Historical Context & Motivation
Humans have been tracking how things increase and decrease for thousands of years. Ancient merchants needed to know how much more grain they could expect from a good harvest, and bankers in Renaissance Italy needed a reliable way to calculate the interest owed on loans. The idea of expressing change as a fraction of 100 — a percent — gave people a universal language for comparing changes of all sizes, whether it was the price of silk rising by a small amount or a city's population doubling over a century.
The word "percent" itself comes from the Latin per centum, meaning "for every hundred." Over time, mathematicians refined this idea into powerful tools — percent change and growth factors — that let us model everything from population booms to radioactive decay. Today these tools underpin economics, biology, environmental science, and even the algorithms that recommend your next video.
The central question this lesson addresses is: When a quantity changes, how do we measure that change in a way that is meaningful, comparable, and useful for prediction? By the end of this lesson, you will be able to calculate percent change, convert it into a growth or decay factor, and interpret what those numbers mean in real-world contexts.
Core Principles & Definitions
Before diving into calculations, it helps to lock in the key vocabulary. Each term below builds on the one before it, so take them in order. Together, they form a toolkit that lets you describe any increase or decrease in precise mathematical language.
Percent Change
Growth Factor
Decay Factor
Original (Initial) Value
New (Final) Value
Visual Explanation — Percent Change on a Number Line
A number line is one of the most intuitive ways to see how percent change works. The diagram below starts with an original value of 200 and shows two scenarios: a 25% increase and a 25% decrease. Notice how the growth factor of 1.25 lands you at 250, while the decay factor of 0.75 drops you to 150. Both changes are 50 units in size, but the factor tells the whole story in one number.
The key insight from this diagram is that every percent change can be rewritten as a single multiplication. Instead of first calculating "how much did it change?" and then adding or subtracting, you just multiply the original value by the appropriate factor. This one-step approach becomes especially powerful when a percent change happens repeatedly — like interest compounding every year — because you can simply raise the factor to a power.
Mathematical Framework
Let's formalize the relationships you just saw visually. There are three core formulas you should be comfortable with. Each one connects percent change, growth/decay factor, and the original and new values.
Growth vs. Decay — A Detailed Breakdown
The diagram below compares two scenarios over four time periods. The blue curve shows a quantity that grows by 20% each period (growth factor 1.20), and the red curve shows a quantity that shrinks by 20% each period (decay factor 0.80). Both start at 100. Notice how growth curves upward faster and faster — that's the nature of exponential growth — while decay curves downward and gets flatter, approaching zero but never quite reaching it.
| Period | Growth Value (×1.20) | Decay Value (×0.80) | Growth Calculation | Decay Calculation |
|---|---|---|---|---|
| 0 | 100 | 100 | Starting value | Starting value |
| 1 | 120 | 80 | 100 × 1.20 | 100 × 0.80 |
| 2 | 144 | 64 | 100 × 1.20² | 100 × 0.80² |
| 3 | 172.8 | 51.2 | 100 × 1.20³ | 100 × 0.80³ |
| 4 | 207.4 | 41.0 | 100 × 1.20⁴ | 100 × 0.80⁴ |
An important pattern emerges from the table: to find the value after n periods, you raise the factor to the nth power. This idea — repeatedly multiplying by the same factor — is the foundation of exponential functions, which you'll study in depth later this year.
Worked Example
Let's walk through a real-world problem from start to finish. Suppose you buy a used car for $12,000, and it loses 18% of its value each year. What is the car worth after 3 years?
Strengths & Limitations — Percent Change vs. Growth Factor
You now have two ways to describe the same change: as a percent change or as a growth/decay factor. Both are correct, but they shine in different situations. The table below compares them so you know when to reach for each tool.
| Feature | Percent Change | Growth / Decay Factor |
|---|---|---|
| Best for | Communicating change to a general audience ("Sales rose 12%") | Calculations, especially when the change repeats over multiple periods |
| Operation | Add/subtract from original | Multiply the original (one step) |
| Combining changes | Cannot simply add percentages (common error) | Multiply factors together to combine successive changes |
| Tells you at a glance | How much the quantity changed relative to the original | Whether the quantity grew (factor > 1) or shrank (factor < 1) |
| Limitation | Misleading if you try to add successive percent changes | Less intuitive for everyday communication |
Connection to Exponential Functions
The growth factor is actually the base of an exponential function. Later in this course — and especially in Algebra 2 — you will study equations of the form y = a · bx, where a is the initial amount and b is the growth factor. Everything you've learned in this lesson directly feeds into that more advanced framework.
| Concept | This Lesson | Exponential Functions (coming soon) |
|---|---|---|
| Notation | New = Original × (factor)ⁿ | y = a · bˣ |
| Initial value | "Original value" | a (y-intercept) |
| Factor | Growth factor or decay factor | b (base of the exponential) |
| Growth | Factor > 1 | b > 1 → exponential growth |
| Decay | Factor between 0 and 1 | 0 < b < 1 → exponential decay |
You will also encounter continuous growth models using the number e ≈ 2.718 in precalculus and calculus. For now, just know that the intuition you're building — multiplying by a factor to move from one value to the next — is the same intuition that powers those more advanced models.
Practice Problems
Lesson Summary
Percent change measures how much a quantity increases or decreases relative to its original value, calculated as (New − Original) / Original × 100%. A positive result means growth; a negative result means decay. The growth factor (equal to 1 + r, where r is the percent change as a decimal) lets you compute a new value in one multiplication step: New = Original × Factor. Factors greater than 1 represent increases; factors between 0 and 1 represent decreases.
When a percent change repeats over multiple periods, raise the factor to a power: New = Original × (Factor)n. To combine successive percent changes, multiply their individual factors — never simply add the percentages. These ideas form the bridge to exponential functions (y = a · bˣ), where the growth factor becomes the base of the function. Mastering percent change and growth factors now gives you a solid foundation for modeling real-world growth and decay in any context.