The Central Idea
A measurable quantity is a property that can be assigned a numerical value by measurement, counting, estimation, or calculation.
A measurement is usually written as
\[ \boxed{\text{quantity}=\text{numerical value}\times\text{unit}}. \]For example,
\[ 12\text{ m} \]means twelve units of length, where the chosen unit is the meter.
The number and the unit work together. The number tells us how much; the unit tells us of what kind and relative to which standard.
Why Units Matter in Modeling
Models are built from quantities such as distance, time, mass, temperature, population, cost, energy, concentration, or probability. Units attach real-world meaning to the symbols used for those quantities.
If a vehicle travels at
\[ 60\text{ mi/h}, \]the unit tells us that the quantity is a rate comparing distance with time. Writing only
\[ 60 \]does not tell us whether the number represents miles per hour, kilometers per hour, meters per second, or something entirely different.
Quantity Versus Numerical Value
A quantity is not the same thing as its numerical value.
The same physical length might be written as
\[ 1\text{ m} \]or
\[ 100\text{ cm}. \]The numerical values are different, but the represented quantity is the same. Changing units changes the number needed to describe the quantity.
Measured, Counted, and Calculated Quantities
Quantities enter models in several ways.
- Measured quantities come from instruments or observations.
- Counted quantities come from discrete totals.
- Calculated quantities are computed from other quantities.
For example, a road length may be measured, the number of vehicles may be counted, and average traffic density may then be calculated from those inputs.
Examples of Measurable Quantities
Common measurable quantities include:
- length;
- area;
- volume;
- time;
- mass;
- temperature;
- speed;
- acceleration;
- energy;
- power;
- pressure;
- population;
- currency value;
- concentration;
- information or data size.
Not every model uses physical units. Some quantities are counts, percentages, scores, indices, or dimensionless ratios. The same habit still applies: identify clearly what each number represents.
Units Are Agreed Standards
A unit is an agreed standard used to express the size of a quantity.
For length, familiar units include meters, centimeters, kilometers, inches, feet, yards, and miles.
For time, familiar units include seconds, minutes, hours, days, and years.
A model should use units consistently so that quantities can be compared and combined without ambiguity.
SI Units
The International System of Units, usually called SI, provides a widely used standard for scientific and technical measurement.
Common SI base units include:
- meter \(\text{m}\) for length;
- kilogram \(\text{kg}\) for mass;
- second \(\text{s}\) for time;
- kelvin \(\text{K}\) for thermodynamic temperature;
- ampere \(\text{A}\) for electric current;
- mole \(\text{mol}\) for amount of substance;
- candela \(\text{cd}\) for luminous intensity.
Many other units are derived from combinations of these base units.
US Customary Units
In the United States, models and measurements also commonly use customary units such as inches, feet, yards, miles, ounces, pounds, gallons, and degrees Fahrenheit.
A model does not automatically become better merely because it uses one unit system rather than another. What matters is that the chosen units are appropriate, explicitly stated, and used consistently.
Symbols Should Carry Meaning
In a model, a variable should usually be defined together with its unit.
For example,
\[ d=\text{distance in meters} \]and
\[ t=\text{time in seconds}. \]Then a formula such as
\[ v=\frac{d}{t} \]immediately implies that the resulting speed has units of meters per second.
Units Behave Algebraically
Units can often be multiplied, divided, canceled, and raised to powers in the same structural way as algebraic factors.
If
\[ d=150\text{ km} \]and
\[ t=3\text{ h}, \]then
\[ v=\frac{150\text{ km}}{3\text{ h}} =50\text{ km/h}. \]The division acts on both the numerical values and the units.
Multiplying Quantities
Multiplying quantities usually multiplies their units as well.
A rectangle with length
\[ 8\text{ m} \]and width
\[ 5\text{ m} \]has area
\[ A=(8\text{ m})(5\text{ m})=40\text{ m}^2. \]The square on the unit comes from multiplying meters by meters.
Dividing Quantities
Dividing one quantity by another creates a ratio of units.
If \(240\) kilometers are traveled in \(4\) hours, then
\[ \frac{240\text{ km}}{4\text{ h}}=60\text{ km/h}. \]Units such as kilometers per hour, dollars per item, people per square kilometer, and joules per second all describe rates or densities formed by division.
Adding and Subtracting Quantities
Addition and subtraction require compatible quantities.
We can add
\[ 3\text{ m}+5\text{ m}=8\text{ m}. \]But an expression such as
\[ 3\text{ m}+5\text{ s} \]has no ordinary measurement meaning because length and time are different kinds of quantities.
Compatible Units May Need Conversion
Even quantities of the same kind may need to be expressed in a common unit before they are added or compared.
For example,
\[ 2\text{ m}+35\text{ cm} \]can be rewritten using meters:
\[ 35\text{ cm}=0.35\text{ m}. \]Therefore,
\[ \boxed{2\text{ m}+35\text{ cm}=2.35\text{ m}}. \]Conversion Factors
A conversion factor is a ratio of equivalent measurements. Because the numerator and denominator represent the same quantity, the ratio is equal to \(1\).
Since
\[ 1\text{ m}=100\text{ cm}, \]both of the following are equal to \(1\):
\[ \frac{100\text{ cm}}{1\text{ m}} \qquad\text{and}\qquad \frac{1\text{ m}}{100\text{ cm}}. \]Choose the orientation that cancels the unit you want to remove.
Example: Converting Length
Convert
\[ 2.4\text{ m} \]to centimeters.
Multiply by a conversion factor:
\[ 2.4\text{ m}\left(\frac{100\text{ cm}}{1\text{ m}}\right). \]The meter units cancel:
\[ \boxed{2.4\text{ m}=240\text{ cm}}. \]Example: Converting a Rate
Convert
\[ 72\text{ km/h} \]to meters per second.
Use
\[ 1\text{ km}=1000\text{ m} \]and
\[ 1\text{ h}=3600\text{ s}. \]Then
\[ 72\frac{\text{km}}{\text{h}} \left(\frac{1000\text{ m}}{1\text{ km}}\right) \left(\frac{1\text{ h}}{3600\text{ s}}\right) =20\frac{\text{m}}{\text{s}}. \]Therefore,
\[ \boxed{72\text{ km/h}=20\text{ m/s}}. \]Derived Quantities
A derived quantity is formed from other quantities by mathematical operations.
Examples include:
- area as length times length;
- volume as length cubed;
- speed as distance divided by time;
- acceleration as change in speed divided by time;
- density as mass divided by volume;
- power as energy divided by time.
The units of a derived quantity follow from the equation defining it.
Example: Density
Suppose an object has mass
\[ m=540\text{ g} \]and volume
\[ V=200\text{ cm}^3. \]Density is
\[ \rho=\frac{m}{V}. \]Therefore,
\[ \rho=\frac{540\text{ g}}{200\text{ cm}^3} =2.7\text{ g/cm}^3. \]The unit communicates the structure of the quantity: grams of mass per cubic centimeter of volume.
Counts Can Also Carry Units
Units are not limited to physics.
A business model might use
\[ 350\text{ customers/day}, \]a demographic model might use
\[ 1200\text{ people/km}^2, \]and a manufacturing model might use
\[ 45\text{ units/hour}. \]Treating these labels as units makes the mathematical structure easier to track.
Currency as a Modeled Quantity
Monetary quantities also require units.
If a product costs
\[ \$18\text{/item} \]and \(25\) items are purchased, then
\[ 25\text{ items}\left(\frac{\$18}{\text{item}}\right)=\$450. \]The item unit cancels, leaving the monetary unit expected for total cost.
Percentages and Dimensionless Quantities
Some quantities are ratios of two quantities with the same units. Their units cancel.
For example, efficiency might be modeled as
\[ \eta=\frac{\text{useful output energy}}{\text{input energy}}. \]If both energies are measured in joules, then
\[ \frac{\text{J}}{\text{J}}=1. \]The resulting quantity is dimensionless. Percentages, probabilities, proportions, and many indices have this form.
Unit Labels Can Reveal a Formula
Units often suggest how quantities should be combined.
Suppose a worker is paid
\[ \$24\text{/h} \]for
\[ 7.5\text{ h}. \]To obtain dollars, multiply:
\[ \left(\frac{\$24}{\text{h}}\right)(7.5\text{ h})=\$180. \]The unit cancellation shows which operation produces the desired result.
Unit Analysis as a Problem-Solving Tool
Before inserting numbers into a formula, inspect the units.
If the desired answer is a distance and you know speed and time, then
\[ \left(\frac{\text{distance}}{\text{time}}\right)(\text{time}) =\text{distance}. \]This points toward the model
\[ d=vt. \]Unit structure does not prove that a model is correct, but it can often expose impossible operations or suggest the proper arrangement of variables.
Unit Consistency in Equations
In a meaningful equation, quantities being added or equated must have compatible units.
Consider
\[ d=d_0+vt. \]If \(d\) and \(d_0\) are distances, then \(vt\) must also be a distance. Indeed,
\[ \left(\frac{\text{m}}{\text{s}}\right)(\text{s})=\text{m}. \]Every term therefore has units of length.
A Unit Mismatch Signals a Problem
Suppose someone proposes
\[ d=v+t. \]If \(d\) is measured in meters, \(v\) in meters per second, and \(t\) in seconds, then the right side attempts to add
\[ \frac{\text{m}}{\text{s}}+\text{s}. \]These are incompatible quantities. The unit mismatch immediately tells us that the formula cannot represent distance in the ordinary way.
Units and Powers
When a quantity is raised to a power, its unit is raised to the same power.
If
\[ r=3\text{ m}, \]then
\[ r^2=(3\text{ m})^2=9\text{ m}^2. \]Likewise,
\[ r^3=(3\text{ m})^3=27\text{ m}^3. \]Squared and cubed units are not decoration; they record how the quantity was constructed.
Units in Scientific Notation
Scientific notation changes the numerical form but not the unit.
For example,
\[ 0.00042\text{ m}=4.2\times10^{-4}\text{ m}. \]The exponent applies to the numerical value, while the unit remains meters.
Metric Prefixes
Metric prefixes scale units by powers of ten.
- kilo- means \(10^3\);
- centi- means \(10^{-2}\);
- milli- means \(10^{-3}\);
- micro- means \(10^{-6}\);
- nano- means \(10^{-9}\).
Thus,
\[ 1\text{ km}=10^3\text{ m} \]and
\[ 1\text{ mm}=10^{-3}\text{ m}. \]Converting Squared Units
Area conversions require the conversion factor to be squared.
Since
\[ 1\text{ m}=100\text{ cm}, \]squaring both sides gives
\[ 1\text{ m}^2=(100\text{ cm})^2=10{,}000\text{ cm}^2. \]Therefore,
\[ \boxed{1\text{ m}^2=10^4\text{ cm}^2}. \]Converting Cubed Units
Volume conversions require cubing the length conversion.
From
\[ 1\text{ m}=100\text{ cm}, \]we get
\[ 1\text{ m}^3=(100\text{ cm})^3=1{,}000{,}000\text{ cm}^3. \]The exponent applies to the entire conversion factor.
Temperature Requires Special Care
Not every unit conversion is accomplished by multiplying by a constant.
Celsius and Fahrenheit use different zero points as well as different degree sizes. Their conversion therefore requires an affine formula such as
\[ F=\frac{9}{5}C+32. \]This is an important reminder: units can differ not only by scale but also by offset.
Absolute Versus Relative Temperature
A temperature value and a temperature difference are not always converted in exactly the same way.
For example, a change of
\[ 10^\circ\text{C} \]corresponds to a change of
\[ 18^\circ\text{F}, \]because temperature differences use the scale factor \(9/5\) but not the offset of \(32\).
Define Units Before Building the Equation
A strong modeling habit is to define each variable and its unit before carrying out algebra.
For example:
- \(t\): time in hours;
- \(P(t)\): population in people;
- \(r\): growth rate in people per hour.
These definitions reduce ambiguity and make later equations easier to inspect.
Choose Units Appropriate to the Scale
The same quantity can often be expressed in many valid units, but some choices are more convenient than others.
A city-to-city distance is usually easier to communicate in kilometers or miles than in millimeters. A laboratory sample may be easier to describe in milligrams than in metric tons.
Good unit choices keep numerical values within a readable and useful range.
Units Should Match the Decision Context
Modelers should also consider how results will be used.
A transportation model might internally calculate speed in meters per second but report it in kilometers per hour. A financial model might calculate daily revenue but summarize results monthly or annually.
The model may use one set of units for computation and another for communication, provided conversions are handled explicitly and consistently.
Example: A Simple Travel Model
Suppose a vehicle travels at a constant speed of
\[ 55\text{ mi/h} \]for
\[ 2.5\text{ h}. \]The distance model is
\[ d=vt. \]Substituting with units,
\[ d=\left(55\frac{\text{mi}}{\text{h}}\right)(2.5\text{ h}) =137.5\text{ mi}. \]The hour units cancel, leaving miles, exactly as a distance should.
Example: Production Rate
A machine produces
\[ 120\text{ parts/h}. \]If it operates for \(6.5\) hours at that constant rate, then
\[ 120\frac{\text{parts}}{\text{h}}(6.5\text{ h}) =780\text{ parts}. \]The units show that multiplying a rate by time produces a total count.
Example: Fuel Use
Suppose a vehicle uses fuel at an average rate of
\[ 8.4\frac{\text{L}}{100\text{ km}}. \]Over \(350\) kilometers, the estimated fuel use is
\[ 350\text{ km} \left(\frac{8.4\text{ L}}{100\text{ km}}\right) =29.4\text{ L}. \]Writing the units directly into the calculation makes the structure of the model visible.
Example: Unit Cost
If \(480\) dollars buys \(24\) identical units, then the average unit cost is
\[ \frac{\$480}{24\text{ units}}=\$20\text{/unit}. \]This new quantity is a rate: dollars per unit.
Units Help Distinguish Similar Numbers
Two numerical values may be equal while representing very different quantities.
The number \(50\) could mean
- \(50\text{ km}\);
- \(50\text{ min}\);
- \$50;
- \(50\text{ people}\);
- \(50\%\).
A model must preserve those distinctions even when the raw numerical values happen to match.
Units and Data Tables
Tables should identify the unit of every measured column.
A clear table heading might use labels such as
- Time (s);
- Distance (m);
- Temperature (°C);
- Revenue ($);
- Flow Rate (L/min).
Recording the unit once in the heading avoids repeatedly writing it in every cell while keeping the data interpretable.
Units and Graphs
Graph axes should also include units.
If time is plotted horizontally and distance vertically, labels such as
\[ \text{Time (s)} \]and
\[ \text{Distance (m)} \]make the graph self-explanatory. The slope then inherits the units
\[ \frac{\text{m}}{\text{s}}, \]which identifies it as a speed.
Units and Slopes
The unit of a slope is the vertical-axis unit divided by the horizontal-axis unit.
If cost in dollars is graphed against production in units, then the slope has units
\[ \frac{\$}{\text{unit}}. \]If temperature is graphed against time, then the slope might have units
\[ ^\circ\text{C}/\text{min}. \]Units often tell us how to interpret the slope before we examine its numerical value.
Units in Integrals and Accumulation
In continuous models, integration combines a rate with an infinitesimal amount of its input variable.
If \(v(t)\) is measured in meters per second and \(dt\) carries seconds, then
\[ v(t)\,dt \]has units of meters. Therefore,
\[ \int v(t)\,dt \]represents accumulated distance or displacement under the appropriate model.
Units in Derivatives and Rates of Change
A derivative compares the unit of the output with the unit of the input.
If position \(x\) is measured in meters and time \(t\) in seconds, then
\[ \frac{dx}{dt} \]has units
\[ \frac{\text{m}}{\text{s}}. \]Differentiating once more gives
\[ \frac{d^2x}{dt^2} \]with units
\[ \frac{\text{m}}{\text{s}^2}. \]Common Error: Dropping Units Too Early
A calculation such as
\[ 80\times3=240 \]may be numerically correct but incomplete in a modeling context.
If the original quantities were
\[ 80\text{ km/h} \]and
\[ 3\text{ h}, \]the complete result is
\[ \boxed{240\text{ km}}. \]Carrying the units through the calculation makes the interpretation explicit.
Common Error: Treating a Conversion as an Addition
Converting between multiplicatively related units should not change the quantity itself.
For example,
\[ 3\text{ m}=300\text{ cm}. \]The conversion multiplies the numerical value by \(100\) while replacing meters with centimeters. It does not mean adding \(100\) or attaching both units at once.
Common Error: Converting Area Like Length
Since
\[ 1\text{ m}=100\text{ cm}, \]it is incorrect to claim
\[ 1\text{ m}^2=100\text{ cm}^2. \]The conversion factor must also be squared:
\[ \boxed{1\text{ m}^2=10{,}000\text{ cm}^2}. \]Common Error: Mixing Units Inside a Formula
Suppose speed is measured in kilometers per hour but time is entered in minutes. Multiplying those values directly will not produce kilometers unless the time is converted to hours first.
For \(30\) minutes,
\[ 30\text{ min}\left(\frac{1\text{ h}}{60\text{ min}}\right) =0.5\text{ h}. \]Only after conversion should the values be combined in a model using hours.
Common Error: Adding Incompatible Quantities
Expressions such as
\[ 5\text{ kg}+7\text{ m} \]or
\[ 4\text{ h}+9\text{ dollars} \]do not represent ordinary additions of like quantities.
A valid model may combine such variables through multiplication, division, or a more complex function, but direct addition requires compatible meanings and units.
A Reliable Procedure for Unit-Aware Modeling
- Identify each quantity in the situation.
- Choose a symbol for each important quantity.
- State the unit associated with each symbol.
- Convert quantities to compatible units when necessary.
- Write the mathematical relationship.
- Carry units through multiplication, division, and powers.
- Check that quantities being added or equated have compatible units.
- Verify that the final unit matches the quantity requested.
- Convert the final result to a useful reporting unit when appropriate.
Example: Full Unit-Aware Workflow
A pump moves water at
\[ 18\text{ L/min}. \]How much water is moved in \(2.5\) hours?
First convert time:
\[ 2.5\text{ h} \left(\frac{60\text{ min}}{1\text{ h}}\right) =150\text{ min}. \]Then multiply rate by time:
\[ 18\frac{\text{L}}{\text{min}}(150\text{ min}) =2700\text{ L}. \]Therefore,
\[ \boxed{2700\text{ L}} \]of water are moved.
Example: Using Units to Find an Unknown
Suppose a trip covers
\[ 210\text{ km} \]at an average speed of
\[ 70\text{ km/h}. \]Time should have units of hours. Dividing distance by speed gives
\[ t=\frac{210\text{ km}}{70\text{ km/h}} =3\text{ h}. \]The kilometers cancel and the reciprocal of \(1/\text{h}\) leaves hours.
Practice 1: Identify the Quantity and Unit
A tank contains \(450\text{ L}\) of water. Identify the measurable quantity, the numerical value, and the unit.
Practice 2: Convert Length
Convert
\[ 3.6\text{ km} \]to meters.
Practice 3: Convert Time
Convert
\[ 135\text{ min} \]to hours.
Practice 4: Rate Times Time
A machine produces \(42\text{ components/min}\) for \(18\) minutes. How many components does it produce?
Practice 5: Unit Cost
A shipment of \(80\) units costs \$1,200. Find the average cost per unit.
Practice 6: Area Conversion
Convert
\[ 2.5\text{ m}^2 \]to square centimeters.
Practice 7: Check the Units
A model proposes
\[ d=v+t, \]where \(d\) is distance, \(v\) is speed, and \(t\) is time. Explain why the equation fails a basic unit check.
Practice 8: Density
A sample has mass \(625\text{ g}\) and volume \(250\text{ cm}^3\). Find its density.
Practice 9: Convert a Speed
Convert
\[ 90\text{ km/h} \]to meters per second.
Practice 10: Interpret a Slope Unit
A graph has cost in dollars on the vertical axis and production in units on the horizontal axis. What unit does the slope have, and what does it represent?
Practice Answers
Practice 1
The measurable quantity is volume, the numerical value is \(450\), and the unit is liters.
Practice 2
\[ 3.6\text{ km} \left(\frac{1000\text{ m}}{1\text{ km}}\right) =\boxed{3600\text{ m}}. \]Practice 3
\[ 135\text{ min} \left(\frac{1\text{ h}}{60\text{ min}}\right) =\boxed{2.25\text{ h}}. \]Practice 4
\[ 42\frac{\text{components}}{\text{min}}(18\text{ min}) =\boxed{756\text{ components}}. \]Practice 5
\[ \frac{\$1200}{80\text{ units}} =\boxed{\$15\text{/unit}}. \]Practice 6
Since \(1\text{ m}=100\text{ cm}\),
\[ 2.5\text{ m}^2 \left(\frac{100\text{ cm}}{1\text{ m}}\right)^2 =\boxed{25{,}000\text{ cm}^2}. \]Practice 7
The terms on the right have units of speed and time, such as \(\text{m/s}+\text{s}\). They cannot ordinarily be added, and their sum does not have units of distance.
Practice 8
\[ \rho=\frac{625\text{ g}}{250\text{ cm}^3} =\boxed{2.5\text{ g/cm}^3}. \]Practice 9
\[ 90\frac{\text{km}}{\text{h}} \left(\frac{1000\text{ m}}{1\text{ km}}\right) \left(\frac{1\text{ h}}{3600\text{ s}}\right) =\boxed{25\text{ m/s}}. \]Practice 10
The slope has units of dollars per unit,
\[ \boxed{\$\text{/unit}}. \]It represents the change in cost associated with each additional unit of production over the interval represented by the slope.
What to Remember
A measurement combines a numerical value with a unit:
\[ \boxed{\text{quantity}=\text{numerical value}\times\text{unit}}. \]- Units give numerical values real-world meaning.
- Equivalent quantities can have different numerical values in different units.
- Multiplication and division combine units algebraically.
- Addition and subtraction require compatible quantities.
- Conversion factors are ratios equal to \(1\).
- Squared and cubed quantities require squared and cubed conversion factors.
- Unit analysis can expose errors and help determine how quantities should combine.
- A model should define the units of its variables before calculations begin.
The key modeling habit is simple: carry the units with the numbers until the meaning of the result is clear.
Where We Go Next
Units identify how quantities are measured and combined. The next step is to ask whether an entire equation is structurally compatible with those measurements.
The next lesson, Dimensional Consistency, develops this idea into a systematic check for equations and models by comparing the dimensions of every term and quantity.