3 Rates
A ☞ rate ☜ is a quantity. The words “rate” and “ratio” are strongly related; a rate is the ratio of two quantities, that is, one quantity divided by another. This simple definition speaks to the how of creating a rate, but is silent about the when and why such a ratio helps gain quantitative insight. This chapter introduces a handful of common uses of rates. Paying attention to the appearance or absence of rates in quantitative arguments is a key skill for quantitative reasoning.
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3.1 Example: Energy for walking
Consider the energetics of walking, how much energy we consume while walking. An AI query 1 about the energetics of human walking produced this result:
Walking a mile burns approximately 60 to 100 calories2, but this varies based on individual weight, pace, terrain, and fitness level. Heavier individuals burn more calories due to the increased energy needed to move their bodies, while faster paces and inclines require more exertion, boosting calorie expenditure. More fit individuals tend to be more efficient, burning fewer calories at a given activity level than less fit individuals.
The first sentence from the AI response is equivalent to, “Walking burns approximately 60 to 100 Calories per mile.” The word ☞ per ☜ signals, as usual, a division of one quantity by another: 60-100 Cal divided by one mile.
Like all rates, 60 Cal per mile is one quantity divided by another quantity. The colloquial English form, “a one-mile walk burns 60 Cals,” is a fine way to express this rate, but we prefer the per form in this book to signal explicitly which quantity is on the bottom of the ratio. More subtly, “60 Cal per mile” emphasizes that we are talking about walking rather than the characteristics of a particular one-mile trail.
The rate 60 Cal per mile is, like all other rates, a quantity. All quantities have a dimension. The dimension of 60 Cal/mile is [energy] / L, that is, M L T-2. (Remember that “[energy]” is pronounced, “the dimension of energy.) In this case, the dimension of the rate is different from the dimensions of either of the quantities that formed the rate.
Here is another rate that describes the same situation of human walking energetics: “a mile per 60 Cal.” In fraction notation we could write this as \[\frac{\text{1 mile}}{\text{60 Cal}},\ \text{which is the same as}\ \ \frac{1}{60}\frac{\text{mile}}{\text{Cal}} = 0.01667\ \frac{\text{mile}}{\text{Cal}}\] Whether to write the numerical part of the rate in quotient or in decimal form is a matter of style.
Notice that even though the rates 60 Cal per mile and 0.01667 mile per Cal describe the same situation, they are not just different formats; they are entirely different quantities. First, the dimension differs: Cal-per-mile (L M T-2 ) versus mile-per-Cal (L^-1 M-1 T2 ).
Which of the two rates to prefer? The answer depends on the context. Here is one possible context:
Suppose you have just come home from a 3-mile walk and want to quench your thirst with a can of soda. In walking 3 miles, you (or, more precisely, the model person described by 60 Cal/mile) used up 180 Cal, that is \[60\ \frac{\text{Cal}}{\text{mile}} \times 3\ \text{miles} = 180 \ \text{Cal}\].
Now consider this different context:
Last night your friend ate a lot of pasta—3000 Cal’s worth—loading up for her participation in a walking marathon. How far will this get her towards the total distance of 26.219 miles? You want to know how many calories the runners will use up. This calculation involves the other rate: \[0.01667\ \frac{\text{mile}}{\text{Cal}} \times 3000\ \text{Cal} = 50\ \text{miles}, \] more than enough!3
You can also frame the calculation another way; how many Cals will propel you the whole distance, 26.219 miles. The end result is to be in Cals, so we need a rate that will transform 26.219 miles into Cals. Here we use the form of the rate with Cals on top and miles on the bottom: 60 Cals per mile. Multiplying the distance times the appropriate rate gives: \[26.219\ \text{miles} \times 60\ \frac{\text{Cals}}{\text{mile}} = 1573.1\ \text{Cals},\]
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Often, a quantity is known only approximately. In the AI answer quoted above, a ☞ range ☜ represents the uncertainty: “60 to 100 Calories per mile.” A range is a pair of quantities. To make sense, the two quantities in the range must have the same dimension. “60 to 100 Calories per mile” is shorthand for “somewhere between 60 Cal/mi and 100 Cal/mi.
In Chapter 7, we will discuss uncertain quantities in more detail.
3.2 Everyday rates
Some examples of rates encountered in everyday life:
comparison shopping: dollars per oz.
dollars per square foot
earnings payment per money in an investment account
flow versus rate of flow: water volume per minute used by a shower
economic product per person
miles per gallon (Compound: ton miles per gallon(truck) or passenger miles per gallon (car or plane).
3.3 Conversion: Flavors of 1
In Chapter 2 we referred to “dimensionless quantities.” For instance, 60 is a dimensionless quantity, a pure number. In keeping track of things as we form rates, however, it is convenient to think of even so-called dimensionless quantities as having a dimension. We denote that dimension [1].
One reason to think about the dimension of a dimensionless number is that even dimensionless numbers can have units. For example, an angle is dimensionless: it is a ratio of two lengths. (See Figure 2.1.) The dimension of the ratio is [length] / [length], that is, L / L or [1].
Dimensionless quantities (that is, dimension [1]) play an important role in converting from one unit to another. In general4, to accomplish the conversion from one unit (say, miles) to another of the same dimension (say, km), multiply the first quantity by a conversion-factor that is dimensionless. We can call such conversion factors “☞ flavors of one ☜.”
A flavor of one is a ratio formed by dividing a quantity by an exactly equal quantity. For example, 1 km is the same length as 0.621371 miles. The conversion factor that converts miles to km will be a ratio with km on top and miles on the bottom. Constructing this ratio from the two equal amounts gives:
\[\frac{1\ \text{km}}{0.621371\ \text{mi}} = \frac{1}{0.621371}\ \frac{\text{km}}{\text{mi}} = 1.609334\ \frac{\text{km}}{\text{mi}}\]
The conversion factor 1.609334 km/mi is a flavor of one; it is the ratio of two equal quantities.
A different, but closely related flavor of one converts km to miles: 0.621371 mi/km. In both cases, the dimension of the flavors of one are L / L = [1]. That is, “dimensionless.”
Every conversion factor is a quantity, having both a pure number and units. Looking at the pure number in isolation won’t tell you which way the conversion goes. For that, you need the units. If the units are mi/km, the conversion goes from km to miles. If the units are km/mi, the conversion goes from miles to km. In general, the units of the conversion factor that translates from into to has units to/from.
Fig 3. 1 shows the result of a Google search on “kilometer to miles.” The boxes labeled “kilometer” and “mile” indicate the units of the two main quantities provided. However, the value in the formula, 1.609, omits the units, which should be listed explicitly as mi/km. Multiplying the km value by 1.609 mi/km turns km into miles.
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3.4 Percent
Section 2.4 introduced the notion that “percent” is a unit sometimes used to report dimensionless quantities. (See percent.) For instance, 13 percent—often written 13%—is the same dimensionless quantity as 0.13. The word “percent” is short for the Latin phrase “per centum.” The word centum means 100.5 Since per means “divided by,” per centum” means “divided by 100.”
Most adults interpret “%” as meaning “a fraction of.” The whole of something is 100% (which is, of course, just 1.00). A store discount of 20% means “we take 20% off of the price.” Subtracting 20% from the whole means the new price is \(100\% - 20\% = 1 - 0.2 = 0.8 = 80\%\) of the original (non-discounted) price. A credit card interest rate of 28% per year means that after a year, if there are no new charges or payments, 28% of the original debt will be added to the original debt. In other words, the end-of-year debt will be \(100\% + 28\% = 1 + 0.28 = 1.28 = 128\%\) of the start-of-year debt.
In the above paragraph, the word “of” has been italicized. “80% of …” is perfectly idiomatic English. It means “0.80 multiplied by ….” The quantitative thinker has to pay careful attention to this use of idiom and translate it properly into the appropriate mathematical operation.
When it comes to division, standard English uses the keyword “per” to signal which quantity is being divided by which. Unfortunately, “of” is not a reliable signal for multiplication; “of” is used in so many other ways in English. By analogy to “per,” we quantitative thinkers might push for general acceptance of a Latin word that provides a clearer signal. Appropriate choices are “ex,” “de,” and “e,” all of which carry the meaning “out of.” With such a convention, we would say “20% ex price” instead of “20% of the price.”
In mathematical theory, the use of percent is extraneous. Anything we write as a percent, we can write in decimal notation. In practice, people have a marked preference for quantities expressed with one or two digits, that is, in the range 0 to \(\pm\) 100. This preference is why it would be absurd to report weather temperatures in the SI base unit kelvin. “Today’s high temperature will be 303\(^\circ\)K,” violates the 2-digit preference, whereas the equivalent “30\(^\circ\)C” or “86\(^\circ\)F” remain within the preferred range.
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“Rhetoric” refers to techniques for speaking persuasively or compellingly. Everyone supports compelling quantitative arguments, but sometimes quantities are presented in ways that create a misleading impression. Here is an example from the New York Times in an article about the distinctive rhetorical style found in AI compositions:
“A.I.s like gesturing at complexity (“intricate” and “tapestry” have surged since 2022), as well as precision and speed: “swift,” “meticulous,” “adept.” But “delve” — in particular the conjugation “delves” — is an extreme case. In 2022, the word appeared in roughly one in every 10,000 abstracts collected in PubMed. By 2024, usage had shot up by 2,700 percent.” — Sam Kriss, “Why does AI write like … that”, New York Times, Dec. 3, 2025
An increase from 1 in 10,000 to 2700 is staggering and suggests that things are getting out of hand. But an increase by 2700 percent corresponds to growing from 1 in 10,000 to 28 in 10,000. Not so staggering. The double-zeros at the end of 2700 cancel out the “percent”: 2700% is the same as 27. “Shooting up” from 1 by 27 gives 28.
Be aware that the practice of using percent as a unit can produce large numbers, like 2700%, that may create a misleading impression. The statement in the article would have been more meaningfully phrased as, “By 2024, 28 in every 10,000 abstracts contained ‘delve’.”
3.5 Rates of change
Earlier, in Section 2.6, we introduced change in a quantity as a simple difference in the value of the quantity. When modeling systems, we focus on the connections between the quantities involved. So why did the value of the quantity change? Typically, it is because something else changed in the system. We use the phrase “with respect to …” to identify what may have changed. For instance, we can look at the change in temperature over an hour; the varying condition is then time and the change is described as being with respect to time. Sticking with the temperature example, we all know that temperature changes with respect to elevation (as we, say, climb a mountain), with respect to the amount of sunlight, or with respect to position.
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The simple idea of change (Section 2.6) involves looking at the shifting value of one primary variable. Rate of change, however, involves two quantities, a primary and a context-setting, “with respect to” variable.
A ☞ rate of change ☜ includes both the primary and the context-setting variable in the calculation. To form a rate of change, divide the quantitative difference in the primary variable by the corresponding quantitative difference in the context-setting variable.
“Primary” and “context-setting” can be thought of in terms of functions. The context-setting variable is the input, while the “primary” variable is the output.
To illustrate, consider the temperature of a cup of coffee over time as the cup sits at room temperature. “Over time” corresponds to the quantitative-thinking concept of function: temperature is a function of time. Fig 3. 2 shows some data collected by Prof. Stan Wagon. He poured near-boiling coffee into a cup instrumented with a recording thermometer. Following the familiar pattern, the coffee cooled.
Blue lines mark the temperature levels recorded at 5 and 35 minutes. The change in temperature with respect to time is the vertical distance shown by the magenta arrow. The change over the 30-minute time interval is 29.9\(^\circ\)C. The dimension of the change is simply 𝛳, that is, temperature, read directly from the vertical axis.
The rate of change over the 30-minute time interval is 29.9\(^\circ\)C / 30 min = 0.997\(^\circ\)C/min. Notice that the dimension of the rate of change is 𝛳 T-1, that is, temperature divided by time.
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Note that we could have calculated the rate of change for any interval, not just the 30-minute interval between times marked by the blue lines. Thus, the rate of change can itself change. The idea of changing rates of change takes some getting used to. But it forms a cornerstone of quantitative reasoning. We come back to it in Chapter 11.
3.6 Compound rates
As stated previously, a rate is a quantity. Calling the quantity a “rate” is merely a way of pointing out how the quantity was formed by the division of one quantity by another. Other than this history of the process of formation, a rate is no different than any other kind of quantity: a pure number plus a unit plus (optionally) a descriptive word.
Rates are often composed by arithmetic on other rates. We call these ☞ compound rates ☜, but this is merely to call attention to the formation of the rate. Like all rates, a compound rate is simply a quantity. Even so, in interpreting a quantity, it is helpful to keep in mind where it comes from. The words “rate” and “compound rate” are helpful in this.
To illustrate, here is the result of an AI search on “running energy per mile”:
Running energy per mile, or calories burned per mile, averages around 100 for a 150-pound person but varies significantly based on individual factors like body weight, fitness level, age, sex, and muscle mass. A commonly cited rule of thumb is to estimate about 100 calories burned per mile; a more accurate estimation involves multiplying body weight (in pounds) by 0.71 to get a general figure, for example, a 200-pound person burns around 142 calories per mile.
“Running energy per mile” is a rate. (Or, perhaps you would like to think of it as a rate of change, since “Calories burned” is a change in the runner’s stock of energy and the mile is really a change in the distance run.) The AI response goes on to discuss the role of body weight in the number of calories burned. Part of this involves a compound rate 0.71 Calories per mile per pound of body weight. Note the two uses of “per” in the phrase. The two occurrences of per signal that this is a compound rate.
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The English grammar of double-pers creates an ambiguity. Two different mathematical interpretations are possible for the English phrase “per mile per pound.” The parentheses of mathematical notation make this clearer. For instance, here are the two interpretations using mathematical parentheses to indicate the order of operations:
- (Calories burned per mile) per pound, or
- Calories burned per (mile per pound),
Alternatively, using the over-under notation of ratios:
- \(\frac{\text{Calories burned / mile}}{\text{pound}}\), which is \(\frac{\text{Calories burned}}{\text{mile} \times \text{pounds}}\).
- \(\frac{\text{Calories burned}}{\text{miles/pound}}\) which resolves to \(\frac{\text{Calories} \times \text{pound}}{\text{mile}}\).
Compound rates (1) and (2) are different quantities. To see this, imagine a 150-pound person running 3 miles, burning 300 Calories in the process. Note that there are three quantities involved:
- 300 Calories burned
- 150 pounds of weight
- 3-mile length run
Compound rate (1) works out to be 300 Calories / (3 mile \(\times\) 150 pounds), that is, 0.667 Calories per (pound-miles). In contrast, compound rate (2) is (300 Calories \(\times\) 150 pounds) / 3 miles, or 15,000 Calorie-miles per pound. (1) and (2) are different numbers, different units, and different dimensions.
It is not possible to tell just from the English phrase “Calories per pound per mile” which compound rate is intended. Instead, we need to look for clues. The AI text gives a clue about the use of the quantity 0.71. It says, “for example, a 200-pound person burns around 142 calories per mile.” Quantity (1) divides the calories-per-mile by pounds, whereas quantity (2) multiplies the calories-per-mile by pounds. We can try both:
\(0.71\, \frac{\text{Cal}}{\text{mile-pounds}} \times 200\, \text{pounds}\) gives 141 Cal per mile. \(0.71\, \frac{\text{Cal}}{\text{mile-per-pounds}} \times 200\, \text{pounds}\) gives 141 Cal pounds2 per mile. Only format (1) gives the result cited in the AI example.
There is a more fundamental way to see why interpretation (1) is the right one. The person is using 0.71 Cals to carry out a task. Which task? To move one pound for one mile. The unit “pound-mile” measures the amount of the task; a 150-pound person needs to perform 150 pound-miles to get themselves a distance of one mile.
Consider this example involving an eco-science task: measuring the number of animals crossing different roads. The technique: position the observer along the road who will spend a couple of hours with binoculars, counting the crosses they see. Making a fair comparison between roads requires adjusting for the time duration of observation. Less obvious, the raw count also needs to be adjusted for the length of roadway visible in binocular range. (On a curved or hilly road, for instance, sightlines might be limited to, say, 100 yards, while on a straight road, a half-mile of road is visible.) The amount of monitoring done by the observer units of mile-hours, that is, miles of road visible times hours spent monitoring. The format for the compound rate that fairly compares roads is animal crossings per mile-hour.
Some other examples of compound rates:
- Shipping costs. Dollars per the amount of shipping. There are two components to “Shipping”: the weight of the goods and the distance covered. The total amount of shipping is the weight times the distance. For railways, this is sometimes presented as “ton-miles.”
- Medicine dosing. Nurses using an intravenous (IV) drip to administer medicine are given the dose as a compound rate: milligrams of medicine per kilogram-hour. The nurse mixing the medicine in the IV bottle needs to know how many hours the bottle will last. The bigger the bottle, or the bigger the patient, the more milligrams needed.
- Rent. Rents for commercial space are often specified in dollars per square-foot-months. Multiply the monthly rent rate by the number of square feet
- Work effort is sometimes tallied using units such as person-hours or person-months.
- Energy use. The amount of energy needed to heat a house depends on the size of the house, the length of the heating season, and the temperature. An AI search on “amount of natural gas to heat a house” produced this response:
Another standard metric is the energy use per square foot per heating degree-day (HDD), where a typical range is 5.0 to 10.0 BTU/HDD/Sq.Ft.
BTU (British Thermal Unit) is a measure of energy. The amount of heating “service” required is the product of duration, coldness, and house size. The standard unit is degree-day-square-feet. So the rate of energy use is BTU per degree-day-square-feet. There are tables showing the average annual number of heating degree-days for various locales. A web search for “annual heating degree days for Minneapolis” yielded an estimate of 6000 degree-days. For a house of 1200 square feet, this comes to a demand of 1200 square-feet \(\times\) 6000 degree-days, or just over 7 million degree-day-square-feet. The typical range for energy use over a year would be 35 to 70 million BTU.
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Notice that the unit “BTU/HDD/Sq.Ft.” quoted by the AI is ambiguous. It should be BTU per HDD-(ft2). Suppose that we misinterpret the quoted range—5.0 to 10.0—as being in units of BTU/(HDD/ft2). For the 1200-square-foot house in the example, we would have 6000 degree-days divided by 1200 square feet, resulting in 5 degree-days per square foot. The total BTU usage for the house would be (mis-)calculated as 5.0 to 10.0 BTU per (HDD/Sq.Ft.) \(\times\) 5 HDD/Sq.Ft. = 25 to 50 BTU. For comparison, bringing 1 liter of water to a boil consumes about 250 BTU. It is hard to imagine that heating a house for a year would use less energy than boiling a liter of water. Mistaking HDD \(\times\) ft2. for HDD per ft2 will cause a lot of trouble when designing or outfitting a house!
3.7 Multiplying quantities
This morning, the author went for his habitual walk, which he estimates consumed 300 Cal. For diet planning or indicating general health, 300 Cal can be a good summary of the exercise. But the estimate combines two different factors: the energetics of walking and the distance walked. We use a rate, Cals per mile, to describe the energetics of walking. For walking, the rate is approximately 80 Cal/mile. Just about anyone on just about any walk can use the rate to find the total calories consumed; multiply the walk’s length by the energetic rate. A two-mile walk consumes 80 Cal mile-1 \(\times\) 2 mile = 160 Cal.6
Using the rate, we can compare the energetics of walking and running, again using a Cal/mile rate.
Take a moment to guess whether running or walking consumes more calories.
Many people guess that running consumes more energy. For instance, the author used to go for a 3-mile run each morning, and people assume that the author’s 4-mile walk is less exercise than the 3-mile run. There is no reason to assume; an internet search shows that running consumes about 100 Cal/mile. Surprisingly, that is similar to the energy rate for walking.
The author’s 4-mile walk uses about 300 Calories, roughly the same amount of energy as the 3-mile run: 100 Cal mile-1 \(\times\) 3 mile = 300 Cal. Once again, knowing the energy rate enables us to calculate a total energy: multiply the rate by the distance.
The pattern that “rate \(\times\) quantity gives total” is encountered every day. Some examples:
| Context | Rate | Multiplier | Result |
|---|---|---|---|
| Buying groceries | 12 dollars per pound | 0.75 pounds | 9 dollars |
| American driving | 35 miles per gallon | 10 gallons | 350 miles |
| European driving | 5 liters per 100 km | 500 km | 50 liters |
| Wages | 17 dollars / hour | 40 hours | 680 dollars |
The ratio/rate (price per pound, liters per 100 km, dollars per hour) describes the context. Multiplying the rate by an amount—in terms of the denominator of the ratio—gives a total in terms of the numerator of the ratio.
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As previously stated, we use dimensionless flavors of one to convert units between two quantities with the same dimension. It is reasonable to view dimensionful rates similarly: as conversion factors between different dimensions. Have gallons and want miles: multiply by the rate miles-per-gallon. Have 3/4 of a pound of cheese and want the price: multiply by the rate dollars-per-pound.
For some people, the result of multiplying two quantities sometimes seems odd. For instance, units of passenger-miles denote the amount of transportation service. One passenger going one mile is one passenger-mile; an airplane carrying 150 passengers for 3000 miles is providing 150 passengers \(\times\) 3000 miles of transit: 450,000 passenger-miles. A water-resources engineer might measure volume in acre-feet: multiplying area in acres by the depth in feet of water. Mechanics working on a car or a bike have to apply the right amount of torque to tighten a bolt without overstressing it. Torque is measured in foot-pounds: a foot of length times a pound of force.
New terms {
Footnotes
Remember, AI queries are typically not completely reproducible, so there is no way to reference them other than by quoting.↩︎
Strictly speaking, the AI should have said “Calorie.” A Cal is 1000 cal. The food labels always report energy content in Cal.↩︎
Taking into account that a person uses about 2000 Cal in an ordinary day’s activities, the excess energy available for the marathon is only 1000 Cal. That supports about 17 miles of walking.↩︎
Temperature is an exception.↩︎
“C-note” is old American slang for a $100 bill, perhaps short for “centum-note. The Latin numeral C also means 100.↩︎
This is roughly the caloric content of one can of (non-diet) soda.↩︎