Percent Increase/Decrease — I can solve percent...

SECTION 1

Where Did Percent Change Come From?

People have always needed to describe how things change. Is a price going up or down? Is a city growing or shrinking? Percent increase and percent decrease give us a simple way to compare changes, no matter how big or small the numbers are.

The word "percent" itself comes from the Latin phrase per centum, meaning "out of one hundred." Over centuries, merchants, scientists, and governments all adopted percentages to communicate changes clearly.

~3000 BCE

Ancient Trade

Babylonian and Egyptian merchants tracked how prices changed between seasons. They used fractions, but had no standard way to compare changes.

~100 BCE

Roman Taxes

The Roman emperor Augustus collected a tax of 1/100 on goods sold at auction. This "one per hundred" idea planted the seed for our modern percent.

1400s

Italian Merchants

Italian traders began writing "per cento" in financial records. They used it to describe profit, loss, and interest on loans.

1800s

Modern Statistics

Governments started using percent change to track population growth, inflation, and economic trends. It became a standard tool in newspapers and reports.

Today

Everyday Use

You see percent change everywhere: store sales, sports stats, social media follower counts, and even your test score improvements!

Here is the big question this lesson answers: When a value changes, how do we describe how big that change is compared to where it started?

SECTION 2

Core Principles & Definitions

Before we start calculating, let's make sure we understand the key ideas behind percent change. These four concepts are the building blocks you'll use in every problem.

Original Amount

This is the starting value — the number before the change happens. It is sometimes called the "initial value" or "original price."

Amount of Change

This is the difference between the new value and the original value. You find it by subtracting: New − Original. If the answer is positive, it went up. If negative, it went down.

Percent Increase

When the new value is larger than the original, the change is an increase. A percent increase tells you how much the value grew, expressed as a percent of the original.

Percent Decrease

When the new value is smaller than the original, the change is a decrease. A percent decrease tells you how much the value shrank, expressed as a percent of the original.

✦ KEY TAKEAWAY

Think of percent change like a score in a video game. Imagine you had 200 coins and now you have 250 coins. You gained 50 coins — but how impressive is that? Percent change answers that question by comparing the gain (50) to where you started (200). That's a 25% increase. If someone else gained 50 coins but started with 1,000, their percent increase is only 5%. Percent change always compares to the original amount.

SECTION 3

Seeing Percent Change

A picture can make percent change much easier to understand. The diagram below shows two bar charts side by side. One shows a percent increase and the other shows a percent decrease. Notice how the amount of change is always compared back to the original bar, not the new bar.

Both examples start at $200. On the left, the value rises by $50 to $250, which is a 25% increase. On the right, the value drops by $50 to $150, which is a 25% decrease. The change ($50) is divided by the original ($200) in both cases.

The key thing to notice is that the same dollar amount ($50) gives the same percent in both directions — but only because the original value is the same ($200). If the originals were different, the percents would be different too.

SECTION 4

The Formulas You Need

There is one main formula for percent change. Once you know it, you can handle both increases and decreases. Let's break it down step by step.

PERCENT CHANGE FORMULA

Percent Change = (Amount of Change ÷ Original Amount) × 100

Amount of Change = New Value − Original Value. If the result is positive, it is an increase. If negative, it is a decrease. Original Amount = the starting value (before the change happened).

You can also think of this formula in two smaller steps.

STEP 1 — FIND THE CHANGE

Amount of Change = New Value − Original Value

Subtract the original from the new. A positive answer means increase; a negative answer means decrease.

STEP 2 — DIVIDE AND CONVERT

Percent Change = (Amount of Change ÷ Original Amount) × 100%

Divide the change by the original. This gives you a decimal. Multiply by 100 to turn it into a percent.

⚠️ Common Mistake Alert

Always divide by the original amount, not the new amount! The original is your starting point, and it is always the denominator (the bottom number in the fraction). Dividing by the wrong number is the #1 error students make.

SECTION 5

Increase vs. Decrease — A Closer Look

Let's look at several real-life examples side by side. The table below shows both increases and decreases so you can practice spotting the difference.

Examples of percent increase and percent decrease in everyday life

Situation	Original	New Value	Change	Type
Sneaker price goes up	$80	$100	+$20 → 25% increase	Increase
Video game on sale	$60	$45	−$15 → 25% decrease	Decrease
Followers grow	400	500	+100 → 25% increase	Increase
Test errors drop	20 errors	12 errors	−8 → 40% decrease	Decrease

Follow these four steps every time you solve a percent change problem. The most important rule is highlighted in Step 3: always divide by the original value.

One helpful tip: if the new value is bigger, you already know it's an increase before you even calculate. If the new value is smaller, it's a decrease. This is a great way to check your answer and make sure it makes sense.

SECTION 6

Worked Example: Solving a Percent Change Problem

Let's walk through a full problem together. Read each step carefully and notice how we follow the flowchart from Section 5.

A bike was originally priced at $120. It is now on sale for $90. What is the percent decrease?

Step 1 — Identify the ValuesThe original price is $120. The new price is $90. Since the new price is lower, we expect a decrease.

Step 2 — Find the Amount of ChangeAmount of Change = New − Original = $90 − $120 = −$30. The negative sign confirms this is a decrease.

Change = −$30

Step 3 — Divide by the OriginalWe divide the amount of change by the original amount: 30 ÷ 120 = 0.25. (We can ignore the negative sign for now and label it as a decrease at the end.)

Decimal = 0.25

Step 4 — Convert to a PercentMultiply the decimal by 100: 0.25 × 100 = 25%. Since the price went down, this is a 25% decrease.

The bike's price decreased by 25%.

Step 5 — Interpret the AnswerA 25% decrease means the sale took away one quarter of the original price. For every dollar the bike used to cost, 25 cents was removed. That's a pretty good deal!

✅ Quick Check

Does the answer make sense? The bike dropped from $120 to $90. Half of $120 is $60, so a 50% decrease would bring it to $60. Our answer of 25% decrease (bringing it to $90) is smaller than 50% decrease, which checks out!

SECTION 7

Helpful Tips & Common Pitfalls

Students often make a few predictable mistakes with percent change problems. Learning about these now will save you a lot of headaches on tests and homework!

Avoid these four common mistakes

Common Mistake	Why It's Wrong	What to Do Instead
Dividing by the new value	Percent change compares to where you started, not where you ended up.	Always divide by the original (starting) value.
Forgetting to multiply by 100	You get a decimal (like 0.25) instead of a percent (25%).	After dividing, multiply by 100 and add the % symbol.
Mixing up increase and decrease	Writing "increase" when the value actually went down (or vice versa).	Compare the new value to the original: bigger = increase, smaller = decrease.
Using the change as the percent	A change of $20 is NOT automatically a 20% change.	You must divide the change by the original to find the actual percent.

✦ KEY TAKEAWAY

Think of the original value as the "home base" in a game of tag. No matter how far you run from home base (the change), your distance is always measured from home base, not from where someone else is standing. The original value is always your home base in percent change.

SECTION 8

Connecting to Future Math Topics

Percent change is not just a stand-alone skill. It connects to many topics you'll see in algebra, finance, and science. The table below shows how this lesson's ideas grow into bigger ones.

How percent change connects to future topics

This Lesson	Where It Leads
Finding percent increase/decrease	Calculating simple and compound interest in personal finance
Comparing change to the original	Proportional reasoning and scale factors in geometry
Writing change as a decimal (0.25)	Using multipliers like 1.25 (increase) or 0.75 (decrease) in algebra
Interpreting the direction of change	Analyzing slope (rate of change) on a graph in algebra

One exciting preview: in algebra, you'll learn a shortcut called the multiplier method. Instead of finding the change and dividing, you can multiply the original by a single number. For a 25% increase, you multiply by 1.25. For a 25% decrease, you multiply by 0.75. This makes repeated percent changes (like interest on a savings account) much faster to compute.

🤯 Fun Fact

If a price increases by 50% and then decreases by 50%, you do NOT end up back where you started! Try it: $100 → +50% → $150 → −50% → $75. You actually end up lower. This surprising result shows why understanding percent change deeply really matters.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. For each one, follow the four steps: identify values, find the change, divide by the original, and convert to a percent.

PROBLEM 1 — CONCEPTUAL

A shirt's price changes from $40 to $50. Is this a percent increase or a percent decrease? How do you know without doing any math?

PROBLEM 2 — BASIC CALCULATION

A class had 30 students last year. This year it has 36 students. What is the percent increase?

PROBLEM 3 — INTERMEDIATE

A store sells a backpack for $45. During a sale, the price drops to $36. What is the percent decrease?

PROBLEM 4 — APPLIED

Maya scored 60 points on her first math quiz and 78 points on her second quiz. She says, "I improved by 18%." Is she correct? Explain your reasoning.

PROBLEM 5 — CRITICAL THINKING

A population of 500 deer increases by 20% one year, then decreases by 20% the next year. Does the population return to 500? Show your work and explain what happens.

SUMMARY

Lesson Summary

Percent change measures how much a value grows or shrinks compared to its starting point. To find it, calculate the amount of change (New − Original), divide by the original amount, and multiply by 100. If the result is positive, it is a percent increase. If negative, it is a percent decrease.

The most important rule to remember is: always divide by the original value. This is what makes percent change a fair comparison. Whether you're tracking prices, scores, populations, or stats, the formula stays the same: Percent Change = (Change ÷ Original) × 100. With practice, you'll be able to spot and solve these problems quickly and confidently.