Rates - PSAT Math

1. A car uses $3.2$ gallons of gasoline to travel $96$ miles. At this consumption rate, how many miles can the car travel on $5$ gallons?
   Answer: $150\text{ miles}$ This is a unit rate problem where you need to find how far the car can travel per gallon, then scale up. When you see consumption rates or "at this rate" language, you're working with proportional relationships. First, find the car's fuel efficiency by calculating miles per gallon. The car travels $96$ miles using $3.2$ gallons, so divide: $\frac{96 \text{ miles}}{3.2 \text{ gallons}} = 30 \text{ miles per gallon}$. Now multiply this rate by $5$ gallons: $30 \times 5 = 150$ miles. The answer is B. You could also solve this using a proportion: $\frac{3.2 \text{ gallons}}{96 \text{ miles}} = \frac{5 \text{ gallons}}{x \text{ miles}}$. Cross-multiplying gives $3.2x = 480$, so $x = 150$ miles. Choice A ($120$ miles) might result from incorrectly calculating the unit rate as $24$ miles per gallon instead of $30$. Choice C ($165$ miles) could come from a computational error when dividing $96$ by $3.2$. Choice D ($180$ miles) might happen if you mistakenly used $36$ miles per gallon as your rate. Strategy tip: For rate problems, always find the unit rate first (miles per gallon, cost per item, etc.), then multiply by the new quantity. Double-check your division when calculating the unit rate, as this is where most errors occur. Setting up a proportion works too, but the unit rate method often feels more intuitive and is less prone to setup mistakes.
2. A cyclist travels at a constant speed. In 18 minutes, the cyclist goes 4.5 miles. At this same rate, how many miles will the cyclist travel in 1 hour?
   Answer: 15 mi This question asks how many miles a cyclist will travel in 1 hour (60 minutes) at a constant speed, given 4.5 miles in 18 minutes. The speed is constant, so distance = rate (in miles per minute) × time. First, find the rate: 4.5 miles / 18 minutes = 0.25 miles per minute; then for 60 minutes, distance = 0.25 miles/min × 60 min = 15 miles. Unit analysis: (miles per minute) × minutes cancels to miles, stressing proper rate and unit setup. A key error is forgetting to convert hours to minutes, like treating 1 hour as 1 unit without adjustment. For time unit mismatches in rates, convert everything to consistent units like minutes to avoid calculation errors.
3. A driver travels $2$ hours at $40$ miles per hour and then $1$ additional hour at $60$ miles per hour. What is the driver's average speed for the entire trip?
   Answer: $47\text{ mph}$ When you encounter average speed problems, remember that average speed isn't simply the arithmetic mean of the speeds—it's total distance divided by total time. Let's calculate the total distance and time for this trip. In the first part, the driver travels for 2 hours at 40 mph, covering $2 \times 40 = 80$ miles. In the second part, they travel for 1 hour at 60 mph, covering $1 \times 60 = 60$ miles. The total distance is $80 + 60 = 140$ miles, and the total time is $2 + 1 = 3$ hours. Therefore, the average speed is $\frac{140 \text{ miles}}{3 \text{ hours}} = 46.67$ mph, which rounds to 47 mph. This confirms answer choice B. Now let's see why the other answers are wrong. Choice A (45 mph) might tempt you if you incorrectly weighted the speeds by time: $\frac{(40 \times 2) + (60 \times 1)}{2 + 1} = \frac{140}{3}$, but then made an arithmetic error. Choice C (50 mph) is the simple arithmetic mean of 40 and 60 mph, which ignores the fact that the driver spent more time at the slower speed. Choice D (55 mph) has no clear mathematical basis but might appeal if you mistakenly thought the average should be closer to the higher speed. The key insight: average speed problems require you to find total distance and total time separately, then divide. Never just average the individual speeds unless the time periods are equal.
4. A water tank is being filled at a constant rate. After 12 minutes, 30 gallons have been added. At this same rate, how many minutes will it take to add 80 gallons total? Choose the answer with the correct time unit.
   Answer: 32 min The question asks how many minutes it will take to add 80 gallons to the water tank at a constant filling rate. The rate relationship is given by 30 gallons added in 12 minutes, so the filling rate is 30 gallons per 12 minutes or 2.5 gallons per minute after dividing both by 12. To find the time, use time = total gallons / rate, so t = 80 gallons / 2.5 gallons per minute. Calculating that: 80 / 2.5 = 32 minutes, with gallons canceling to leave minutes. Ensure unit consistency by keeping everything in gallons and minutes without unnecessary conversions. A common error is setting up the proportion inversely, like confusing gallons over time with time over gallons, leading to fractions like 8/3. For rate problems, write out the units explicitly in your setup to avoid inversion errors.
5. A recipe uses 2.5 cups of flour to make 20 muffins. At this same rate, how many cups of flour are needed to make 32 muffins?
   Answer: 4.0 cups The question asks how many cups of flour are needed to make 32 muffins at the same rate. The rate is flour usage, in cups per muffin. Calculate the rate: 2.5 cups / 20 muffins = 0.125 cups per muffin. Then, for 32 muffins: 32 * 0.125 cups/muffin = 4 cups, with muffins canceling to leave cups. Unit analysis is key to verify the setup, ensuring you're multiplying by the per-muffin rate. A common mistake is inverting the rate, like dividing instead, which might give muffins per cup, such as 8, leading to wrong scalings. In recipe rate problems, write the ratio clearly to avoid confusion between scaling up or down.
6. A hose fills a tank at a constant rate. It takes 18 minutes to fill 45 gallons. At this same rate, how many gallons will the hose fill in 26 minutes? Be careful not to divide 26 by 45 or to treat 45 as gallons per minute without dividing by 18.
   Answer: $65\text{ gallons}$ We need to find how many gallons the hose fills in 26 minutes at a constant rate. First, calculate the filling rate: 45 gallons ÷ 18 minutes = 2.5 gallons per minute. Then multiply by the new time: 2.5 gallons/minute × 26 minutes = 65 gallons. Common errors include dividing 26 by 45 (which gives a meaningless ratio) or treating 45 as the rate without dividing by 18. Always set up rates as quantity per unit time, then multiply by the desired time.
7. A cyclist rides at a constant speed of 14 miles per hour. At this rate, how many feet does the cyclist travel in 30 minutes? (Use $1\text{ mile}=5{,}280\text{ ft}$ and $60\text{ min}=1\text{ hr}$.)
   Answer: 36,960 ft The question asks how many feet the cyclist travels in 30 minutes at a constant speed of 14 miles per hour, requiring unit conversions. The rate relationship is 14 miles per hour, but we need to convert to feet and adjust for 30 minutes, noting 1 mile = 5,280 feet and 60 minutes = 1 hour. First, find distance in miles: in 30 minutes or 0.5 hours, distance = 14 mph 0.5 hr = 7 miles. Then convert to feet: 7 miles 5,280 feet per mile = 36,960 feet, with units chaining as miles cancel and hours are accounted for in the time. A key error is forgetting to convert minutes to hours or applying the conversion factors incorrectly, such as using 30 minutes without dividing by 60. In problems with unit conversions, perform them step-by-step and check that all units cancel appropriately to match the desired output.
8. A job is completed at a constant rate. Worker A can paint a room in 6 hours, and Worker B can paint the same room in 8 hours. If they work together at their constant rates, how long will it take them to paint 1 room?
   Answer: $\tfrac{24}{7}$ hr The question asks for the time to paint 1 room when two workers collaborate. Worker A's rate is 1/6 room per hour, and Worker B's rate is 1/8 room per hour. When working together, add their rates: 1/6 + 1/8 = 4/24 + 3/24 = 7/24 rooms per hour. To find time for 1 room, divide: 1 room ÷ (7/24 rooms/hour) = 1 × 24/7 = 24/7 hours. This equals 3 3/7 hours. The key insight is that rates add when workers collaborate, not times. Always work with rates (rooms per hour) rather than times (hours per room) when combining efforts.
9. At a grocery store, $3.2$ pounds of apples cost $5.76. At the same price per pound, what is the cost of $7$ pounds of apples?
   Answer: $$12.60 This is a unit rate problem where you need to find the price per pound first, then use it to calculate the cost of a different quantity. Start by finding the price per pound of apples. Divide the total cost by the weight: $\frac{\$5.76}{3.2 \text{ pounds}} = \$1.80$ per pound. Now multiply this unit rate by the desired quantity: $\$1.80 \times 7 \text{ pounds} = \$12.60$. This confirms that answer choice B is correct. Let's examine why the other options are wrong. Choice A ($11.20) results from incorrectly calculating the unit rate as $1.60 per pound, which happens if you make an arithmetic error in the division step. Choice C ($13.44) comes from using an incorrect unit rate of approximately $1.92 per pound, possibly from rounding errors or calculation mistakes. Choice D ($14.50) suggests a unit rate that's too high, around $2.07 per pound, which could result from misreading the original price or weight. For unit rate problems on the PSAT, always set up your work in two clear steps: first find the rate (cost per unit), then multiply by the new quantity. Double-check your division in the first step, as this is where most errors occur. You can verify your answer makes sense by noting that $7$ pounds should cost a bit more than double what $3.2$ pounds costs, since $7 > 2 \times 3.2$.
10. A recipe uses 2.5 cups of flour to make 12 muffins. At the same ratio, how many cups of flour are needed to make 30 muffins? Round to the nearest tenth of a cup if necessary.
    Answer: 6.3 cups The question asks how many cups of flour are needed to make 30 muffins at the same ratio as 2.5 cups for 12 muffins, rounded to the nearest tenth if necessary. The rate relationship is the constant flour per muffin, given as 2.5 cups per 12 muffins or approximately 0.2083 cups per muffin. To calculate, multiply the rate by 30: (2.5 / 12) 30 = 75 / 12 = 6.25 cups, which rounds to 6.3 cups. Units are consistent as cups per muffin times muffins cancels to cups. A key error is setting up the proportion backwards, like 12/2.5 30, which would give an incorrect amount. For ratio problems, write the rate with units and cross-multiply carefully to avoid inversion.
11. A car travels at a constant speed. On a highway, it goes 96 miles in 1 hour 36 minutes. What is the car's speed, in miles per hour? (Hint: convert 36 minutes to hours.)
    Answer: 60 mph The question asks for the car's speed in miles per hour, given it travels 96 miles in 1 hour 36 minutes at a constant speed. The rate relationship is distance in miles per hour of time. First, convert 1 hour 36 minutes to hours: 36 minutes ÷ 60 minutes per hour = 0.6 hours, so total time = 1.6 hours. Then, speed = 96 miles ÷ 1.6 hours = 60 miles per hour, with units miles over hours. A key error is forgetting to convert minutes to hours, leading to incorrect time like 1.36 hours. Another mistake is treating 36 minutes as 0.36 hours without proper division. For test-taking, always convert all times to the same unit before calculating rates to ensure accurate unit analysis.
12. A grocery store sells almonds for $7.80 per 3 pounds. At this rate, how much will 5 pounds of almonds cost? Round to the nearest cent.
    Answer: $13.00 We need to find the cost of 5 pounds of almonds when 3 pounds cost $7.80. First, find the unit rate: $7.80 ÷ 3 pounds = $2.60 per pound. Then multiply by 5 pounds: $2.60/pound × 5 pounds = $13.00. The key is recognizing this as a unit rate problem where we find dollars per pound first. A common mistake is trying to scale directly without finding the unit rate, which can lead to calculation errors.
13. A landscaping crew can mow 3 lawns in 2 hours at a constant rate. Assuming the same rate and no breaks, how long will it take the crew to mow 10 lawns? Give your answer in hours.
    Answer: $\frac{20}{3}$ hr The question asks how long it will take the landscaping crew to mow 10 lawns at a constant rate, with the answer in hours. The rate relationship is 3 lawns in 2 hours, so the mowing rate is $3$ lawns per $2$ hours or $1.5$ lawns per hour after dividing. To find the time, use time = total lawns / rate, so t = $10$ lawns / $1.5$ lawns per hour. Calculating that: $10 / 1.5 = \frac{20}{3}$ hours, with lawns canceling to leave hours. Units are consistent as long as we stay in lawns and hours. A common error is inverting the rate, like using hours per lawn instead, leading to fractions like 3/5. Always label your rate with units to ensure you're dividing correctly in work rate problems.
14. A grocery store sells almonds in bulk. A customer pays $7.80 for $1.2$ pounds of almonds. At the same price per pound, how much will $3.5$ pounds cost? Watch out for using $7.80 as a unit price without dividing by $1.2$.
    Answer: $22.75 We need to find the cost of 3.5 pounds of almonds at the same price per pound. First, find the unit price: $7.80 ÷ 1.2 pounds = $6.50 per pound. Then multiply by the desired amount: $6.50/pound × 3.5 pounds = $22.75. A common mistake is using $7.80 as the price per pound without dividing by 1.2, which would give $27.30. Always calculate the unit rate first by dividing total cost by total quantity before scaling up.
15. A water pump drains a tank at a constant rate. It removes 18 gallons every 4 minutes. At this rate, how many minutes will it take to drain 63 gallons?
    Answer: $14\text{ min}$ We need to find how long it takes to drain 63 gallons when the pump removes 18 gallons every 4 minutes. First, find the drainage rate: 18 gallons ÷ 4 minutes = 4.5 gallons per minute. Then find the time: 63 gallons ÷ 4.5 gallons/minute = 14 minutes. The key is recognizing this as a rate problem where we need gallons per minute first. A common mistake is setting up the proportion backwards, which would give minutes per gallon instead.
16. A line graph shows total cost versus number of tickets purchased. The line passes through $(0,0)$ and $(6,54)$. What is the unit rate, in dollars per ticket, represented by the slope of the line?
    Answer: $9 per ticket The question asks for the unit rate in dollars per ticket, represented by the slope of the line passing through (0,0) and (6,54). The rate is the slope, in dollars per ticket. Calculate the slope: change in cost / change in tickets = (54 - 0) dollars / (6 - 0) tickets = 9 dollars per ticket. The units confirm dollars divided by tickets give the per-ticket rate. Emphasizing unit analysis helps interpret graph slopes as rates correctly. A key error is using only one coordinate, like 54/6 without considering the origin, but here it's correct; another is misreading the axes. For graph-based rates, always use two points and check if it passes through the origin for proportional relationships.
17. A hose fills a tank at a constant rate of 12 gallons per minute. How many seconds will it take to fill 90 gallons at this rate?
    Answer: 450 seconds The question asks how many seconds it will take to fill 90 gallons at the given rate of 12 gallons per minute. The rate is filling speed, in gallons per minute. First, find the time in minutes: 90 gallons / 12 gallons per minute = 7.5 minutes, where gallons cancel to leave minutes. Convert to seconds: 7.5 minutes * 60 seconds per minute = 450 seconds, ensuring unit consistency throughout. Unit analysis is crucial here to handle the conversion from minutes to seconds properly. A common mistake is forgetting the unit conversion, leading to answers in minutes like 7.5, or misapplying it, such as dividing by 60. When rates involve time units, always convert to the requested unit step-by-step to avoid errors.
18. A runner completes 5 kilometers in 24 minutes at a constant pace. At this pace, how many minutes will it take to run 8 kilometers? (Round to the nearest whole minute.)
    Answer: 38 minutes The question asks how many minutes it will take to run 8 kilometers at the same constant pace, rounded to the nearest whole minute. The rate is pace, in minutes per kilometer or equivalently kilometers per minute. First, find the pace: 24 minutes / 5 kilometers = 4.8 minutes per kilometer. Then, for 8 kilometers: 8 km * 4.8 min/km = 38.4 minutes, which rounds to 38 minutes since 0.4 is less than 0.5. Unit analysis shows kilometers cancel, leaving minutes, emphasizing the importance of consistent units. A key error is setting up the proportion backwards, like using kilometers per minute incorrectly, leading to times like 15 minutes. For rounding in rate problems, always apply standard rules after calculating to ensure accuracy.
19. A factory produces $450$ identical parts in $9$ hours at a constant rate. Approximately how many minutes does it take the factory to produce one part?
    Answer: $1.2\text{ min}$ This is a unit rate problem that tests your ability to convert between time units and calculate rates. When you see questions asking "how much time per item" or "how many items per unit time," you're working with rates that require careful attention to units. To find how long it takes to produce one part, you need to divide the total time by the number of parts produced. First, convert the time to minutes: $9 \text{ hours} \times 60 \text{ minutes/hour} = 540 \text{ minutes}$. Then calculate the time per part: $\frac{540 \text{ minutes}}{450 \text{ parts}} = 1.2 \text{ minutes per part}$. The answer is B) $1.2\text{ min}$. Looking at the wrong answers: A) $1.0\text{ min}$ would result from incorrectly calculating $\frac{450}{540}$ instead of $\frac{540}{450}$ - you've flipped the fraction and gotten parts per minute rather than minutes per part. C) $1.5\text{ min}$ might come from calculation errors or rounding mistakes in the division. D) $2.0\text{ min}$ could result from forgetting to convert hours to minutes and using $\frac{9}{450}$ directly, then making additional errors. Remember to always check your units carefully in rate problems. Ask yourself: "Am I looking for time per item or items per time?" Then set up your fraction accordingly, with the desired unit in the numerator. Converting to consistent units (like minutes) before calculating will help you avoid common mistakes.
20. A roller-coaster car travels the entire $1.2$-kilometer track in $90$ seconds at a constant average speed. What is this speed, to the nearest mile per hour?
    Answer: $30\text{ mph}$ This problem tests your ability to convert between different units of speed - a skill you'll see repeatedly on standardized tests. When you encounter speed conversion problems, always identify what units you're starting with and what units you need to end up with. First, calculate the speed in kilometers per second: $\frac{1.2 \text{ km}}{90 \text{ s}} = \frac{1.2}{90} = \frac{2}{150} = \frac{1}{75}$ km/s. Now convert this to miles per hour using two key conversion factors: 1 kilometer ≈ 0.621 miles and 1 hour = 3600 seconds. $\frac{1}{75} \text{ km/s} \times 0.621 \frac{\text{miles}}{\text{km}} \times 3600 \frac{\text{s}}{\text{hour}} = \frac{0.621 \times 3600}{75} = \frac{2235.6}{75} ≈ 29.8 \text{ mph}$ Rounded to the nearest mile per hour, this gives us 30 mph, which is choice B. Looking at the wrong answers: Choice A (24 mph) likely results from using an incorrect conversion factor or calculation error. Choice C (36 mph) might come from forgetting to convert kilometers to miles and only doing the time conversion. Choice D (42 mph) could result from using the wrong conversion factor entirely or making multiple computational errors. Strategy tip: For unit conversion problems, always set up your conversion factors as fractions so unwanted units cancel out. Write out each step clearly - rushing through conversions is where most errors occur. Also, memorize that 1 km ≈ 0.621 miles, as this conversion appears frequently on standardized tests.

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