Use rates and unit rates

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Difficulty: Beginner
Estimated time: 37 minutes
Domains: Problem Solving and Data Analysis
Techniques: Unit-ratesUnit-orderConstant-rateUnit-priceAverage-rate

What you’ll learn

Tell a ratio, a rate and a unit rate apart.
Write a rate in the order the question asks for.
Find a rate for one unit and say what it means.
Use a constant rate to find an amount or how long something takes.
Compare choices by their unit rates, in the same units.
Find an average rate from the total amount and the total time.
Decide, once the setup is right, whether to do the arithmetic by hand or type it straight into Desmos.

Prerequisites

Reason with ratios and proportions0%
Keep the comparison order and scale equivalent relationships.

Why this matters on the SAT

Let the units point to the operation

SAT rate questions show up in travel, factories, prices, density and population. The question might ask for an amount, a time or a rate for one unit, like the cost of one ticket. Before you calculate, write down the units the answer should have, in order. That tells you what goes on the top of the fraction and what goes on the bottom.

Solution to the example

“Readings per minute” means readings divided by minutes:

\frac{936\text{ readings}}{8\text{ minutes}}.

Type 936/8 into Desmos and you get $117$ , so the answer is B. That’s $117$ readings per minute. It’s a unit rate, because the bottom of the fraction is one minute.

SAT example

A research sensor records $936$ readings in $8$ minutes at a constant rate. What is the sensor's rate, in readings per minute?

A
$8$
B
$117$
C
$928$
D
$944$

Keep the rate in the order the question asks for

In ratios and proportions, a comparison had to keep the same order all the way through. Rates work the same way.

A ratio is any comparison by division. A rate is a ratio between quantities with different units, like dollars and tickets. A unit rate is a rate for one unit of whatever’s on the bottom, like dollars for one ticket.

Say $54$ dollars buys $6$ tickets. You can divide in either order:

\frac{54\text{ dollars}}{6\text{ tickets}} =9\frac{\text{dollars}}{\text{ticket}},

\frac{6\text{ tickets}}{54\text{ dollars}} =\frac19\frac{\text{ticket}}{\text{dollar}}.

The first is the cost per ticket. The second, its reciprocal, is the number of tickets per dollar. Both are correct math, but only one of them answers a given question.

So how do you know which order you need? Look for the word per. It works like the fraction bar:

\text{pages per minute} = \frac{\text{pages}}{\text{minutes}}.

The unit before “per” goes on top, in the numerator. The unit after “per” goes on the bottom, in the denominator.

Separate the setup from the arithmetic

Every rate problem has two parts, the setup and the arithmetic. Use this hybrid rule throughout the lesson:

Work out the setup yourself: the unit order, whether to multiply or divide, and when each part happens if the timing matters.
If the arithmetic is quick, like $54\div6$ , finish it by hand.
If it isn’t, type the whole expression into Desmos at once. Then attach the unit and say what the number means.

Say a filter processes $18$ liters every $5$ minutes, and you want the time for $73.8$ liters. The time is

\frac{73.8}{18/5}

minutes. Type 73.8/(18/5) as one expression. Desmos returns $20.5$ , so it takes $20.5$ minutes. Desmos did the arithmetic, but the units told you to divide and what the answer means.

Here are some rates you’ll often see on the SAT:

dollars per item, a unit price
miles per hour, a speed
liters per minute, a flow rate
grams per cubic centimeter, a density
people per square kilometer, a population density

Check your understanding:

A machine uses $42$ kilowatt-hours of energy to make $700$ parts. Write (1) the energy used per part and (2) the parts made per kilowatt-hour. Include units.

Common mistake:

It’s tempting to divide the numbers in the order they appear in the problem. If that isn’t the order the question asks for, you get the reciprocal. Write the target units as a fraction first, put each number next to its unit, and check that your final unit reads exactly like the question.

Spot a rate question

You’re working with a rate when a question looks like one of these:

It asks “how many per one,” “for each” or “at this rate.”
It gives an amount and the time it took, like $936$ readings in $8$ minutes, and asks for the rate.
It gives a steady rate, like $55$ kilometers per hour, and asks for an amount, like a distance, or a time.
It asks which package, plan or machine gives the better value per unit.
It asks for the average speed, or another average rate, over a whole interval, like a full trip or work period.

Two kinds of question can look like this but work differently:

$3$ red tiles for every $5$ blue tiles compares two parts with the same unit, tiles. That’s a ratio, and Reason with ratios and proportions shows how to work it.
If the main job is changing units with conversion factors, especially through a chain of units or with squared and cubed units, that’s a unit conversion, and Convert units with dimensional analysis shows how.

One quick time change doesn’t stop a question from being about a rate. In the shuttle example below, you’ll write a $30$ -minute rest as $\frac{30}{60}$ hour, and everything else is rate work.

Check your understanding:

Which of these is a rate question? (1) Compare the cost per ounce of two packages. (2) Convert $4.2$ cubic meters to cubic centimeters. (3) Use a $2:7$ ratio and a total of $270$ objects to find one group.

Calculator loads as you approach

Liters divided by liters per minute leaves minutes, so the time is 20.5 minutes.

Use a constant rate in either direction

Once you know a constant rate, you can use it to find an amount or a time. The units tell you whether to multiply or divide.

Say a pump moves $7.5$ liters per minute for $12$ minutes. Multiply:

7.5\frac{\text{liters}}{\text{minute}} (12\text{ minutes}) =90\text{ liters}.

The minutes cancel, and liters are left.

Now say a vehicle travels $165$ kilometers at $55$ kilometers per hour. To find the time, divide:

\frac{165\text{ kilometers}} {55\frac{\text{kilometers}}{\text{hour}}} =3\text{ hours}.

Dividing kilometers by kilometers per hour leaves hours.

Here’s the same pattern with letters. Call the quantity before “per” $A$ , since it goes on top, and the one after “per” $B$ , since it goes on the bottom. If $r$ is the constant rate, then

r=\frac{A}{B}.

Rearrange that, and you can find either quantity:

\boxed{A=rB}

and

\boxed{B=\frac{A}{r}}.

So divide $A$ by $B$ to get the rate. Multiply the rate by an amount of $B$ to get $A$ , as with the pump. Divide $A$ by the rate to get $B$ , as with the trip.

Check your understanding:

A filter processes $22$ liters every $4$ minutes at a constant rate. Find (1) the unit rate in liters per minute and (2) the time it takes to process $82.5$ liters.

Try it yourself:

Stuck between multiplying and dividing? Cover the numbers and look only at the units. Liters per minute and minutes? Multiply, and the minutes cancel. Liters and liters per minute? Divide, and minutes are left.

Compare choices using one common unit

To compare two rates fairly, put both in the same unit order. For prices, that usually means the unit price:

\text{unit price} = \frac{\text{total price}}{\text{number of units}}.

If one package is measured in kilograms and the other in grams, pick one unit for both first. If both are already in kilograms, divide each price by its number of kilograms.

With friendly numbers, go by hand. $36 for $12$ kilograms is $3 per kilogram, and opening a calculator would only add a step. Awkward decimals, like the ones below, are a different story.

Supplier A charges $34.65 for $18$ kilograms of material. Supplier B charges $52.80 for $30$ kilograms. A workshop needs $275$ kilograms.

The question is how much the workshop saves on $275$ kilograms, so start with dollars per kilogram for each supplier. The savings is the difference in unit price times $275$ . Those quotients are messy, and the answer needs rounding, so type the whole thing into Desmos at once. That way nothing gets rounded until the end:

\boxed{275\left(\frac{34.65}{18}-\frac{52.80}{30}\right)}

Desmos returns $45.375$ . It’s positive, so Supplier A costs $45.375 more. The workshop saves $45.38 by choosing Supplier B, to the nearest cent.

Notice what Desmos couldn’t do for you. It didn’t pick dollars per kilogram, decide which price to subtract from which, or say which supplier is cheaper. Those calls were yours, and you made them before typing anything.

Related: When a calculation really does take several steps, Build ratio, rate, and unit chains in Desmos gives more practice. A one-step unit rate like $36\div12$ doesn’t need it.

Check your understanding:

A third supplier charges $28.50 for $15$ kilograms. Is its unit price lower than Supplier B’s $1.76 per kilogram? Show the comparison.

Calculator loads as you approach

The whole savings in one expression:

45.375

dollars, before rounding to the nearest cent.

Common mistake:

Comparing total prices alone can make a bigger package look worse only because it holds more. Divide each price by its own quantity, use the same units for both, and then compare. Check the direction too: for cost per unit, the smaller positive rate is the better price.

Example: Find an average rate over the whole trip

Try this one yourself first. Two of the wrong choices are very tempting.

Worked example

A shuttle travels $60$ miles at a constant speed of $40$ miles per hour. It then rests for $30$ minutes before returning along the same $60$ -mile route at $60$ miles per hour. What is the shuttle's average speed for the entire trip?

A
$36$ miles per hour
B
$40$ miles per hour
C
$48$ miles per hour
D
$50$ miles per hour

Step 1

Add up the whole distance

Average speed is total distance divided by total time. The shuttle goes $60$ miles each way, so

\text{total distance}=60+60=120\text{ miles}.

Step 2

Find every piece of the time

The time comes in three pieces: the drive out, the rest and the drive back. The rest is given in minutes, so write it in hours, as $\frac{30}{60}$ hour:

\frac{60}{40}\text{ hours}, \qquad \frac{30}{60}\text{ hour}, \qquad \frac{60}{60}\text{ hour}.

The question asks about the entire trip, so the rest counts. Leave the pieces as they are. They all go into one expression in the next step.

Step 3

Divide total by total

Average speed is miles per hour, so it’s total miles over total hours. Type the whole quotient into Desmos:

\text{average speed} =\frac{60+60}{60/40+30/60+60/60} =40\text{ miles per hour}.

The answer is B. Choice C leaves out the rest, so it’s the average speed only while the shuttle is moving. Choice D takes the simple mean of $40$ and $60$ , but the shuttle spends different amounts of time at those speeds, and it also rests.

Check your understanding:

A machine makes $80$ parts in its first $4$ hours and $180$ parts in its next $6$ hours. What is its average production rate over all $10$ hours? Why isn’t averaging the two separate rates reliable?

Common mistake:

Averaging the listed speeds or rates ignores how long each one lasted. Go back to total over total: total amount divided by total time, or by whatever the rate is “per.” Count every interval that words like “entire trip” include, like a rest. Then check that your final unit matches the one the question asks for.

Practice problems

Before you touch the numbers in each problem, write the units the answer needs as a fraction.

Use a unit rate

Practice problem

A laboratory dispenser releases $18$ milliliters of solution every $3$ seconds at a constant rate. How many milliliters of solution does it release in $25$ seconds?

Calculator loads as you approach

Find milliliters per second first.

18\div3

and

6\times25

are both quick by hand, so the calculator is only for a check.

Compare two unit prices

Practice problem

Data plan A costs $42 for $16$ gigabytes of data. Data plan B costs $67.50 for $25$ gigabytes of data.

How much less, in dollars per gigabyte, is the lower unit price than the higher unit price?

Calculator loads as you approach

Find dollars per gigabyte for each plan, then subtract in one expression.

Two machines, two running times

Practice problem

Inspection machine A checks $35$ components every $14$ minutes at a constant rate. Inspection machine B checks $48$ components every $16$ minutes at a constant rate.

Machine A begins operating. Machine B begins $20$ minutes later. Both machines stop $1$ hour $20$ minutes after machine A begins.

How many components do the two machines check in total?

Your answer

Calculator loads as you approach

Start with how long each machine runs.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A rate compares quantities with different units. A unit rate is that comparison for one unit of the bottom quantity.
Write the units the question wants first. The unit before “per” goes on top.
If $r=\frac{A}{B}$ , then $A=rB$ finds the top quantity and $B=\frac{A}{r}$ finds the bottom one.
Compare choices only after putting their rates in the same units.
Average rate is total over total, not the simple mean of the listed rates.
Settle the units, the operation and the timeline yourself. Then do quick arithmetic by hand, and type anything messier into Desmos as one expression and attach its unit.

Related lessons

To review same-unit part comparisons and equivalent ratios, go back to Reason with ratios and proportions. Later, Find slope and rate of change connects the change in one variable for each unit of change in another to tables, graphs and linear equations.

Next lesson

Convert units with dimensional analysis

Arrange one or more conversion factors so unwanted units cancel.

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