flyers are produced in 30 minutes, then in 60 minutes (1 hour), the machine produces 450×2=900450 \times 2 = 900 flyers per hour. Next, determine how long it takes to produce 2,700 flyers: 2,700900=3\frac{2,700}{900} = 3 hours exactly. Since the question asks for "complete hours needed to produce at least 2,700 flyers," and we need exactly 3 hours to produce 2,700 flyers, the answer is 3 hours (B). Looking at the wrong answers: (A) 2 hours would only produce 900×2=1,800900 \times 2 = 1,800 flyers, which falls short of the 2,700 needed. (C) 4 hours would produce 900×4=3,600900 \times 4 = 3,600 flyers, which exceeds the requirement but represents overestimating the time needed. (D) 5 hours would produce 900×5=4,500900 \times 5 = 4,500 flyers, which significantly overshoots the target. The key insight is that "at least 2,700" means the minimum number of complete hours needed. Since we need exactly 3 hours to reach 2,700 flyers, 3 complete hours is the answer. Strategy tip: In rate problems asking for "at least" a certain amount, calculate the exact time needed first, then determine if you need to round up to the next whole unit. Here, no rounding was necessary since the calculation yielded a whole number.

Question 5

A recipe for 6 servings requires 1121\frac{1}{2} cups of sugar. How many cups of sugar are needed for 9 servings of the same recipe?

  1. 22 cups
  2. 2142\frac{1}{4} cups (correct answer)
  3. 2122\frac{1}{2} cups
  4. 2342\frac{3}{4} cups
  5. 33 cups
Explanation: When you encounter a recipe problem asking you to scale ingredients up or down, you're dealing with proportional reasoning. The key insight is that all ingredients must be scaled by the same factor to maintain the recipe's balance. First, determine the scaling factor by comparing servings: you need to go from 6 servings to 9 servings. Set up the proportion: 9 servings6 servings=32=1.5\frac{9 \text{ servings}}{6 \text{ servings}} = \frac{3}{2} = 1.5. This means you need 1.5 times as much of every ingredient. Now multiply the original sugar amount by this factor: 112×1.51\frac{1}{2} \times 1.5. Convert the mixed number to an improper fraction: 112=321\frac{1}{2} = \frac{3}{2}. Then calculate: 32×32=94=214\frac{3}{2} \times \frac{3}{2} = \frac{9}{4} = 2\frac{1}{4} cups. Looking at the wrong answers: Choice A (2 cups) results from incorrectly adding 12\frac{1}{2} cup to the original amount instead of scaling proportionally. Choice C (2122\frac{1}{2} cups) comes from mistakenly doubling the recipe rather than scaling by 1.5. Choice D (2342\frac{3}{4} cups) might result from calculation errors when working with mixed numbers or fractions. The correct answer is B: 2142\frac{1}{4} cups. Strategy tip: For recipe scaling problems, always find the ratio between the target servings and original servings first, then multiply every ingredient by that same ratio. Convert mixed numbers to improper fractions before multiplying to avoid calculation mistakes.

Question 6

Jenny walks 2.4 miles in 36 minutes. At this same pace, how many minutes will it take her to walk 4 miles?

  1. 45 minutes
  2. 54 minutes
  3. 60 minutes (correct answer)
  4. 66 minutes
  5. 72 minutes
Explanation: This is a rate and proportion problem where you need to find how long it takes to walk a different distance at the same pace. When you see questions about maintaining a constant rate or pace, set up a proportion to compare the two scenarios. First, find Jenny's walking rate. She walks 2.4 miles in 36 minutes, so her rate is 2.4 miles36 minutes=115 miles per minute\frac{2.4 \text{ miles}}{36 \text{ minutes}} = \frac{1}{15} \text{ miles per minute}. Now you can find how long 4 miles takes. If she walks 115\frac{1}{15} miles per minute, then to walk 4 miles: 4 miles÷115 miles per minute=4×15=60 minutes4 \text{ miles} ÷ \frac{1}{15} \text{ miles per minute} = 4 × 15 = 60 \text{ minutes}. Alternatively, set up a proportion: 2.4 miles36 minutes=4 milesx minutes\frac{2.4 \text{ miles}}{36 \text{ minutes}} = \frac{4 \text{ miles}}{x \text{ minutes}}. Cross multiply: 2.4x=4×36=1442.4x = 4 × 36 = 144, so x=1442.4=60x = \frac{144}{2.4} = 60 minutes. Choice A (45 minutes) likely comes from incorrectly thinking that since 4 is about 1.67 times 2.4, you multiply 36 by 1.25 instead of the correct factor. Choice B (54 minutes) might result from calculation errors when setting up the proportion. Choice D (66 minutes) could come from rounding errors or incorrect cross multiplication. The correct answer is C) 60 minutes. Study tip: For rate problems, always identify what stays constant (the rate) and set up your proportion carefully. Double-check by verifying that faster rates mean less time, and longer distances mean more time.

Question 7

A factory produces 1,800 widgets in 4 days. At this rate, how many widgets will the factory produce in 3 weeks?

  1. 8,100 widgets
  2. 9,450 widgets (correct answer)
  3. 10,800 widgets
  4. 12,150 widgets
  5. 13,500 widgets
Explanation: This is a rate problem that tests your ability to convert units and maintain proportional relationships. When you see questions asking about production "at this rate," you need to find the unit rate first, then scale it to the new time period. Start by finding the daily production rate. If the factory produces 1,800 widgets in 4 days, then it produces 1,8004=450\frac{1,800}{4} = 450 widgets per day. Now convert 3 weeks to days: 3×7=213 \times 7 = 21 days. Finally, multiply the daily rate by the number of days: 450×21=9,450450 \times 21 = 9,450 widgets. Let's examine why each wrong answer represents a common error. Choice A (8,100) results from incorrectly calculating 18 days instead of 21 days for 3 weeks, then multiplying 450×18450 \times 18. Choice C (10,800) comes from using 24 days instead of 21, perhaps confusing weeks with some other time unit. Choice D (12,150) appears to use 27 days, possibly from miscounting or confusing the conversion. The key strategy for rate problems is to always establish your unit rate clearly, then carefully convert all time units to the same measurement before multiplying. Double-check your unit conversions—many SSAT rate problems include trap answers based on common conversion mistakes. Remember: 1 week = 7 days, and always verify your arithmetic by checking if your answer makes sense relative to the given information.

Question 8

A copy machine makes 12 copies in 20 seconds. At this rate, how many copies can it make in 5 minutes?

  1. 150 copies
  2. 180 copies (correct answer)
  3. 200 copies
  4. 240 copies
  5. 300 copies
Explanation: This is a rate problem that asks you to find how many copies can be made in a different time period. When you see questions about rates, think about finding the unit rate first - how much work gets done per unit of time. Start by finding the copy rate per second. The machine makes 12 copies in 20 seconds, so the rate is 12 copies20 seconds=0.6 copies per second\frac{12 \text{ copies}}{20 \text{ seconds}} = 0.6 \text{ copies per second}. Next, convert 5 minutes to seconds: 5×60=300 seconds5 \times 60 = 300 \text{ seconds}. Now multiply the rate by the total time: 0.6 copies per second×300 seconds=180 copies0.6 \text{ copies per second} \times 300 \text{ seconds} = 180 \text{ copies}. This confirms answer choice B is correct. Let's examine why the other answers are wrong. Choice A (150 copies) likely comes from incorrectly calculating the rate as 0.5 copies per second instead of 0.6. Choice C (200 copies) might result from miscalculating the conversion to seconds or rounding errors in the rate calculation. Choice D (240 copies) could come from using 4 minutes instead of 5 minutes, or from setting up an incorrect proportion. When solving rate problems on the SSAT, always establish a clear unit rate first, then carefully convert all time units to match. Double-check your time conversions - mixing up minutes and seconds is a common mistake. Setting up the problem as "copies per second × total seconds" gives you a reliable framework for these questions.

Question 9

A delivery truck uses 12 gallons of fuel to travel 180 miles. How many gallons of fuel will the truck use to travel 300 miles at the same rate?

  1. 18 gallons
  2. 20 gallons (correct answer)
  3. 22 gallons
  4. 24 gallons
  5. 25 gallons
Explanation: This is a rate problem where you need to find how much fuel is needed for a different distance. When you see questions about consistent rates (like fuel consumption per mile), set up a proportion to maintain that same relationship. First, find the fuel consumption rate. The truck uses 12 gallons for 180 miles, so the rate is 12 gallons180 miles=115\frac{12 \text{ gallons}}{180 \text{ miles}} = \frac{1}{15} gallons per mile. Now you can find how much fuel is needed for 300 miles: 300×115=20300 \times \frac{1}{15} = 20 gallons. Alternatively, set up a proportion: 12 gallons180 miles=x gallons300 miles\frac{12 \text{ gallons}}{180 \text{ miles}} = \frac{x \text{ gallons}}{300 \text{ miles}}. Cross-multiply: 12×300=180x12 \times 300 = 180x, so 3600=180x3600 = 180x, which gives you x=20x = 20 gallons. Looking at the wrong answers: Choice A (18 gallons) might result from incorrectly calculating the rate or making an arithmetic error in the proportion. Choice C (22 gallons) could come from rounding errors or miscalculating the cross-multiplication. Choice D (24 gallons) might result from setting up the proportion incorrectly or using faulty mental math shortcuts. The correct answer is B) 20 gallons. Strategy tip: For rate problems on the SSAT, always double-check your setup by asking "does this make sense?" Since 300 miles is less than double 180 miles, the fuel needed should be less than double 12 gallons (24). This quick reasonableness check can help you eliminate obviously wrong answers.

Question 10

A car uses 8 gallons of gas to travel 280 miles. At this rate, how many miles can the car travel with 15 gallons of gas?

  1. 480 miles
  2. 525 miles (correct answer)
  3. 560 miles
  4. 600 miles
  5. 640 miles
Explanation: When you encounter a problem asking "at this rate," you're dealing with a unit rate or proportion problem. The key is finding how much the car can do per unit (per gallon) and then scaling up. First, find the car's fuel efficiency by dividing total miles by total gallons: 280 miles8 gallons=35 miles per gallon\frac{280 \text{ miles}}{8 \text{ gallons}} = 35 \text{ miles per gallon}. Now you can calculate how far the car travels with 15 gallons: 35×15=525 miles35 \times 15 = 525 \text{ miles}. You can also solve this using proportions: 8 gallons280 miles=15 gallonsx miles\frac{8 \text{ gallons}}{280 \text{ miles}} = \frac{15 \text{ gallons}}{x \text{ miles}}. Cross-multiplying gives 8x=280×15=42008x = 280 \times 15 = 4200, so x=525x = 525 miles. Looking at the wrong answers: Choice A (480 miles) likely comes from incorrectly calculating 32 miles per gallon instead of 35, then multiplying by 15. Choice C (560 miles) might result from misreading the original data or making an arithmetic error in the division step. Choice D (600 miles) suggests someone calculated 40 miles per gallon, possibly by rounding 35 up incorrectly or making a computational mistake. The correct answer is B) 525 miles. For rate problems like this, always identify what stays constant (the rate of fuel consumption) and set up your calculation methodically. Double-check your unit rate calculation first, since any error there will carry through to your final answer. Converting to "per unit" makes these problems much more manageable.

Question 11

A water tank is 2/3 full and contains 80 gallons. At the rate of 5 gallons per minute, how long will it take to completely fill the tank?

  1. 6 minutes
  2. 8 minutes (correct answer)
  3. 12 minutes
  4. 16 minutes
  5. 24 minutes
Explanation: When you encounter a tank-filling problem, you need to determine both the tank's total capacity and how much more water is needed, then calculate the time based on the filling rate. Since the tank is 23\frac{2}{3} full and contains 80 gallons, you can find the total capacity by setting up the equation: 23×total capacity=80\frac{2}{3} \times \text{total capacity} = 80 gallons. Solving for total capacity: total capacity=80÷23=80×32=120\text{total capacity} = 80 \div \frac{2}{3} = 80 \times \frac{3}{2} = 120 gallons. Now you know the tank holds 120 gallons total, and it currently has 80 gallons. To completely fill it, you need 120−80=40120 - 80 = 40 more gallons. At a rate of 5 gallons per minute, the time required is 40÷5=840 \div 5 = 8 minutes. This confirms answer choice B. Looking at the incorrect choices: A) 6 minutes would only add 30 gallons (6×5=306 \times 5 = 30), leaving the tank at 110 gallons, which isn't full. C) 12 minutes would add 60 gallons (12×5=6012 \times 5 = 60), overfilling the tank to 140 gallons total. D) 16 minutes would add 80 gallons (16×5=8016 \times 5 = 80), which would double the current amount rather than fill the remaining space. Remember this three-step approach for tank problems: find the total capacity using the given fraction, calculate how much more is needed, then divide by the rate. Don't confuse "time to fill completely" with "time to double the current amount" — a common trap in these questions.

Question 12

The school cafeteria serves 450 students in 75 minutes. At this rate, how many students can be served in 2 hours?

  1. 600 students
  2. 720 students (correct answer)
  3. 900 students
  4. 1,080 students
  5. 1,200 students
Explanation: When you encounter rate problems like this one, you're working with the relationship between quantities and time. The key is finding the rate (how much per unit of time) and then applying it to a new time period. First, let's find the serving rate. The cafeteria serves 450 students in 75 minutes, so the rate is 450 students75 minutes=6 students per minute\frac{450 \text{ students}}{75 \text{ minutes}} = 6 \text{ students per minute}. Now we need to convert 2 hours to minutes: 2 hours=2×60=120 minutes2 \text{ hours} = 2 \times 60 = 120 \text{ minutes}. At 6 students per minute for 120 minutes: 6×120=720 students6 \times 120 = 720 \text{ students}. This confirms answer B is correct. Let's examine why the other answers are wrong. Answer A (600 students) comes from incorrectly calculating the rate as 450 ÷ 75 = 6, then multiplying by 100 instead of 120 minutes—this suggests confusion about how many minutes are in 2 hours. Answer C (900 students) results from finding the correct rate but then multiplying by 150 minutes instead of 120, possibly mixing up time conversions. Answer D (1,080 students) comes from multiplying 450 by 2.4, which suggests trying to scale up by the ratio of 180÷75 instead of properly converting to the correct time frame. Remember: in rate problems, always establish your rate clearly (amount per time unit), convert all times to the same units, then multiply. Double-check your time conversions—mixing up hours and minutes is a common trap on standardized tests.

Question 13

The cost of Internet service is $45 for 3 months. At this rate, what would be the cost for 8 months of service?

  1. $105
  2. $120 (correct answer)
  3. $135
  4. $150
  5. $180
Explanation: This is a classic rate problem where you need to find a unit rate and then scale it up. When you see questions asking "at this rate," you're being asked to find a constant rate of change and apply it to a different quantity. First, find the monthly cost by dividing the total cost by the number of months: $453 months=$15 per month\frac{\$45}{3 \text{ months}} = \$15 \text{ per month}. Now multiply this rate by 8 months: $15×8=$120\$15 \times 8 = \$120. You can also solve this using proportional reasoning. Set up the proportion: $453 months=x8 months\frac{\$45}{3 \text{ months}} = \frac{x}{8 \text{ months}}. Cross multiply: 45×8=3x45 \times 8 = 3x, so 360=3x360 = 3x, which gives you x=$120x = \$120. Looking at the wrong answers: (A) $105 represents a common error where students might have calculated $45×83−15\frac{45 \times 8}{3} - 15 ormadeanarithmeticmistakeinthecrossmultiplication.(C)or made an arithmetic mistake in the cross multiplication. (C)135 could result from incorrectly adding the original 45toamiscalculatedmonthlyrate,orfromcomputationalerrors.(D)45 to a miscalculated monthly rate, or from computational errors. (D) 150 might come from incorrectly thinking the rate is 45for2monthsinsteadof3,thencalculating45 for 2 months instead of 3, then calculating 452×8=180\frac{45}{2} \times 8 = 180 $ and making an additional error. Remember that rate problems always follow the same pattern: find the unit rate first, then multiply by the new quantity. Double-check your arithmetic, especially when cross multiplying in proportions.

Question 14

A gardener plants 18 flowers in 45 minutes. If she continues at this rate, how many flowers will she plant in 2122\frac{1}{2} hours?

  1. 45 flowers
  2. 54 flowers
  3. 60 flowers (correct answer)
  4. 72 flowers
  5. 90 flowers
Explanation: This is a rate problem where you need to find how many flowers can be planted in a different time period. When you see questions asking "at this rate," you're looking for a unit rate (flowers per unit of time) that you can then scale up or down. First, find the rate of flower planting. The gardener plants 18 flowers in 45 minutes, so her rate is 18 flowers45 minutes=25\frac{18 \text{ flowers}}{45 \text{ minutes}} = \frac{2}{5} flowers per minute (dividing both numerator and denominator by 9). Next, convert 2122\frac{1}{2} hours to minutes: 212×60=2.5×60=1502\frac{1}{2} \times 60 = 2.5 \times 60 = 150 minutes. Now multiply the rate by the new time period: 25×150=3005=60\frac{2}{5} \times 150 = \frac{300}{5} = 60 flowers. Looking at the wrong answers: Choice A (45 flowers) likely comes from confusing the original time period (45 minutes) with the answer. Choice B (54 flowers) might result from incorrectly calculating the rate or making an arithmetic error in the conversion. Choice D (72 flowers) could come from using an incorrect rate calculation or failing to properly convert hours to minutes. The correct answer is C: 60 flowers. Strategy tip: For rate problems, always establish your unit rate first, then make sure all your time units match before multiplying. Convert everything to the same unit (usually the smaller one, like minutes) to avoid mistakes with mixed units like hours and minutes.

Question 15

The ratio of red marbles to blue marbles in a jar is 3:7. If there are 21 red marbles, how many blue marbles are there?

  1. 35 blue marbles
  2. 42 blue marbles
  3. 49 blue marbles (correct answer)
  4. 63 blue marbles
  5. 70 blue marbles
Explanation: When you encounter ratio problems, you're looking at proportional relationships between quantities. The key insight is that ratios tell you the relative sizes of groups, and you can use this to find actual quantities when given one piece of information. The ratio 3:7 means that for every 3 red marbles, there are 7 blue marbles. Since you know there are 21 red marbles, you need to figure out what multiple of 3 gives you 21. Dividing: 21÷3=721 \div 3 = 7. This means the ratio has been scaled up by a factor of 7. If the red marbles were scaled up by 7 (from 3 to 21), then the blue marbles must also be scaled up by the same factor: 7×7=497 \times 7 = 49 blue marbles. Looking at the wrong answers: Choice A (35) represents what you'd get if you mistakenly thought the scaling factor was 5, perhaps by incorrectly calculating 21÷321 \div 3. Choice B (42) is what you'd get if you accidentally doubled the number of red marbles instead of applying the ratio correctly. Choice D (63) results from multiplying 21 by 3, which confuses the ratio relationship entirely. The correct answer is C) 49 blue marbles. Strategy tip: With ratio problems, always find the scaling factor first by dividing the known quantity by its ratio number, then multiply that factor by the unknown ratio number. Set up your work as: "If 3 becomes 21, then 7 becomes what?" This systematic approach prevents calculation errors.

Question 16

Store A sells 3 pounds of apples for $4.50, while Store B sells 4 pounds of apples for $5.60. Store C sells apples at the same unit price as the store with the better deal. How much would 7 pounds of apples cost at Store C?

  1. $9.80 (correct answer)
  2. $10.15
  3. $10.50
  4. $11.20
  5. $12.25
Explanation: When you encounter unit price comparison problems, you need to find the cost per unit for each option, then determine which offers the better deal. Let's calculate the unit price (cost per pound) for each store. Store A charges $4.50 for 3 pounds, so the unit price is $\frac{\4.50}{3} = $1.50 per pound. Store B charges 5.60 for 4 pounds, so the unit price is $$\frac{\5.60}{4} = $1.40$$ per pound. Since 1.40<1.40 < 1.50, Store B offers the better deal. Store C uses the same unit price as the store with the better deal, so Store C charges 1.40perpound.For7poundsatStoreC:1.40 per pound. For 7 pounds at Store C: 7 \times \1.40 = $9.80 . Looking at the wrong answers: Choice B (10.15)doesn′tcorrespondtoeitherstore′sunitpriceappliedto7pounds.ChoiceC(10.15) doesn't correspond to either store's unit price applied to 7 pounds. Choice C (10.50) equals 7 pounds at Store A's unit price ( 7 \times $1.50 = $10.50 ), which represents the mistake of choosing the worse deal instead of the better one. Choice D ($11.20) is too high and doesn't match any logical calculation from the given information. The correct answer is A. Study tip: In unit price problems, always calculate cost per unit for all options first, identify the best deal, then apply that rate to the final quantity. Watch out for answer choices that use the worse deal's unit price—this is a common trap designed to catch students who mix up "better" and "worse."

Question 17

A water tank can be filled by Pipe A in 12 minutes and by Pipe B in 15 minutes. If both pipes work together, how many minutes will it take to fill 34\frac{3}{4} of the tank?

  1. 5 minutes (correct answer)
  2. 6236\frac{2}{3} minutes
  3. 7127\frac{1}{2} minutes
  4. 8138\frac{1}{3} minutes
  5. 10 minutes
Explanation: When you encounter pipe or work rate problems, think in terms of how much of the job gets completed per unit of time. This approach transforms what seems like a complex scenario into straightforward fraction arithmetic. First, find each pipe's rate. Pipe A fills the tank in 12 minutes, so it completes 112\frac{1}{12} of the tank per minute. Pipe B fills the tank in 15 minutes, so it completes 115\frac{1}{15} of the tank per minute. When working together, you add their rates: 112+115\frac{1}{12} + \frac{1}{15}. To add these fractions, find a common denominator. The LCM of 12 and 15 is 60: 112=560\frac{1}{12} = \frac{5}{60} and 115=460\frac{1}{15} = \frac{4}{60}. Together, they complete 560+460=960=320\frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} of the tank per minute. To fill 34\frac{3}{4} of the tank at a rate of 320\frac{3}{20} per minute: 34÷320=34×203=5\frac{3}{4} ÷ \frac{3}{20} = \frac{3}{4} × \frac{20}{3} = 5 minutes. Choice A (5 minutes) is correct. Choice B (6236\frac{2}{3} minutes) might result from incorrectly finding the time to fill the entire tank, then multiplying by 34\frac{3}{4}. Choice C (7127\frac{1}{2} minutes) could come from averaging the individual pipe times incorrectly. Choice D (8138\frac{1}{3} minutes) might stem from calculation errors when finding the combined rate. Remember: for combined work rates, always add the individual rates first, then solve for the time needed.

Question 18

A swimming pool can be filled by a hose in 6 hours. At this rate, what fraction of the pool will be filled in 2 hours and 15 minutes?

  1. 14\frac{1}{4}
  2. 38\frac{3}{8} (correct answer)
  3. 25\frac{2}{5}
  4. 12\frac{1}{2}
  5. 58\frac{5}{8}
Explanation: When you encounter rate problems like this, think about finding the unit rate first - how much work gets done per unit of time. Since the hose fills the entire pool in 6 hours, you can find the rate by dividing: 1 pool6 hours=16\frac{1 \text{ pool}}{6 \text{ hours}} = \frac{1}{6} of the pool per hour. Now convert 2 hours and 15 minutes to the same units. Since 15 minutes = 1560=14\frac{15}{60} = \frac{1}{4} hour, the total time is 214=942\frac{1}{4} = \frac{9}{4} hours. Multiply the rate by the time: 16×94=924=38\frac{1}{6} \times \frac{9}{4} = \frac{9}{24} = \frac{3}{8} of the pool will be filled. Looking at the wrong answers: Choice A (14\frac{1}{4}) likely comes from incorrectly using just the 15 minutes as 14\frac{1}{4} without properly converting the total time or calculating the rate. Choice C (25\frac{2}{5}) might result from converting 2 hours 15 minutes incorrectly as 2.25 hours, then making calculation errors. Choice D (12\frac{1}{2}) could come from roughly estimating that 2+ hours is about half of 6 hours without doing precise calculations. The correct answer is B. Study tip: For rate problems, always establish your unit rate first (work per time), convert all times to the same units, then multiply rate × time = work completed. Double-check your fraction conversions, especially when dealing with minutes and hours.

Question 19

Based on the table, which person has the fastest typing speed?

  1. Alice has the fastest speed
  2. Bob has the fastest speed
  3. Charlie has the fastest speed (correct answer)
  4. Diana has the fastest speed
  5. All four people type at the same speed
Explanation: Calculate each person's words per minute: Alice: 480 ÷ 8 = 60 wpm. Bob: 350 ÷ 5 = 70 wpm. Charlie: 540 ÷ 6 = 90 wpm. Diana: 630 ÷ 9 = 70 wpm. Charlie has the highest rate at 90 wpm. Choice (A) only considers total words typed. Choice (B) and (D) might result from calculating 70 wpm for both Bob and Diana and stopping there. Choice (E) would result from not calculating unit rates properly.

Question 20

Brand X yogurt costs $3.75 for 5 cups, and Brand Y costs $4.80 for 6 cups. Which option offers a better unit price?

  1. Brand Y has the better unit price.
  2. Brand X has the better unit price. (correct answer)
  3. They have the same unit price.
  4. Brand X is better because $3.75 is less total.
Explanation: This question tests middle school math skills: using unit rates to compare quantities. Unit rates help compare different quantities by standardizing one of the variables, such as price per unit or speed per hour. In this scenario, students are asked to determine which brand of yogurt offers the better unit rate based on provided data. The correct answer is B because it demonstrates a clear understanding of unit rate calculation, showing that Brand X has a lower price per cup of $0.75 compared to $0.80 for Brand Y (3.75 ÷ 5 = 0.75 and 4.80 ÷ 6 = 0.80). A common error is D, where students compare total costs without considering quantity, often due to overlooking the need for per-unit comparison. To teach this skill, focus on real-world applications like shopping or travel. Encourage students to practice with everyday examples, ensuring they check units and context. Emphasize the importance of precise calculation and reasoning.