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Rates and Unit Rates

Rates connect two quantities, such as miles per hour, dollars per pound or parts per minute, and they drive a large share of Arithmetic Reasoning word problems. This 34-minute core lesson explains what a rate means, how to find unit rates, and how distance, rate and time relate. Sections on converting time units and on finding distance, rate and time separately build the basics.

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Video lesson: Rates and Unit Rates

Video transcript

0:00 Okay, let's just dive right into this. If you're prepping for the ASVAB, you already know that the arithmetic reasoning section is, well, it's a pretty big deal. Today, we're zeroing in on rates and unit rates. And honestly, the best part, we're going to do the whole thing without touching a calculator. We'll build up those exact mental math muscles you need to completely ace these questions with total confidence. We've got a really clear, straightforward roadmap for this explainer. We'll start with the bare-bones basics, move into some real-world stuff, tackle combined work, and then get into the really good stuff. Mental math shortcuts and actual step-by-step examples, before wrapping up with some practice questions solved entirely by hand. All right, section one, basic rates and formulas. We've got to set the foundation first. So what is a unit rate? It simply tells us how much of something happens in just one single minute, one hour or, you know, one ounce. Finding that per one number is basically the secret key that unlocks almost every problem you'll run into. For example, if a pump moves 450 gallons in 15 minutes, you just divide 450 by 15. A super-quick mental math trick here, 45 divided by 15 is 3, right?

1:10 So just slap that zero back on the end and you've got 30 gallons per minute. And here are your core motion formulas. Distance equals rate times time. It's a classic for a reason, showing exactly how these three pieces fit perfectly together. You can easily shuffle this equation around depending on what you actually need to find. Need the rate? Just divide distance by time. Looking for the time? Divide distance by the rate. Just make absolutely sure your units always match up. Moving on to section two, let's see how we apply these rates to real contexts, like grocery shopping and production. This scenario is a classic ASVAB trap. To find the better buy, you can't just look at the overall sticker price. You have to compute the price of one single unit. Take a six-pack for $4.20. Mentally, just think, okay, 42 divided by 6 is 7, so that's 70 cents a bottle. But a 10-pack for $6.50, that's an easy 65 cents a bottle. The 10 pack is actually the cheaper deal per unit, even though it costs a little more up front. Okay, section three, combined work and relative motion. This is where things get kind of fun, because we're dealing with multiple things moving at the exact same time.

2:16 The rule of thumb here is really all about context. When things are helping each other out, like two hoses filling up a pool, or when two cars are driving in opposite directions, and that gap between them is opening or closing super fast, you add their rates together. But if they're working against each other, say, a leak draining that same pool, or one car chasing another down the highway, you subtract the rates. Section four, error checks and shortcuts. Since you absolutely won't have a calculator on test day, let's get into some mental math mastery. Memorizing these simple fractions of an hour is a total game changer. It's going to save you so much time and keep you from falling into the trap of multiplying by raw minutes. Let's say you drive 40 miles an hour for 30 minutes. If you multiply by a fraction, 0.5 hours, you get the right answer, 20 miles. But if you panic and just multiply 40 by 30 minutes, you'd get 1,200 miles, which obviously is a wildly wrong answer. All right, section five, step by step, worked examples. Let's put all of this into practice with the exact kinds of questions you're going to see.

3:21 First up, finding distance. Let's say a tour bus travels at 58 miles per hour for three hours and 30 minutes. Step one, convert that 30 minutes into a decimal, so 3.5 hours. Now you can multiply, 58 times 3.5. Doing this in your head is actually pretty easy if you break it into two steps. First, 58 times 3 is 174, then just find half of 58, which is 29. And 174 and 29 together, and boom, 203 miles. Finding the rate of a runner uses the exact same logic. They cover 13.5 kilometers in 1.5 hours, so distance divided by time. Now dividing 13.5 by 1.5 looks messy, right? But here's a brilliant little mental math trick. Just double both numbers in your head to ditch the decimals. Suddenly, you're just doing 27 divided by 3, which gives us a nice, clean, 9 kilometers per hour. Number three is all about finding the best buy. Brand X is $3.36 for 24 ounces, and Brand Y is $4.16 for 32 ounces. We just find the unit price for both.

4:27 The math shows us Brand Y is 13 cents an ounce, making it a cheaper option compared to Brand X at 14 cents an ounce. So always run the division to be sure. For production problems, like a press stamping 135 brackets in 9 minutes, you always want to find your per-minute rate first. You need 135 by 9. Break it apart mentally. 90 divided by 9 is 10, and 45 divided by 9 is 5. Add those up, and you're stamping 15 brackets a minute. To find out how many are stamped in 40 minutes, just multiply that 15 by 40, which scales up to 600 brackets. Now combined work. One crew clears a lot in 10 hours. Another takes 15. When they work together, you add their fractional rates. So one-tenth plus one-fifteenth. Find a common denominator, which is 30. It gives us 3/30 plus 2/30, which equals 5/30. Reduce that down, and it's one-sixth. This means they finish one-sixth of the job every single hour, so the whole job takes them six hours total. In our final worked example, let's look at opposing rates. Two cyclists start 66 miles apart, and ride right toward each other at 12 and 10 miles per hour.

5:36 Because they're heading toward each other, that gap is shrinking fast. So we add their speeds together to get 22 miles per hour. Then just divide the total distance of 66 by that combined speed of 22, and you get exactly three hours. Section six, we've got some common traps and a little practice. It's time to test your skills. But before we do, keep your eyes peeled for these common traps. The absolute biggest one, mixed units. Forgetting to convert your minutes to hours will literally make your answer 60 times too large. Also, never ever average the times in combined work problems. Always add those fractions. Oh, and in fill and drain scenarios, always make sure you use the net rate, which is just the inflow minus the outflow. Okay, check yourself on this one. Question one, a pump moves 840 gallons of water in 35 minutes. What is its rate in gallons per minute? I'm going to pause for a few seconds so you can mentally work this out. All right, if you picked D, 24 gallons per minute, you nailed it. You just divide the 840 total gallons by the 35 minutes to find the unit rate.

6:41 And hey, if dividing by 35 in your head sounded awful, remember to simplify first. Divide both numbers by seven. That turns it into 120 divided by five, which is a much, much easier 24. Let's try one more. Question two, a delivery truck travels at 52 miles per hour for two hours and 45 minutes. How far does it travel? Take a second to solve this one. The correct answer here is A. To dodge the time conversion trap, you have to convert that 45 minutes into three quarters of an hour or 0.75. So your total time is 2.75 hours. Then multiply 52 by 2.75. In your head, just do 52 times two, which is 104. Then find three quarters of 52, which is 39. Add 104 and 39 together, and you've got your total distance of 143 miles. You see, math really is just logic. Mental math and step-by-step thinking are way more reliable than reaching for a calculator anyway. You've got the foundation down now. But you are totally ready for this part of the test. Keep practicing, trust your logic, and I will catch you in the next explainer.

Visual lesson

Lesson slides

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Lesson Goal

By the end of this lesson you will be able to read a rate as a comparison of two different units, reduce it to a unit rate, and use it to find distance, speed or time. You will also compare package prices, scale up production, combine workers or pipes, and handle objects that move apart or toward each other.

What a Rate Means

A rate compares two quantities measured in different units, such as miles and hours, dollars and pounds, or gallons and minutes. The word "per" is the signal. If a pump moves 450 gallons in 15 minutes, its rate is 450 gallons per 15 minutes. A rate lets you predict how much will happen over a different amount of time, as long as the pace stays steady.

Unit Rates

A unit rate tells how much of the first quantity goes with exactly one unit of the second. Divide the first quantity by the second. The pump that moves 450 gallons in 15 minutes has a unit rate of

450 \div 15 = 30 gallons

per minute. Multiply that unit rate by any number of minutes to predict the total.

Double number line: 450 gallons in 15 minutes is 30 gallons in 1 minute.

Distance, Rate, and Time

Motion problems rest on one relationship: distance equals rate times time,

d = r \times t

Rearranged, it gives

r = d \div t and t = d \div r

Write the formula, fill in the two known values, and solve for the third. Keep units matched: miles, miles per hour and hours.

d-r-t triangle: d = r × t, r = d ÷ t, t = d ÷ r.

Converting Time Units

Speeds are usually stated per hour, but problems often give time in minutes. Divide minutes by 60 to get hours: 45 minutes is 0.75 hour and 20 minutes is one-third of an hour. A mixed time such as 1 hour 30 minutes becomes 1.5 hours. Converting first avoids errors like multiplying 40 mph by 30 minutes and getting 1,200 miles instead of the correct 20 miles.

Hour bars of 60 minutes: 45 min shaded = 45/60 = 0.75 h; 20 min shaded = one of three equal thirds = 1/3 h; 1 h 30 min = 1.5 h; so 40 mph × 0.5 h = 20 mi, not 40 × 30 = 1,200 mi.

Finding Distance

When rate and time are known, multiply them. A cyclist riding at 14 mph for 3.5 hours covers

14 \times 3.5 = 49 miles

The hours cancel and miles remain. A quick estimate, 14 times 3 is 42, confirms the answer is in the right range.

Finding Rate

To find a rate, divide the distance by the time. A delivery van covers 186 miles in 3 hours, so its average rate is

186 \div 3 = 62 mph

If the time is not a whole number, write it as a decimal first: 99 miles in 1 hour 30 minutes gives

99 \div 1.5 = 66 mph

Finding Time

To find time, divide distance by rate.

t = \frac{d}{r}

A boat crossing 36 miles of lake at 24 mph needs

36 \div 24 = 1.5 hours

or 1 hour 30 minutes. Multiply any decimal part of an hour by 60 to express it in minutes.

Comparing Unit Prices

To find the better buy, compute the price of one unit in each package and compare. A 6-pack of juice for 4.20 dollars costs 0.70 dollar per bottle. A 10-pack for 6.50 dollars costs 0.65 dollar per bottle. The 10-pack is cheaper per bottle even though its total price is higher. Compare cost per unit, never the sticker price alone.

6-pack at $0.70 per bottle versus 10-pack at $0.65 per bottle.

Production and Consumption Rates

Machines produce, and engines consume, at steady rates. A bottling line fills 1,200 bottles in 8 minutes, which is 150 bottles per minute, so in 20 minutes it fills 3,000. A generator that burns 2.5 gallons of fuel per hour uses 15 gallons in 6 hours. Find the unit rate first, then scale it.

Rate × Time in Real Contexts

The rate times time idea fits many settings. A clerk who processes 18 forms per hour completes 144 forms in an 8-hour shift. A faucet dripping 0.2 liter per hour wastes 4.8 liters in a day. A printer running 25 pages per minute needs 12 minutes for 300 pages. Identify the rate, the time, and what is being counted.

Combined Rates

When two sources work on the same task at the same time, add their rates. Two hoses delivering 8 and 5 gallons per minute together supply 13 gallons per minute, so 117 gallons takes

117 \div 13 = 9 minutes

If one source works against the other, subtract instead of adding.

Two hoses pour down into the open top of a tank holding 117 gallons: hose A 8 gal/min and hose B 5 gal/min, together 13 gal/min; 117 ÷ 13 = 9 min.

Combined Work With Whole-Job Rates

If a worker finishes a job in h hours, the rate is $\frac{1}{h}$ job per hour. Painters who need 4 hours and 12 hours alone have a combined rate of

\frac{1}{4} + \frac{1}{12} = \frac{1}{3}

so together they finish in 3 hours.

Each hour the painters finish 3/12 + 1/12 = 1/3 of the job, so 3 hours in all.

Closing and Filling Problems

In fill-and-drain problems, inflow adds and outflow subtracts. An inlet pours in 20 liters per minute while a leak loses 4 liters per minute, so the tank gains a net 16 liters per minute. Gaining 400 liters therefore takes

400 \div 16 = 25 minutes

Always divide by the net rate, not the inflow alone.

Tank with an inlet pouring in 20 L/min and a leak losing 4 L/min; a dashed target level 400 L above the water marks the gain; net gain 16 L/min, 20 − 4 = 16 and 400 ÷ 16 = 25 min.

Relative Motion Basics

When two objects move, focus on how fast the gap between them changes. If they move in opposite directions or toward each other, add their speeds: trains at 70 mph and 55 mph heading apart separate at 125 mph. If they move in the same direction, subtract: a car at 65 mph gains on a truck at 50 mph at 15 mph. Divide the gap by that combined speed to get the time.

Trains moving apart at 55 and 70 mph separate at 125 mph; a 65 mph car gains on a 50 mph truck at 15 mph.

Use Units as an Error Check

Attach units to every number and let them cancel like factors. Miles per hour times hours leaves miles, so that product is a distance. Miles divided by miles per hour leaves hours, a time. If your setup produces miles times hours, or hours per mile when you wanted speed, you multiplied where you should have divided. This quick check removes many wrong choices before you commit.

No-Calculator Rate Shortcuts

Simplify before you divide. For 252 miles in 4.5 hours, double both numbers to clear the decimal:

504 \div 9 = 56 mph

Learn easy fractions of an hour: 15 minutes is one-quarter, 20 minutes one-third, 40 minutes two-thirds. At 60 mph, minutes and miles are equal. Estimate first so you can discard choices that are far off.

Worked Example 1 — Distance

Problem: A tour bus travels at 58 mph for 3 hours 30 minutes. How far does it go?

Bus at 58 mph for 3 h 30 min: three 58-mile hours plus a half hour; total distance = ?

Step 1: The speed is in miles per hour, so write the time in hours. 30 minutes is half an hour.

3 h 30 min = 3.5 h

Step 2: Use distance = rate × time.

58 \times 3.5

Step 3: Break it into easy pieces: 3 hours plus half an hour.

58 \times 3 = 174

58 \times 0.5 = 29

174 + 29 = 203

Answer: The bus travels 203 miles.

Check: Estimate with 60 mph:

60 \times 3.5 = 210 miles

The real speed is a little less, so 203 miles is reasonable.

Worked Example 2 — Rate

Problem: A runner covers 13.5 kilometers in 1.5 hours. What is her average speed?

Step 1: Speed (rate) is distance divided by time.

r = 13.5 \div 1.5

Step 2: Get rid of the decimals by multiplying both numbers by 10. This does not change the answer.

13.5 \div 1.5 = 135 \div 15

Step 3: Divide.

15 \times 9 = 135

135 \div 15 = 9

Answer: Her average speed is 9 km per hour.

Check: Multiply back: 9 km per hour × 1.5 hours =

9 + 4.5 = 13.5 km

That matches the distance.

Worked Example 3 — Best Buy

Problem: Brand X sells 24 ounces for 3.36 dollars and Brand Y sells 32 ounces for 4.16 dollars. Which is the better buy?

Step 1: Find each brand's unit price: cost divided by ounces. Brand X:

$3.36 \div 24 = $0.14

(

24 \times 0.10 = 2.40

and the 0.96 left over is 24 × 0.04, so

0.10 + 0.04 = 0.14

)

Step 2: Brand Y:

$4.16 \div 32 = $0.13

(

32 \times 0.10 = 3.20

and the 0.96 left over is 32 × 0.03, so

0.10 + 0.03 = 0.13

)

Step 3: Compare. Brand X costs 14 cents per ounce and Brand Y costs 13 cents per ounce. Lower is better.

Answer: Brand Y is the better buy.

Check: Multiply back:

24 \times 0.14 = 3.36 and 32 \times 0.13 = 4.16

Both match the prices.

Worked Example 4 — Production

Problem: A press stamps 135 brackets in 9 minutes. How many brackets does it stamp in 40 minutes?

Step 1: Find the unit rate, the number of brackets in one minute.

135 \div 9 = 15

(

9 \times 15 = 135

) So the press stamps 15 brackets per minute.

Step 2: Multiply the unit rate by the new time.

15 \times 40 = 600

(

15 \times 4 = 60

then add a zero.)

Answer: The press stamps 600 brackets in 40 minutes.

Check: 40 minutes is a bit more than 4 times 9 minutes (36), so the answer should be a bit more than

4 \times 135 = 540

600 fits.

Worked Example 5 — Combined Work

Problem: One crew can clear a lot in 10 hours and a second crew in 15 hours. How long does it take if they work together?

Step 1: Write each crew's rate as the fraction of the job done in one hour. The first crew does $\frac{1}{10}$ of the job per hour; the second does $\frac{1}{15}$ .

Step 2: Add the rates. Use the common denominator 30, since 10 and 15 both divide into 30.

\frac{1}{10} = \frac{3}{30}, \frac{1}{15} = \frac{2}{30}

A job bar of 30 thirtieths: crew 1 does 1/10 = 3/30 per hour and crew 2 does 1/15 = 2/30 per hour, shaded; time together = ?.

\frac{3}{30} + \frac{2}{30} = \frac{5}{30}

Step 3: Simplify by dividing top and bottom by 5.

\frac{5}{30} = \frac{1}{6}

Step 4: Together they finish one-sixth of the job each hour, so the whole job takes 6 hours.

Answer: Working together, the crews take 6 hours.

Check: In 6 hours the first crew does 6/10 and the second does

6/15 = 4/10

Together

6/10 + 4/10 = 10/10

one whole job. Also, 6 hours is faster than the quicker crew alone (10 hours), as it should be.

Worked Example 6 — Opposing Rates

Problem: Two cyclists start 66 miles apart and ride toward each other at 12 mph and 10 mph. When do they meet?

Two cyclists 66 miles apart riding toward each other at 12 mph and 10 mph; meeting point = ? and time to meet = ?.

Step 1: Riding toward each other, both close the gap at once, so add the speeds.

12 + 10 = 22 mph

Step 2: Time = distance ÷ rate. Divide the gap by the closing speed.

66 \div 22 = 3 hours

Answer: They meet after 3 hours.

Check: In 3 hours, the first rides

12 \times 3 = 36 miles

and the second rides

10 \times 3 = 30 miles

36 + 30 = 66 miles

Together they cover the full 66 miles.

Cyclists 66 miles apart at 12 and 10 mph meet after 3 hours, 36 and 30 miles from their starts.

Common Rate Problems

Mixed units: multiplying a speed in miles per hour by a time in minutes makes the answer 60 times too large. Convert minutes to hours first.

Averaging times in combined work: workers needing 6 and 3 hours do not need 4.5 hours together. Add their rates instead; a team is always faster than its fastest member.

Comparing total prices: a bigger package can cost more yet be cheaper per unit. Divide to get unit prices.

Wrong relative speed: subtract speeds for same-direction travel, add them for opposite or approaching travel.

Ignoring a drain: use the net rate, inflow minus outflow.

Quick Review

A rate compares two different units; a unit rate gives the amount for one unit and comes from division. For motion,

d = r \times t r = d \div t and t = d \div r

with time in hours when speed is per hour. Compare deals by price per unit. Scale production by multiplying the unit rate by time. For workers on one job, add rates of $\frac{1}{h}$ and invert the sum. For fill and drain, use the net rate. Add speeds when the gap opens or closes from both sides, subtract when one object chases another, and check units at the end.

Quick review

What is a unit rate and how do you find it?: The amount of the first quantity for exactly one unit of the second; divide the first by the second. 450 gallons in 15 minutes is $450 \div 15 = 30$ gallons per minute.
What is the distance, rate and time formula with its rearrangements?: $d = r \times t$ , $r = \frac{d}{t}$ , $t = \frac{d}{r}$ , where d is distance (miles), r is rate (miles per hour) and t is time (hours). Keep the units matched.
A car goes 40 mph for 30 minutes. How far does it travel?: 20 miles. Convert minutes to hours first (30 minutes = 0.5 hour). Multiplying 40 by 30 gives 1,200, which is 60 times too large.
How do you decide which package is the better buy?: Compare the price per unit, not the sticker price. 6 bottles for 4.20 dollars is 0.70 dollar each; 10 for 6.50 dollars is 0.65 dollar each, so the bigger pack is cheaper per bottle.
Two painters need 4 hours and 12 hours alone. How long together?: Add rates of $\frac{1}{h}$ job per hour: $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$ , so 3 hours. A team is always faster than its fastest member.
An inlet adds 20 L/min and a leak loses 4 L/min. Time to gain 400 L?: Use the net rate: $20 - 4 = 16$ liters per minute, so $400 \div 16 = 25$ minutes. Dividing by the inflow alone is the trap.
In relative motion, when do you add speeds and when do you subtract?: Add when the objects move toward each other or apart in opposite directions; subtract when one follows the other in the same direction. Then time = gap divided by that combined speed.
How can units reveal a setup error in a rate problem?: Let units cancel like factors: miles per hour times hours leaves miles; miles divided by miles per hour leaves hours. A result in miles times hours, or hours per mile for a speed, means you mixed up multiply and divide.

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About this lesson

You then apply rates to comparing unit prices, production and consumption rates, and rate times time in real contexts. The lesson introduces combined rates, combined work with whole-job rates, closing and filling problems and relative motion basics. Using units as an error check and a set of no-calculator rate shortcuts help you work quickly on this AFQT subtest.

Six worked examples cover distance, rate, a cheaper-buy comparison, production, combined work and opposing rates. Finish with the common rate problems and the Quick Review, then take the Rates and Unit Rates lesson test.

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