Percent Change & Growth Factors — I can interpret percent change and growth/decay factor in context. - Grade 9 Math

SECTION 1

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.

~300 BCE • Ancient Proportional Reasoning

Greek and Egyptian mathematicians used ratios and proportions to solve trade and taxation problems, laying the groundwork for percentage thinking.

1400s • Rise of "Per Cento" in Italy

Italian merchants began expressing interest rates and profits as parts per hundred, and the "%" symbol evolved from abbreviations of per cento.

1600s–1700s • Compound Interest & Growth

Jacob Bernoulli and other mathematicians studied how repeatedly applying a percent increase leads to exponential growth, giving rise to the concept of growth factors.

1900s–Today • Growth Factors Everywhere

Percent change and growth/decay factors now model inflation, epidemics, battery drain, carbon emissions, and much more — making them essential tools in every STEM field.

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.

SECTION 2

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

The ratio of the amount of change to the original value, expressed as a percentage. A positive result means an increase; a negative result means a decrease.

Growth Factor

The single number you multiply the original amount by to get the new amount. For a 20% increase, the growth factor is 1.20. It equals 1 + (percent change as a decimal).

Decay Factor

A growth factor that is less than 1, representing a decrease. For a 15% decrease, the decay factor is 0.85. It equals 1 − (percent decrease as a decimal).

Original (Initial) Value

The starting quantity before the change occurs. This is always the denominator in the percent change formula and provides the basis for comparison.

New (Final) Value

The quantity after the change has occurred. You obtain it by multiplying the original value by the growth (or decay) factor.

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KEY TAKEAWAY

Think of the growth factor like a zoom setting on a photo. A factor of 1.30 zooms in by 30% (the image gets bigger), a factor of 1.00 means no change, and a factor of 0.70 zooms out by 30% (the image gets smaller). The factor tells you the multiplier that takes you from the old quantity to the new one in a single step.

SECTION 3

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.

Starting from 200, multiplying by the growth factor 1.25 (green arrow) gives 250, while multiplying by the decay factor 0.75 (pink arrow) gives 150. The factor captures both the direction and the size of the change.

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.

SECTION 4

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.

PERCENT CHANGE

Percent Change = ((New Value − Original Value) / Original Value) × 100%

A positive result indicates a percent increase; a negative result indicates a percent decrease. The original value is always the denominator — it is your reference point.

GROWTH / DECAY FACTOR

Growth Factor = 1 + r (where r is the percent change written as a decimal)

If r = 0.25 (a 25% increase), then the growth factor = 1.25. If r = −0.25 (a 25% decrease), then the factor = 0.75. When the factor is less than 1, we often call it a decay factor.

NEW VALUE FORMULA

New Value = Original Value × Growth Factor

This is the single-step shortcut. For example, to find the value after a 10% increase: New = Original × 1.10. To find the value after a 10% decrease: New = Original × 0.90.

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Common Mistake Alert

Students sometimes confuse a 50% increase followed by a 50% decrease as "getting back to the original." It doesn't! A 50% increase on 100 gives 150. Then a 50% decrease on 150 gives 75, not 100. The growth factors multiply: 1.50 × 0.50 = 0.75, which means the combined effect is a 25% decrease.

SECTION 5

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.

Both curves start at 100. The blue growth curve multiplies by 1.20 each period (100 → 120 → 144 → 172.8 → 207.4), while the red decay curve multiplies by 0.80 each period (100 → 80 → 64 → 51.2 → 41.0). After four periods, the growing quantity has more than doubled while the decaying quantity has lost nearly 60% of its value.

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

Growth vs. Decay over four periods, starting from 100

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.

SECTION 6

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?

Car Depreciation — Finding Value After Repeated Decay

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Step 1 — Identify Given Values

Original value = $12,000. The car loses 18% per year, so the percent change is −18%, or r = −0.18. The number of time periods is n = 3 years.

Original = $12,000, r = −0.18, n = 3
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Step 2 — Calculate the Decay Factor

Growth/decay factor = 1 + r = 1 + (−0.18) = 0.82. Since this is less than 1, it confirms that the car is decreasing in value.

Decay factor = 0.82
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Step 3 — Apply the Factor for 3 Years

New Value = Original × (Decay Factor)ⁿ = $12,000 × (0.82)³. First compute (0.82)³: 0.82 × 0.82 = 0.6724, then 0.6724 × 0.82 = 0.551368.

(0.82)³ ≈ 0.5514
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Step 4 — Compute the Final Value

New Value = $12,000 × 0.5514 = $6,616.80.

After 3 years, the car is worth approximately $6,616.80
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Step 5 — Interpret in Context

The car lost $12,000 − $6,616.80 = $5,383.20 in three years. That's a total percent decrease of ($5,383.20 / $12,000) × 100% ≈ 44.9%. Notice this is not simply 3 × 18% = 54% — the losses compound on a shrinking base each year, so the total percentage lost is less than you might expect from simple multiplication.

Total depreciation ≈ 44.9% (not 54%)

SECTION 7

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

Comparison of percent change and growth/decay factor

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KEY TAKEAWAY

Think of percent change as the headline and the growth factor as the engine. The headline tells people what happened ("prices up 8%!"), but when you need to crunch numbers — especially repeatedly — the engine (factor of 1.08) does the heavy lifting. Master converting between the two and you'll always pick the right tool for the job.

SECTION 8

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 · b^x, 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

How this lesson's concepts map to exponential functions

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A friend says, "My investment had a growth factor of 0.92 last year." Did the investment increase or decrease in value? Explain your reasoning using the relationship between the growth factor and percent change.

Show answer

The investment decreased in value. A growth factor of 0.92 is less than 1, which always signals a decrease. To find the percent change, subtract 1 from the factor: 0.92 − 1 = −0.08, which corresponds to a −8% change (an 8% loss). Any time the growth factor is between 0 and 1, the quantity is decaying; only factors greater than 1 indicate growth.

Question 2: BASIC CALCULATION

A town's population was 5,400 last year and is now 5,832. What is the percent change, and what is the growth factor?

Show answer

First find the percent change: (5,832 − 5,400) / 5,400 × 100% = 432 / 5,400 × 100% = 8%. The population increased by 8%. The growth factor is 1 + 0.08 = 1.08. You can verify: 5,400 × 1.08 = 5,832, confirming the answer.

Question 3: INTERMEDIATE

A laptop originally costs $950. During a sale it is marked down 15%, and then an additional 10% is taken off the sale price. What is the final price? What single percent change is equivalent to both discounts combined?

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Apply the two decay factors in sequence. The first discount gives a factor of 0.85 and the second gives 0.90. The combined factor is 0.85 × 0.90 = 0.765. So the final price = $950 × 0.765 = $726.75. The single equivalent percent change is 0.765 − 1 = −0.235, or a 23.5% total decrease. Notice this is less than 15% + 10% = 25% because the second discount applies to an already-reduced price.

Question 4: APPLIED

A biologist tracks a bacterial colony that doubles every 4 hours. If the colony starts with 500 bacteria, write an expression for the number of bacteria after 20 hours. Then calculate the value and state the overall percent increase.

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"Doubles" means the growth factor per 4-hour period is 2. In 20 hours there are 20 / 4 = 5 periods. The expression is 500 × 2⁵. Computing: 2⁵ = 32, so the colony has 500 × 32 = 16,000 bacteria. The overall percent increase is (16,000 − 500) / 500 × 100% = 15,500 / 500 × 100% = 3,100%. This dramatic increase illustrates why exponential growth is so powerful even over relatively short time spans.

Question 5: CRITICAL THINKING

Stock A gained 40% one year and then lost 40% the next year. Stock B lost 40% one year and then gained 40% the next year. Without calculating, predict whether both stocks end at the same value. Then verify your prediction and explain why the result makes sense using growth factors.

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Both stocks end at the same value. For Stock A the combined factor is 1.40 × 0.60 = 0.84. For Stock B the combined factor is 0.60 × 1.40 = 0.84. Multiplication is commutative (order doesn't matter), so the final result is identical: both end at 84% of their starting value, a 16% total loss. This demonstrates two key lessons. First, multiplying growth factors is order-independent. Second, a percent gain followed by an equal percent loss (or vice versa) never returns you to the original value — the net effect is always a loss, because the gain and loss apply to different base amounts.

SUMMARY

Lesson Summary

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.