Rates and Unit Rates Lesson Test

Answer: D

Correct answer: 24 gallons per minute. Unit rate = gallons divided by minutes:

per minute.

• A: This subtracts the minutes from the gallons; a unit rate comes from dividing.
• B: This adds the two numbers instead of dividing gallons by minutes.
• C: This misplaces the decimal point in the quotient of 840 and 35.
• D (correct): Dividing 840 gallons by 35 minutes gives the amount for one minute, 24 gallons.

Answer: A

Correct answer: 143 miles. Convert the time: 45 minutes is 0.75 hour, so the time is 2.75 hours. Distance:

• A (correct): 2 hours 45 minutes is 2.75 hours, and 52 times 2.75 is 143 miles.
• B: This uses only the 2 whole hours and drops the 45 minutes.
• C: This rounds the time up to 3 hours before multiplying.
• D: This writes 2 hours 45 minutes as 2.45 hours instead of 2.75 hours.

Answer: C

Correct answer: 2 hours 30 minutes. Time = distance divided by rate:

Multiply the decimal part by 60: 0.5 times 60 = 30 minutes.

• A: This reads 0.5 hour as 5 minutes; half an hour is 30 minutes.
• B: This divides the speed by the distance, which gives hours per mile turned upside down.
• C (correct): 45 divided by 18 is 2.5 hours, and 0.5 hour is 30 minutes.
• D: This reads the decimal 2.5 hours as 2 hours 50 minutes instead of converting 0.5 hour to minutes.

Answer: C

Correct answer: $0.23. Small jar: $3.00 divided by 12 = $0.25 per ounce. Large jar: $4.60 divided by 20 = $0.23 per ounce. The large jar is the better buy at $0.23 per ounce.

• A: This averages the two unit prices instead of picking the lower one.
• B: This is the price per ounce of the 12-ounce jar, which is the more expensive choice per ounce.
• C (correct): The 20-ounce jar costs $4.60 divided by 20, or $0.23 per ounce, less than the $0.25 per ounce of the small jar.
• D: This divides the large jar's price by the small jar's 12 ounces, mixing the two packages.

Answer: B

Correct answer: 14 minutes. Time = amount divided by rate:

• A: This misplaces the decimal point in 490 divided by 35.
• B (correct): Dividing 490 pages by 35 pages per minute gives 14 minutes.
• C: This multiplies the pages by the rate; pages divided by pages per minute gives minutes.
• D: This subtracts the rate from the number of pages.

Answer: C

Correct answer: 28 miles. Convert: 45 minutes is

so the time is 1.75 hours. Distance:

• A: This treats 45 minutes as half an hour, using 1.5 hours.
• B: This writes the time as 1.45 hours instead of 1.75 hours.
• C (correct): 1 hour 45 minutes is 1.75 hours, and 16 times 1.75 is 28 miles.
• D: This multiplies miles per hour by 105 minutes without converting minutes to hours.

Answer: A

Correct answer: 3,600 bottles. Unit rate:

bottles per minute. Total:

bottles.

• A (correct): The unit rate is 960 divided by 12, or 80 bottles per minute, and 80 times 45 is 3,600.
• B: This adds 45 to 960 instead of scaling the rate by the time.
• C: This scales the unit rate by 60 minutes, a full hour, instead of 45 minutes.
• D: This sets up the ratio upside down, multiplying 960 by 12 and dividing by 45.

Answer: B

Correct answer: 27.9 gallons. From 6:00 a.m. to 9:30 p.m. is 15 hours 30 minutes, or 15.5 hours. Fuel used:

• A: This miscounts the run as 16.5 hours.
• B (correct): The run lasts 15.5 hours, and 1.8 times 15.5 is 27.9 gallons.
• C: This writes 15 hours 30 minutes as 15.3 hours instead of 15.5 hours.
• D: This counts only 3.5 hours by subtracting the clock numbers 6 and 9:30 without switching to p.m. time.

Answer: A

Correct answer: 15 minutes. Combined rate:

per minute. Time:

• A (correct): Working together, the hoses deliver 9 + 6 = 15 gallons per minute, and 225 divided by 15 is 15 minutes.
• B: This is the time for the 6-gallon hose working alone.
• C: This subtracts the rates; both hoses add water, so their rates should be added.
• D: This is the time for the 9-gallon hose working alone.

Answer: C

Correct answer: 30 minutes. Net rate:

per minute. Time:

• A: This divides by the leak rate instead of by the net rate.
• B: This adds the leak to the inflow; water lost through a leak must be subtracted.
• C (correct): The net rate is 18 - 6 = 12 liters per minute, and 360 divided by 12 is 30 minutes.
• D: This uses the inflow alone and ignores the leak.

Answer: D

Correct answer: 305 miles. The gap grows at

per hour. Time is 2.5 hours, so the distance apart is

• A: This is the distance covered by the faster truck alone.
• B: This subtracts the speeds, which applies only when the trucks travel in the same direction.
• C: This uses only 2 hours and drops the extra 30 minutes.
• D (correct): Moving apart, they separate at 58 + 64 = 122 miles per hour, and 122 times 2.5 is 305 miles.

Answer: A

Correct answer: 4 hours. Add the whole-job rates:

of the job per hour. Invert: the job takes 4 hours.

• A (correct): Their rates are one-fifth and one-twentieth of the job per hour, which add to one-fourth, so the job takes 4 hours.
• B: This averages the two times; a team is always faster than its fastest member.
• C: This adds the two times instead of adding the two rates.
• D: The size of the floor is not needed, because each painter's rate is stated as a fraction of the whole job per hour.

Answer: D

Correct answer: 1 hour 36 minutes. Same direction, so subtract:

per hour. Time:

Convert: 0.6 times 60 = 36 minutes, for 1 hour 36 minutes.

• A: This adds the speeds; when one vehicle chases another, the gap closes at the difference of the speeds.
• B: This divides the gap by the car's full speed, ignoring that the truck keeps moving away.
• C: This reads 1.6 hours as 1 hour 6 minutes instead of converting 0.6 hour to 36 minutes.
• D (correct): The car gains 15 miles per hour, so 24 divided by 15 is 1.6 hours, and 0.6 hour is 36 minutes.

Answer: D

Correct answer: 20 minutes. Gallons still needed:

Net rate:

per minute. Time:

• A: This ignores the drain and divides 400 gallons by the inflow of 25 gallons per minute.
• B: This uses the full 480 gallons and ignores the 80 gallons already in the tank.
• C: This uses only the 10-gallon pipe; both inlets run at the same time.
• D (correct): The tank needs 400 more gallons, and the net rate is 15 + 10 - 5 = 20 gallons per minute, so 20 minutes.

Answer: B

Correct answer: $1.50. Unit prices: $4.47 divided by 3 = $1.49 per pound; $7.20 divided by 5 = $1.44 per pound. For 30 pounds:

Saving:

• A: This compares the two bag prices, which hold different amounts of rice.
• B (correct): Store B costs $1.44 per pound and Store A $1.49, so 30 pounds costs $43.20 instead of $44.70, a saving of $1.50.
• C: This buys 10 bags at each store, but 10 of the larger bags would be 50 pounds, not 30.
• D: This is the difference in price per pound, not the saving on all 30 pounds.

Answer: C

Correct answer: 52 mph. Convert the time: 15 minutes is

so the trip takes 2.25 hours.

Then

• A: This divides by 2 hours and ignores the extra 15 minutes.
• B: This treats 15 minutes as half an hour, dividing by 2.5 hours instead of 2.25 hours.
• C (correct): 2 hours 15 minutes is 2.25 hours, and 117 divided by 2.25 is 52.
• D: This multiplies distance by time; speed comes from dividing distance by time.

Answer: D

Correct answer: 58 words per minute. A unit rate is the first quantity divided by the second:

words per minute.

• A: This divides by 60, as if the typing took a full hour instead of 30 minutes.
• B: The division is right but the decimal point is misplaced; 30 times 5.8 is only 174.
• C: This is the rate per hour (58 times 60), not the rate per minute.
• D (correct): Dividing 1,740 words by 30 minutes gives 58 words for each minute.

Answer: D

Correct answer: The 12-pack, by $0.05 per battery. Find each unit price:

per battery.

The 12-pack is cheaper by

per battery.

• A: The difference is right, but the 4-pack is the more expensive one per battery.
• B: This compares sticker prices; the 4-pack holds far fewer batteries, so price per battery must be compared.
• C: Three 4-packs cost $0.60 more than one 12-pack, but that is the saving on 12 batteries, not on each one.
• D (correct): The 4-pack costs $1.29 per battery and the 12-pack costs $1.24, so the 12-pack is cheaper by $0.05 each.

Answer: A

Correct answer: 2,340 rolls. Unit rate:

rolls per minute.

Time in minutes: 1 hour 30 minutes = 90 minutes, so the total is

rolls.

• A (correct): The rate is 26 rolls per minute, and 1 hour 30 minutes is 90 minutes, so 26 times 90 is 2,340.
• B: This multiplies the per-minute rate by 1.5 hours, mixing minutes and hours.
• C: This covers only 60 minutes and leaves out the extra 30 minutes.
• D: This multiplies 390 by 15 instead of dividing to get the per-minute rate first.

Answer: A

Correct answer: 45 minutes. Use

Convert to minutes:

• A (correct): Time is 27 divided by 36, which is 0.75 hour, and 0.75 times 60 is 45 minutes.
• B: This reads 0.75 hour as 75 minutes instead of multiplying the decimal by 60.
• C: This divides the speed by the distance, 36 by 27, which inverts the formula.
• D: The quotient 0.75 is in hours because the speed is per hour, not per minute.

Answer: C

Correct answer: 2 hours. Each pump's rate is 1 divided by its time. Add the rates:

At one-half job per hour, the whole job takes 2 hours.

• A: This stops at the combined rate of 1/2 job per hour and reads it as half an hour instead of inverting it.
• B: This averages the three times; working together must be faster than even the fastest pump alone.
• C (correct): The rates 1/4, 1/6 and 1/12 add to 1/2 job per hour, so the job takes 2 hours.
• D: This adds the times, but it is the rates that add when pumps work together.

Answer: D

Correct answer: 1:00 p.m. Head start: 9:00 to 10:30 is 1.5 hours, so the freight train leads by

Same direction, so subtract speeds:

Time to close the gap:

after 10:30 a.m., which is 1:00 p.m.

• A: The 2.5-hour chase is counted from 9:00 a.m., but the passenger train does not start until 10:30 a.m.
• B: This uses a 40-mile lead, as if the head start were 1 hour instead of 1 hour 30 minutes.
• C: This divides the 60-mile lead by the freight train's 40 mph instead of by the closing speed of 24 mph.
• D (correct): The freight train has a 60-mile lead, the gap closes at 24 mph, so the catch takes 2.5 hours after 10:30 a.m.

Answer: D

Correct answer: 1 hour 30 minutes. Outflow minus inflow gives the net rate:

per minute leaving.

which is 1 hour 30 minutes.

• A: This divides by the drain rate of 30 alone and ignores the water the hose keeps adding.
• B: This adds the hose rate to the drain rate, but the hose works against the drain.
• C: This divides by the hose rate of 6 gallons per minute instead of the net drain rate.
• D (correct): The net drain rate is 30 - 6 = 24 gallons per minute, and 2,160 divided by 24 is 90 minutes.

Answer: A

Correct answer: 8 miles. Use

Hours cancel, leaving miles.

• A (correct): Distance is rate times time: 3.2 times 2.5 is 8.
• B: This adds the rate and the time instead of multiplying them.
• C: This divides the rate by the time; distance needs multiplication.
• D: This uses only the 2 whole hours and leaves out the last half hour.

Answer: D

Correct answer: 23 miles per gallon. The word per signals division: miles per gallon =

• A: The digits are right, but the decimal point is misplaced; 9 times 2.3 is only 20.7.
• B: This multiplies miles by gallons; a per-gallon rate comes from dividing.
• C: This divides gallons by miles, which gives gallons per mile, not miles per gallon.
• D (correct): Miles per gallon is miles divided by gallons: 207 divided by 9 is 23.

Answer: D

Correct answer: 40 miles. Convert the time: 40 minutes is

so the time is

Then

• A: This writes 1 hour 40 minutes as 1.4 hours; 40 minutes is two-thirds of an hour, not 0.4.
• B: This uses only the 40 minutes and leaves out the first full hour.
• C: This multiplies miles per hour by 100 minutes without converting minutes to hours.
• D (correct): 1 hour 40 minutes is 5/3 of an hour, and 24 times 5/3 is 40.

Answer: B

Correct answer: 8 hours. Add the rates:

One-eighth of the job per hour means the job takes 8 hours.

• A: This averages the two times; a team is always faster than its faster member.
• B (correct): The rates 1/10 and 1/40 add to 5/40 = 1/8 job per hour, so the job takes 8 hours.
• C: This assumes a second worker halves the faster worker's time, which is true only if both work equally fast.
• D: This adds the times instead of adding the rates.

Answer: C

Correct answer: 11:00 a.m. In the first hour only the bus moves:

remain at 9:00 a.m.

Moving toward each other, add speeds:

Time:

after 9:00 a.m., so they meet at 11:00 a.m.

• A: The 2 hours of closing time are counted from 8:00, but the car does not start until 9:00.
• B: This divides all 245 miles by 100 mph and ignores the 45 miles the bus covered alone.
• C (correct): By 9:00 the bus has gone 45 miles, leaving 200 miles that close at 100 mph, so they meet 2 hours after 9:00.
• D: This ignores the bus's head start and also counts from 8:00 instead of 9:00.

Answer: D

Correct answer: 3 hours 30 minutes. Use

Convert the decimal part:

so the flight takes 3 hours 30 minutes.

• A: This reads 3.5 hours as 3 hours 50 minutes; the decimal part must be multiplied by 60.
• B: This reads the .5 as 5 minutes instead of half of 60 minutes.
• C: This divides the speed by the distance, which inverts the time formula.
• D (correct): 1,575 divided by 450 is 3.5 hours, and 0.5 hour is 30 minutes.

Answer: C

Correct answer: $0.12. 64-ounce bottle:

dollar per ounce.

150-ounce bottle with coupon:

dollar per ounce. The lowest price is $0.12 per ounce.

• A: This is the 150-ounce bottle's price per ounce without the coupon.
• B: This is the 64-ounce bottle's price per ounce, which is the highest of the three per-ounce prices.
• C (correct): With the coupon the 150-ounce bottle costs $18.00, and 18.00 divided by 150 is $0.12 per ounce.
• D: This adds the $1.50 to the price instead of subtracting it: 21.00 divided by 150.

Answer: D

Correct answer: 25.2 liters. Match the time unit to the rate: 1 week =

Waste =

• A: This is the waste for one day (24 hours), not a full week.
• B: This multiplies by 7 days, but the rate is per hour, so days must be converted to hours.
• C: The decimal point is misplaced; 0.15 times 168 is 25.2.
• D (correct): A week has 7 times 24 = 168 hours, and 0.15 times 168 is 25.2.

Answer: A

Correct answer: 3 hours. Time = amount divided by rate:

The rate was per minute, so the answer is in minutes; convert:

• A (correct): 8,100 divided by 45 is 180 minutes, and 180 divided by 60 is 3 hours.
• B: 180 is the time in minutes; it still has to be divided by 60 to give hours.
• C: This treats 180 minutes as 1.80 hours; minutes convert to hours by dividing by 60, not 100.
• D: The decimal is misplaced; 180 minutes divided by 60 is 3, not 30.

Answer: C

Correct answer: $4.40. Divide the cost by the number of pounds:

so the beef costs $4.40 per pound.

• A: This rounds the weight up to 4 pounds before dividing, which understates the price per pound.
• B: This multiplies the price by the weight instead of dividing.
• C (correct): Price per pound is dollars divided by pounds: 15.40 divided by 3.5 is 4.40.
• D: This divides pounds by dollars, which gives pounds per dollar, not dollars per pound.

Answer: B

Correct answer: 48 mph. Find each leg's time:

Average speed = total distance divided by total time:

• A: This averages the two speeds, but the driver spends more time at the slower speed, so that shortcut fails.
• B (correct): The trip is 240 miles in 2 + 3 = 5 hours, and 240 divided by 5 is 48.
• C: This adds the two speeds; a single driver's average speed cannot exceed the faster speed.
• D: This divides only the one-way distance of 120 miles by the total time of 5 hours.

Answer: C

Correct answer: $0.79. Unit price = total cost divided by number of units:

or $0.79 per pen.

• A: This multiplies the price by 10 instead of dividing.
• B: This divides 10 by 7.90, which gives pens per dollar, not dollars per pen.
• C (correct): Divide the cost by the number of pens: 7.90 divided by 10 is 0.79.
• D: This moves the decimal point two places instead of one when dividing by 10.

Answer: B

Correct answer: 4 hours 40 minutes. In 2 hours the painter does

of the job, leaving $\frac{2}{3}$.

Together:

job per hour, so the rest takes

or 2 hours 40 minutes.

Total: 2 hours + 2 hours 40 minutes = 4 hours 40 minutes.

• A: This is only the time the two work together; the painter's first 2 hours must be added.
• B (correct): After 2 hours one-third is done; the other two-thirds at a combined 1/4 job per hour takes 2 hours 40 minutes more.
• C: This adds 2 hours to the full 4-hour team time, ignoring the third of the job already done.
• D: This is the time if both worked together from the start, ignoring the helper's late arrival.

Answer: B

Correct answer: 22. A unit rate divides the first quantity by the second:

applications per day.

• A: This multiplies 132 by 6 instead of dividing.
• B (correct): Divide applications by days: 132 divided by 6 is 22.
• C: The decimal point is misplaced; 6 times 2.2 is only 13.2.
• D: This divides days by applications, which gives days per application.

Answer: D

Correct answer: 15 minutes. Moving toward each other around the loop, add speeds:

Together they cover the full 3-mile loop when they meet.

• A: This subtracts the speeds, which applies only when one jogger chases the other in the same direction.
• B: This reads 0.25 hour as 25 minutes instead of multiplying by 60.
• C: This multiplies the loop length by the combined speed; time requires dividing distance by speed.
• D (correct): Running in opposite directions they close the 3-mile loop at 12 mph, which takes 0.25 hour, or 15 minutes.

Answer: C

Correct answer: 900 flyers. Total = rate × time =

flyers. Minutes cancel, leaving flyers.

• A: This adds the rate and the time instead of multiplying.
• B: This divides the rate by the time; the total comes from multiplying.
• C (correct): Rate times time: 75 times 12 is 900.
• D: This adds an extra zero; 75 times 12 is 900.

Answer: A

Correct answer: 15 minutes. Combined rate:

pages per minute.

Time:

• A (correct): Together they print 68 pages per minute, and 1,020 divided by 68 is 15.
• B: This subtracts the rates; machines working on the same job add their rates.
• C: This uses the average rate of 34 pages per minute instead of the combined rate of 68.
• D: This uses only the faster copier and ignores the second machine's work.

Answer: B

Correct answer: 3 hours. Toward each other, add speeds:

• A: This subtracts the speeds; approaching objects close the gap at the sum of their speeds.
• B (correct): Walking toward each other they close the gap at 6.5 mph, and 19.5 divided by 6.5 is 3.
• C: This uses the average speed of 3.25 mph instead of the combined speed.
• D: This uses only the slower hiker's speed, as if the other hiker stood still.

Answer: B

Correct answer: 2,520 boxes. The rate is per minute, so convert the time to minutes:

Total:

boxes.

• A: This multiplies the per-minute rate by about 2.33 hours, mixing hours with a per-minute rate.
• B (correct): 2 hours 20 minutes is 140 minutes, and 18 times 140 is 2,520.
• C: This writes 2 hours 20 minutes as 2.2 hours, or 132 minutes, instead of 140 minutes.
• D: This counts only the 2 full hours and leaves out the last 20 minutes.

Answer: A

Correct answer: 48 mph. 20 minutes is one-third of an hour, so the time is $\frac{4}{3}$ hours.

• A (correct): 1 hour 20 minutes is 4/3 of an hour, and 64 divided by 4/3 is 48.
• B: This writes 1 hour 20 minutes as 1.2 hours; 20 minutes is one-third of an hour.
• C: This divides by 80 minutes, which gives miles per minute, not miles per hour.
• D: This multiplies the distance by the time instead of dividing.

Answer: B

Correct answer: 14 hours. Unit rate:

per hour.

Time:

Gallons divided by gallons per hour leaves hours.

• A: This multiplies 21 gallons by 1.5 gallons per hour; the units show this is not a time.
• B (correct): The heater burns 1.5 gallons per hour, and 21 divided by 1.5 is 14.
• C: This divides 21 by 4 hours, skipping the step of finding the burn rate.
• D: This divides 21 by 6 and forgets that the 6 gallons last 4 hours, not 1 hour.

Answer: C

Correct answer: 38 minutes. 60 miles per hour is 60 miles in 60 minutes, or 1 mile per minute.

So 38 miles takes 38 minutes. Check: $38÷60$ hour

• A: This reads 38 divided by 60, about 0.63 hour, as 63 minutes.
• B: This multiplies 38 by 60 instead of using the one-mile-per-minute pace.
• C (correct): At 60 mph the car covers 1 mile each minute, so 38 miles takes 38 minutes.
• D: This subtracts 38 from 60, which has no meaning as a time.

Answer: A

Correct answer: 1 hour. First 20 minutes:

leaving

After the leak, the net rate is

per minute, so

Total:

or 1 hour.

• A (correct): The first 20 minutes add 300 gallons; the other 480 gallons at a net 12 gallons per minute take 40 minutes, for 60 minutes in all.
• B: This uses the net rate of 12 gallons per minute for the entire fill, though the leak starts only after 20 minutes.
• C: This ignores the leak and divides 780 by the inlet rate of 15 alone.
• D: This is only the time after the leak opens; the first 20 minutes must be added.

Answer: C

Correct answer: $0.10. Ream: 500 sheets is 5 hundreds, so

per 100 sheets.

Case: 2,500 sheets is 25 hundreds, so

per 100 sheets. The case is cheaper by

per 100 sheets.

• A: This compares sticker prices, but the case holds five times as much paper.
• B: This is the saving on all 2,500 sheets compared with five reams, not the saving per 100 sheets.
• C (correct): The ream costs $1.25 per 100 sheets and the case costs $1.15 per 100 sheets, a difference of $0.10.
• D: This is the case's price per 100 sheets, not how much less it costs.

Answer: B

Correct answer: 24 minutes. Same direction, so subtract speeds:

• A: This reads 0.4 hour as 40 minutes instead of multiplying by 60.
• B (correct): The gap closes at 75 - 60 = 15 mph, and 6 divided by 15 is 0.4 hour, or 24 minutes.
• C: This adds the speeds; when one vehicle chases another, the gap closes at the difference of their speeds.
• D: This divides the gap by the police vehicle's full speed, ignoring that the car is also moving away.

Answer: A

Correct answer: (330 ÷ 55) × 60. Let the units guide the setup: miles ÷ (miles per hour) = hours, so

To express hours in minutes, multiply by 60:

• A (correct): Miles divided by miles per hour leaves hours, and multiplying hours by 60 gives minutes.
• B: This gives the time in hours, not minutes; it still needs to be multiplied by 60.
• C: Multiplying miles by miles per hour leaves miles squared per hour, which is not a time.
• D: Dividing the speed by the distance gives a rate in reciprocal hours, the inverse of the time.

Answer: A

Correct answer: 7 hours 30 minutes. Inflow adds and outflow subtracts:

At two-fifteenths of a tank per hour, the fill takes

or 7 hours 30 minutes.

• A (correct): The net rate is 1/3 - 1/5 = 2/15 tank per hour, so filling takes 15/2 = 7.5 hours.
• B: This adds the two rates, treating the drain as if it also filled the tank.
• C: This subtracts the times, 5 - 3; for opposing pipes it is the rates that subtract.
• D: This averages the two times, which does not account for the drain working against the inlet.