PSAT Math Quiz: Rates

20 questions · exam conditions

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RatesQuestion 1 of 20

A runner completes one $400\text{ m}$ lap in $1\text{ minute }20\text{ seconds}$ . What is the runner's average speed during the lap, in meters per second?

4.5\text{ m/s}

5\text{ m/s}

5.5\text{ m/s}

6\text{ m/s}

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PSAT Math Quiz

PSAT Math Quiz: Rates

Practice Rates in PSAT Math 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 Rates, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT Math.

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 runner completes one $400\text{ m}$ lap in $1\text{ minute }20\text{ seconds}$ . What is the runner's average speed during the lap, in meters per second?

$4.5\text{ m/s}$
$5\text{ m/s}$ (correct answer)
$5.5\text{ m/s}$
$6\text{ m/s}$

Explanation: When you encounter speed problems, remember that average speed equals distance divided by time. The key is making sure your units match what the question asks for. Here, you need to find speed in meters per second. You have the distance (

400 \text{ m}

) and time (

1 \text{ minute } 20 \text{ seconds}

), but first convert the time to seconds only:

1 \text{ minute } 20 \text{ seconds} = 60 + 20 = 80 \text{ seconds}

. Now apply the formula: Average speed

= \frac{\text{distance}}{\text{time}} = \frac{400 \text{ m}}{80 \text{ s}} = 5 \text{ m/s}

This confirms that (B)

5 \text{ m/s}

is correct. The wrong answers represent common calculation errors. Choice (A)

4.5 \text{ m/s}

might result from incorrectly using 90 seconds instead of 80, perhaps by miscounting the time conversion. Choice (C)

5.5 \text{ m/s}

could come from using approximately 73 seconds, possibly from rounding errors in the time conversion. Choice (D)

6 \text{ m/s}

might result from using about 67 seconds, which could happen if you mistakenly converted the time incorrectly or used mental math shortcuts that introduced errors. Always double-check your time conversions in speed problems—this is where most mistakes occur. Convert everything to the same units before calculating, and verify that your final answer's units match what the question requests. Speed problems on the PSAT often test unit conversion as much as the actual speed calculation.

Question 2

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?

$1.0\text{ min}$
$1.2\text{ min}$ (correct answer)
$1.5\text{ min}$
$2.0\text{ min}$

Explanation: 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.

Question 3

A car travels $200$ kilometers on $16$ liters of fuel. At the same rate of fuel consumption, about how many liters would be needed for the car to travel $350$ kilometers?

$20\text{ L}$
$24\text{ L}$
$28\text{ L}$ (correct answer)
$32\text{ L}$

Explanation: This is a classic proportional reasoning problem that tests your ability to set up and solve ratios. When you see a question asking about maintaining the "same rate" of something, think proportions. First, find the car's fuel efficiency rate. The car uses 16 liters to travel 200 kilometers, so its rate is

\frac{16 \text{ liters}}{200 \text{ km}} = 0.08

liters per kilometer. To find how much fuel is needed for 350 kilometers, multiply:

350 \times 0.08 = 28

liters. Alternatively, you can set up a proportion:

\frac{16 \text{ liters}}{200 \text{ km}} = \frac{x \text{ liters}}{350 \text{ km}}

. Cross-multiplying gives

16 \times 350 = 200x

, so

5600 = 200x

, which means

x = 28

liters. Looking at the wrong answers: Choice A (20 L) is too low and might result from incorrectly assuming the car becomes more fuel-efficient over longer distances. Choice B (24 L) could come from calculation errors or using an incorrect proportion setup. Choice D (32 L) is too high and might result from overestimating fuel consumption or arithmetic mistakes in the cross-multiplication. The correct answer is C (28 L). Strategy tip: For proportion problems, always check that your setup makes sense by ensuring the units align properly. Also, do a quick reasonableness check—since 350 km is 1.75 times the original 200 km distance, the fuel needed should be 1.75 times the original 16 liters, which equals 28 liters.

Question 4

A car is traveling at a constant speed of $60$ miles per hour. What is this speed in feet per second?

$60\text{ ft/s}$
$96\text{ ft/s}$
$120\text{ ft/s}$
$88\text{ ft/s}$ (correct answer)

Explanation: Unit conversion problems like this test your ability to systematically convert between different measurement systems using conversion factors. The key is setting up the conversion so that unwanted units cancel out, leaving you with the desired units. To convert 60 miles per hour to feet per second, you need two conversion factors: miles to feet (1 mile = 5,280 feet) and hours to seconds (1 hour = 3,600 seconds). Set up the conversion as a chain of fractions:

60 \frac{\text{miles}}{\text{hour}} \times \frac{5,280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3,600 \text{ seconds}}

The miles cancel out, and the hours cancel out, leaving feet per second:

60 \times \frac{5,280}{3,600} = 60 \times \frac{22}{15} = 60 \times 1.467 = 88 \text{ ft/s}

Choice A (60 ft/s) represents the trap of assuming the numerical value stays the same regardless of units—a common error when students don't perform any conversion. Choice B (96 ft/s) likely comes from using an incorrect conversion factor or making an arithmetic error in the calculation. Choice C (120 ft/s) might result from doubling the original speed or using a completely wrong conversion approach. For unit conversion success on the PSAT, always write out the conversion setup with units included—this helps you verify that unwanted units cancel properly. Memorize key conversions like 1 mile = 5,280 feet and 1 hour = 3,600 seconds, as these appear frequently in math problems.

Question 5

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.

$50\text{ gallons}$
$65\text{ gallons}$ (correct answer)
$78\text{ gallons}$
$117\text{ gallons}$

Explanation: 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.

Question 6

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?

$10.5\text{ min}$
$12\text{ min}$
$14\text{ min}$ (correct answer)
$28\text{ min}$

Explanation: 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.

Question 7

A recipe uses 2.5 cups of flour to make 20 muffins. At this rate, how many cups of flour are needed to make 32 muffins?

$3.2\text{ cups}$
$4.0\text{ cups}$ (correct answer)
$6.25\text{ cups}$
$8.0\text{ cups}$

Explanation: We need to find how much flour is needed for 32 muffins when 2.5 cups make 20 muffins. First, find the rate of flour per muffin: 2.5 cups ÷ 20 muffins = 0.125 cups per muffin. Then multiply by 32 muffins: 0.125 cups/muffin × 32 muffins = 4.0 cups. This is a scaling problem where we find the unit rate first. Watch out for the temptation to use mental math shortcuts that might introduce rounding errors.

Question 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?

$3\tfrac{3}{7}$ hr
$\tfrac{7}{24}$ hr
$\tfrac{24}{7}$ hr (correct answer)
7 hr

Explanation: 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.

Question 9

A student types 540 words in 12 minutes at a constant rate. At this rate, how many words will the student type in 25 minutes?

$900\text{ words}$
$1125\text{ words}$ (correct answer)
$1350\text{ words}$
$1620\text{ words}$

Explanation: We need to find how many words the student types in 25 minutes when typing 540 words in 12 minutes. First, find the typing rate: 540 words ÷ 12 minutes = 45 words per minute. Then multiply by 25 minutes: 45 words/minute × 25 minutes = 1125 words. The key is maintaining consistent units throughout the calculation. A common error is trying to set up a proportion without first finding the unit rate, which can lead to inverted fractions.

Question 10

A line graph shows total distance traveled by a cyclist versus time. The points lie on a straight line through $(0,0)$ and $(4, 60)$ . What is the cyclist's speed, in miles per hour, represented by the slope of the line? Avoid using $60/4$ with the units reversed.

$10\text{ mph}$
$12\text{ mph}$
$15\text{ mph}$ (correct answer)
$20\text{ mph}$

Explanation: We need to find the cyclist's speed represented by the slope of the line through (0,0) and (4,60). The slope equals rise over run: (60-0) miles ÷ (4-0) hours = 60 ÷ 4 = 15 miles per hour. The slope of a distance-time graph always represents speed when distance is on the y-axis and time is on the x-axis. A common error is reversing the division to get 4/60, which would give hours per mile instead of miles per hour. Remember that slope = Δy/Δx, and check your units match what's being asked.

Question 11

A pump fills a tank at a rate of $300$ liters per minute, but a leak in the tank lets out water at $40$ liters per minute. If the tank is initially empty, how many minutes will it take to add $7{,}800$ liters of water to the tank?

$20$
$24$
$30$ (correct answer)
$39$

Explanation: This is a classic net rate problem where two processes work in opposite directions. When you encounter scenarios with simultaneous filling and draining, filling and emptying, or similar opposing forces, always calculate the net effect first. The pump adds water at

300

liters per minute while the leak removes water at

40

liters per minute. The net rate of water accumulation is

300 - 40 = 260

liters per minute. To find the time needed to accumulate

7{,}800

liters, divide the target amount by the net rate:

\frac{7{,}800}{260} = 30

minutes. Let's examine why the other answers are incorrect. Choice (A)

20

minutes represents the trap of using only the pump rate:

\frac{7{,}800}{300} = 26

minutes, which rounds to

20

if you make calculation errors. Choice (B)

24

minutes might result from incorrectly adding the rates instead of finding their difference:

\frac{7{,}800}{300 + 40} = \frac{7{,}800}{340} ≈ 23

. Choice (D)

39

minutes could come from using just the leak rate by mistake:

\frac{7{,}800}{40} = 195

minutes, though this doesn't directly yield

39

—it's likely a calculation error combined with conceptual confusion. Remember: in rate problems involving opposing forces, always subtract the rates to find the net effect. Don't get distracted by the individual rates—focus on what's actually being accomplished overall. This pattern appears frequently on standardized tests.

Question 12

Painter X can paint a house alone in $6$ days, while Painter Y can paint the same house alone in $9$ days. Working together without changing pace, about how many days will it take them to paint the house?

$3.0$
$3.6$ (correct answer)
$4.5$
$5.0$

Explanation: When you encounter work rate problems, think in terms of how much work each person completes per unit of time. This approach turns a seemingly complex scenario into straightforward arithmetic. Painter X completes the job in 6 days, so X's rate is

\frac{1}{6}

of the house per day. Painter Y completes the job in 9 days, so Y's rate is

\frac{1}{9}

of the house per day. When working together, you add their rates:

\frac{1}{6} + \frac{1}{9}

. To add these fractions, find a common denominator. The least common multiple of 6 and 9 is 18, so:

\frac{1}{6} = \frac{3}{18}

and

\frac{1}{9} = \frac{2}{18}

. Their combined rate is

\frac{3}{18} + \frac{2}{18} = \frac{5}{18}

of the house per day. If they complete

\frac{5}{18}

of the house per day, then the time to complete the entire house is

\frac{1}{\frac{5}{18}} = \frac{18}{5} = 3.6

days. This confirms answer choice B. Choice A (3.0) likely comes from averaging the two times incorrectly:

\frac{6+9}{2} = 7.5

, then making additional errors. Choice C (4.5) might result from taking half of 9 days. Choice D (5.0) could come from subtracting 6 from 9, then adding back some arbitrary amount. Remember: in work rate problems, always convert individual completion times to rates (work per unit time), add the rates when people work together, then take the reciprocal to find the total time needed.

Question 13

Printer A can produce $30$ pages per minute, and Printer B can produce $20$ pages per minute. Operating together at these constant rates, how many pages can the two printers produce in $12$ minutes?

$360\text{ pages}$
$450\text{ pages}$
$600\text{ pages}$ (correct answer)
$720\text{ pages}$

Explanation: When you encounter a combined work rate problem, you need to add the individual rates together to find the total output rate, then multiply by the time period. First, find each printer's rate: Printer A produces 30 pages per minute, and Printer B produces 20 pages per minute. When working together, their combined rate is

30 + 20 = 50

pages per minute. To find the total pages produced in 12 minutes, multiply the combined rate by the time:

50 \text{ pages/minute} \times 12 \text{ minutes} = 600 \text{ pages}

. This confirms answer choice C is correct. Looking at the wrong answers: Choice A (360 pages) represents a common error where students might calculate only one printer's output over 12 minutes (

30 \times 12 = 360

), forgetting to include the second printer entirely. Choice B (450 pages) could result from incorrectly averaging the two rates first (

\frac{30+20}{2} = 25

), then multiplying by 12 to get

25 \times 12 = 300

, though this doesn't match exactly—it may represent a calculation error along this flawed path. Choice D (720 pages) likely comes from multiplying the rates instead of adding them (

30 \times 20 = 600

), then making an additional error, or from some other computational mistake. Remember: In combined rate problems, always add the individual rates to get the total rate. Think of it as "how much work gets done per unit time when everyone works together." This approach works for any scenario involving multiple workers, machines, or processes operating simultaneously.

Question 14

A rideshare driver earns a base fee plus a constant amount per mile. On Monday, a 6-mile trip cost $14.50, and a 10-mile trip cost $22.50. Assuming the pricing is linear, what is the driver's charge per mile (the unit rate for miles) in dollars per mile?

$1.25 per mile
$2.00 per mile (correct answer)
$3.75 per mile
$0.50 per mile

Explanation: This question asks for the driver's charge per mile in dollars per mile, given a base fee and total costs for 6-mile and 10-mile trips in dollars. The total cost follows a linear relationship: cost = base fee + (miles) × (rate in dollars per mile). To find the rate, set up the equations base + 6m = 14.50 and base + 10m = 22.50, then subtract to eliminate the base: 4m = 8, so m = 2 dollars per mile. Unit analysis confirms dollars divided by miles yields dollars per mile, ensuring the rate is properly set up. A common error is dividing total cost by miles without accounting for the base fee, such as 14.50 / 6 ≈ 2.42, which ignores the fixed component. For linear rate problems with a y-intercept, always use the difference in costs over difference in miles to find the slope accurately.

Question 15

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?

9 mi
12 mi
15 mi (correct answer)
18 mi

Explanation: 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.

Question 16

A cyclist rides 18 miles in 1.5 hours on a flat trail, then keeps the same speed for the rest of the ride. At this rate, how long will it take the cyclist to ride a total of 42 miles? (Be careful: 1.5 hours is not 1 hour 50 minutes.)

2.3 hours
3.5 hours (correct answer)
4.0 hours
5.0 hours

Explanation: The question asks how long it will take the cyclist to ride a total of 42 miles, given that 18 miles were already ridden in 1.5 hours at a constant speed, with the answer in hours. The rate relationship is speed in miles per hour, which remains constant throughout the ride. First, calculate the speed: 18 miles ÷ 1.5 hours = 12 miles per hour, ensuring units are miles over hours. The remaining distance is 42 miles - 18 miles = 24 miles, so the time for the remaining distance is 24 miles ÷ 12 miles per hour = 2 hours; adding the initial 1.5 hours gives a total of 3.5 hours. A key error to avoid is misinterpreting 1.5 hours as 1 hour and 50 minutes instead of 1 hour and 30 minutes, which could lead to incorrect speed calculation. Another common mistake is forgetting to subtract the initial distance and calculating time for the full 42 miles. As a test-taking strategy, always double-check unit conversions and what the question is asking for—total time includes the initial ride.

Question 17

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.

20 min
32 min (correct answer)
48 min
$\frac{8}{3}$ min

Explanation: 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.

Question 18

A delivery van travels 84 miles in 2.5 hours at a constant speed. If the van keeps the same speed for the next leg of the trip, how long will it take to travel 126 miles? Be careful to use miles per hour (not hours per mile) when setting up the rate.

$2.1\text{ hours}$
$3.0\text{ hours}$
$3.75\text{ hours}$ (correct answer)
$4.5\text{ hours}$

Explanation: We need to find how long it takes to travel 126 miles at the same constant speed. First, calculate the van's speed: rate = distance ÷ time = 84 miles ÷ 2.5 hours = 33.6 miles per hour. To find the time for 126 miles, use time = distance ÷ rate = 126 miles ÷ 33.6 mph = 3.75 hours. A common error is to set up the rate as hours per mile (2.5/84) instead of miles per hour, which would give an incorrect answer. When working with rates, always check that your units make sense—speed should be distance per time, not time per distance.

Question 19

Two machines package granola bars at constant rates. Machine X packages 180 bars in 12 minutes. Machine Y packages 260 bars in 20 minutes. Which machine has the greater unit rate, in bars per minute? Be careful not to compare total bars without accounting for time.

Machine X, $15\text{ bars/min}$ (correct answer)
Machine X, $0.067\text{ min/bar}$
Machine Y, $13\text{ bars/min}$
Machine Y, $20\text{ bars/min}$

Explanation: We need to compare the packaging rates of two machines in bars per minute. Machine X: 180 bars ÷ 12 minutes = 15 bars/minute. Machine Y: 260 bars ÷ 20 minutes = 13 bars/minute. Since 15 > 13, Machine X has the greater unit rate at 15 bars per minute. A common error is comparing total bars (260 > 180) without accounting for the different time periods. Always convert to the same unit rate before comparing—here, bars per minute makes the comparison straightforward.

Question 20

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 $1.2$ .

$22.75 (correct answer)
$19.50
$27.30
$18.20

Explanation: 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.