From Amount to Rate per Unit of Amount
When a quantity changes, its absolute rate tells us how much it changes per unit of time. For a population \(P(t)\), the derivative \(dP/dt\) measures the instantaneous change in the number of people per unit of time. But the same absolute increase can mean very different things for a small population and a large one.
An increase of 20 people per year is substantial for a population of 500, but modest for a population of 20,000. To compare change relative to the amount already present, divide the instantaneous rate by the current amount. This gives the instantaneous relative rate.
For example, a relative rate of \(0.03\) per year means that, at the instant being considered, the rate of change is \(0.03\) times the current amount each year. It is often described as a 3% instantaneous relative rate per year. This is a rate at an instant, not a claim that the amount increases by exactly 3% over every full year.
Constant Relative Rate and Proportional Change
A quantity has a constant relative rate when the ratio of its instantaneous rate to its current amount has the same value throughout the time interval being modeled. Let that fixed value be \(k\). Then, wherever \(P(t)\ne 0\),
Multiplying by \(P(t)\) gives the differential equation
This equation says that the instantaneous rate of change is proportional to the current amount. The proportionality factor is the same \(k\) at every time in the interval. In particular, if the current amount is multiplied by a factor while \(k\) stays fixed, the rate specified by the model is multiplied by that same factor.
The quantity \(P\) has units of amount, such as people or dollars. The derivative \(dP/dt\) has units of amount per time. Therefore \(k\) has units of reciprocal time: multiplying \(k\) by \(P\) must produce the units of \(dP/dt\). The value of \(k\) is a rate per unit of current amount, not an absolute amount added per unit of time.
Why the Equation Expresses Proportionality
The relationship between a constant relative rate and the equation \(dP/dt=kP\) follows directly from the definition. The equivalence applies at times when \(P(t)\ne 0\).
Suppose the relative rate is the same fixed number \(k\). By definition, \(P'(t)/P(t)=k\).
Since \(P(t)\ne 0\), multiplying both sides by \(P(t)\) gives \(P'(t)=kP(t)\). The instantaneous rate is thus \(k\) times the current amount.
If \(P'(t)=kP(t)\) and \(P(t)\ne 0\), divide both sides by \(P(t)\) to obtain \(P'(t)/P(t)=k\). The relative rate is constant.
This also explains why the equation represents proportional change rather than a fixed absolute change. For two nonzero amounts \(P_1\) and \(P_2\) at times when the same constant \(k\) applies, the model assigns rates \(kP_1\) and \(kP_2\). If \(P_2=2P_1\), then
So the larger amount has twice the instantaneous rate under this model. This proportional scaling is the defining feature of constant relative rate.
There is a small but important qualification at \(P(t)=0\). The quotient \(P'(t)/P(t)\) is undefined there, so relative rate cannot be defined by that quotient at a zero amount. The equation \(P'=kP\), however, is still meaningful at \(P=0\): it specifies \(P'=0\) at that instant. Thus the differential equation can include zero values, but the relative-rate interpretation is stated only where the amount is nonzero.
Worked Example: Interpreting a Population’s Relative Rate
Worked Example: Interpreting a Population’s Relative Rate
At a particular instant, a town has \(1{,}200\) people and its population is increasing at \(36\) people per year. Find the instantaneous relative rate. If the same constant relative rate applies at an instant when the population is \(2{,}000\), find the model’s instantaneous rate then.
At the first instant, divide the absolute rate by the current population:
Thus \(k=0.03\) per year. The corresponding model is \(P'=0.03P\). At an instant when \(P=2{,}000\) people, substitute that amount:
Check the relative rate at each amount. For the first instant, \(36/1{,}200=0.03\) per year. For the second, \(60/2{,}000=0.03\) per year. Both rates have the same ratio to the current population, as required.
Worked Example: Relative Rate for a Savings Balance
Worked Example: Relative Rate for a Savings Balance
A savings balance is \(8{,}000\) dollars at an instant when its balance is increasing at \(20\) dollars per month. Suppose a constant relative rate model is being considered. Find its rate constant and the rate of change the model assigns when the balance is \(12{,}000\) dollars.
First calculate the instantaneous relative rate:
The units of dollars cancel, leaving reciprocal months. Under the constant-relative-rate model, \(P'=0.0025P\), where \(P\) is the balance in dollars. At a balance of \(12{,}000\) dollars:
Check: \(20/8{,}000=0.0025\) per month, and \(30/12{,}000=0.0025\) per month. Both balances therefore have the same modeled instantaneous relative rate. This calculation interprets the model; it does not assume a fixed dollar amount is added each month.
Worked Example: Same Absolute Rate, Different Relative Rates
Worked Example: Same Absolute Rate, Different Relative Rates
Two populations have instantaneous rates of increase of \(20\) people per year. Population A has \(500\) people, while Population B has \(2{,}000\) people. Compare their instantaneous relative rates.
For Population A:
For Population B:
The absolute rates are equal, but the relative rates are not: \(0.04\ne 0.01\). Checking by multiplying each amount by its relative rate gives \(0.04\times 500=20\) people per year and \(0.01\times 2{,}000=20\) people per year. The same absolute increase represents a larger fraction of the smaller population.
Relative Change Is Not Absolute Change
An absolute rate describes the amount of change per unit of time. A relative rate describes that change per unit of the current amount. The distinction is not merely a change in wording: the two rates answer different questions.
| Rate | Expression | Interpretation |
|---|---|---|
| Absolute rate | \(P'(t)\) | Amount gained or lost per unit of time |
| Relative rate | \(P'(t)/P(t)\), when \(P(t)\ne 0\) | Rate of change per unit of current amount |
| Constant relative rate model | \(P'(t)=kP(t)\) | Absolute rate scales with the current amount |
A fixed absolute rate does not usually imply a fixed relative rate. For example, if an amount increases at \(20\) units per year, the relative rate is \(20/P(t)\) per year. It changes when \(P(t)\) changes. Conversely, a fixed relative rate generally gives different absolute rates at different amounts, because the model multiplies the amount by \(k\).
When the Model Is a Reasonable Choice
The constant-relative-rate assumption can be useful when the mechanism producing change acts in proportion to the amount currently present. In a population model, for instance, a larger population may have more individuals contributing to births and deaths; a simple model may express the net effect as proportional to the population. For a savings balance, a model may treat the instantaneous interest accumulation as proportional to the current balance.
These are modeling assumptions, not universal facts. A population’s net change can also depend on migration, age structure, resource limits, or other factors. A savings balance can be affected by deposits, withdrawals, fees, or changing interest conditions. If those effects matter, a single constant \(k\) may not describe the observed rate well. The model is appropriate only to the extent that its assumptions reasonably represent the situation over the interval of interest.
A Common Mistake: Confusing a Percentage Rate with an Amount per Time
Suppose a population is \(1{,}000\) people and its instantaneous increase is \(30\) people per year. A common error is to say that its relative rate is \(30\) per year, treating the absolute rate as though it were already a percentage rate. The units reveal the problem: \(30\) people per year is an amount per time, while a relative rate must have units of reciprocal time.
The correction is to divide by the current amount:
Checking the result in the model gives \(0.03\text{ per year}\times 1{,}000\text{ people}=30\text{ people/year}\), which recovers the original absolute rate. Keeping the amount and relative rate distinct prevents this error.
Summary
The instantaneous relative rate is \(P'/P\) wherever \(P\ne 0\). If this ratio is constant and equal to \(k\), then multiplying by \(P\) gives \(P'=kP\). Conversely, when \(P\ne 0\), that equation implies a constant relative rate \(k\). It models proportional instantaneous change: the absolute rate depends on the current amount, rather than being a fixed amount added or removed per unit of time.
Check Your Understanding
Use the definition of relative rate and the meaning of proportional change to answer these questions.
- A population is \(600\) people and is increasing at \(18\) people per year. What is its instantaneous relative rate, including units?
- If a savings balance doubles while the same constant relative rate applies, what happens to the model’s instantaneous absolute rate?
- Two quantities have the same instantaneous absolute rate but different positive current amounts. Must their relative rates be equal? Explain.
- Why is \(P'/P\) undefined at \(P=0\), even though the equation \(P'=kP\) can still be evaluated there?
- Name one population or savings situation that could make a constant-relative-rate model less appropriate, and explain why.
