Historical Context & Motivation
People have been tracking how things grow and shrink for thousands of years. Ancient merchants noticed that their investments grew not by a fixed amount each year, but by a fraction of whatever they already had. A farmer who stored grain might lose a constant percentage to pests each month. This idea — that change happens as a constant percent of the current amount — is one of the most important patterns in mathematics.
Understanding percent growth and decay helped mathematicians, scientists, and bankers solve real problems. From calculating compound interest to predicting how diseases spread, recognizing this pattern gives you a powerful tool for understanding the world.
The central question this lesson addresses is: How do you recognize when a quantity is growing or decaying by a constant percent rate, and how does that differ from growing or shrinking by a fixed amount?
Core Principles & Definitions
Before diving into examples, let's nail down the key ideas. When we say a quantity changes by a constant percent rate per unit interval, we mean that in every equal time period, the quantity is multiplied by the same factor. This is different from adding or subtracting the same number each time.
Percent Growth
Percent Decay
Growth Factor (b)
Linear vs. Exponential
Visual Explanation — Linear vs. Exponential
The best way to see the difference between constant-amount change and constant-percent change is to graph them side by side. The diagram below shows two quantities that both start at 100. The blue line grows by a constant amount of 20 each period (linear). The emerald curve grows by a constant 20% each period (exponential). Notice how the curve starts slowly, then accelerates past the straight line.
Notice that in the first couple of time periods the two lines are very close together. The exponential curve doesn't look dramatically different at first. But as time goes on, the gap widens because each period's increase is larger than the last. That's the signature of constant percent growth: the actual amount of change keeps getting bigger, even though the rate stays the same.
Mathematical Framework
Now let's put the math behind the pattern. Every situation involving constant percent change per unit interval can be modeled with an exponential function. Here is the general form.
The key is figuring out b from the percent rate. If a quantity grows by r percent per period, convert r to a decimal by dividing by 100, then add 1. If it decays, subtract the decimal from 1.
How to Recognize Percent Change in Tables & Scenarios
You won't always be handed a formula. Often you'll see a table of values, a word problem, or a graph and need to figure out whether the pattern is constant percent change. Here's the trick: instead of looking at the differences between consecutive values (which tells you about linear change), look at the ratios. If each value divided by the previous value gives the same number, you have constant percent change.
| Year | Value | Difference (Value − Previous) | Ratio (Value ÷ Previous) |
|---|---|---|---|
| 0 | 200 | — | — |
| 1 | 240 | 40 | 1.20 |
| 2 | 288 | 48 | 1.20 |
| 3 | 345.6 | 57.6 | 1.20 |
| 4 | 414.72 | 69.12 | 1.20 |
You can also recognize percent change from word problems. Key phrases include "increases by 5% per year," "loses one-third of its value each decade," or "doubles every 6 hours." Whenever the rate of change is described as a percentage or fraction of the current amount, you're dealing with exponential growth or decay.
Worked Example
Let's walk through a full problem. A town has a population of 8,000 people and is growing at a rate of 6% per year. We want to write the exponential model and find the population after 5 years.
Comparing Growth and Decay
Growth and decay are two sides of the same coin. They both involve multiplying by a constant factor. The only difference is whether that factor is greater than 1 or less than 1. The table below summarizes the comparison.
| Feature | Percent Growth | Percent Decay |
|---|---|---|
| Direction | Values increase over time | Values decrease over time |
| Growth factor (b) | b > 1 | 0 < b < 1 |
| Formula for b | b = 1 + r | b = 1 − r |
| Graph shape | Curve rises steeply (J-shape) | Curve falls and flattens (approaching 0) |
| Real-world example | Investment earning interest | Car losing value (depreciation) |
| Key signal words | "grows by," "increases by," "appreciates" | "decreases by," "loses," "depreciates" |
Connection to Advanced Concepts
Recognizing constant percent change is the gateway to more advanced ideas in Algebra 2, Precalculus, and beyond. Here's a preview of how this foundation extends.
| What You Learn Now | Where It Leads |
|---|---|
| y = a × bᵗ with b = 1 + r or b = 1 − r | The natural exponential function y = a × eʳᵗ, used in calculus and science |
| Recognizing exponential patterns in tables | Exponential regression — fitting curves to real data using technology |
| Growth factor b > 1 or 0 < b < 1 | Logarithms — the inverse operation that "undoes" exponential functions |
| Simple percent growth problems | Compound interest with different compounding frequencies; half-life problems in chemistry |
The skill you're building right now — spotting whether a situation involves constant percent change — is used every day in finance, biology, environmental science, and medicine. Whenever you hear about "exponential growth" in the news (like the spread of a virus or the growth of social media users), this is the exact concept they're describing.
Practice Problems
Lesson Summary
A quantity exhibits constant percent growth when it is multiplied by a factor b = 1 + r (where b > 1) each time period, and constant percent decay when multiplied by b = 1 − r (where 0 < b < 1). Both situations are modeled by the exponential function y = a × bᵗ. The key to recognizing this pattern is to check whether consecutive values in a table have a constant ratio rather than a constant difference.
In word problems, phrases like "increases by 8% per year" or "loses one-fifth of its value each month" signal constant percent change. Unlike linear functions (which add or subtract the same amount each period), exponential functions produce curves that accelerate upward in growth or flatten toward zero in decay. Mastering this distinction prepares you for compound interest, population modeling, half-life calculations, and many other real-world applications.