Percent Increase and Decrease

Introduction

Percent Increase and Decrease helps you describe how much a quantity grows or shrinks compared with where it started. You meet percent change in discounts, prices, populations, sports statistics, science data, grades, and many other real-life situations. A percent is a ratio out of 100, so percent change gives a fair way to compare changes in quantities of different sizes.

This course is designed for Grades 7–8. By the end, you should be able to identify an original value and a new value, calculate an amount of change, find a percent increase or percent decrease, use multipliers, solve reverse problems, explain why equal percent increases and decreases do not usually cancel, and apply the ideas to real situations.

A percent sign tells you that a number is being compared with 100. For example, 25% means 25 out of 100, which is the same as 0.25 or one quarter.

The video above gives a step-by-step overview of percent change. As you watch, listen for the words original value, new value, and difference. These are the key quantities in almost every percent-change problem.

Understanding Percent Change

Percent Means Per Hundred

The word percent means “per hundred.” A percentage can be written as a fraction with denominator 100 or as a decimal. For example, 40% = 40/100 = 0.40. This connection matters because calculations with percent are often easiest when you turn the percent into a decimal.

A percentage diagram can help you see that 100% represents one whole. If a quantity becomes 120% of its original value, it is the whole original amount plus an extra 20%. If it becomes 75% of its original value, 25% has been removed.

Original Value, New Value, and Change

Every basic percent-change problem begins with two values:

Original value: The starting amount before the change.
New value: The amount after the change.
Amount of change: The difference between the new value and the original value.

For an increase, use:

Amount of increase = new value − original value

Percent increase = amount of increase ÷ original value × 100%

For a decrease, use:

Amount of decrease = original value − new value

Percent decrease = amount of decrease ÷ original value × 100%

The most important rule is that the change is compared with the original value. The original value is the reference or baseline.

A tape diagram is one way to visualize an increase. It makes the original whole visible and shows the extra part as a fraction of that original whole.

Why the Original Value Is the Denominator

Suppose one game has 50 players and later has 60 players. The increase is 10 players. Because the game started with 50 players, the relative increase is 10 ÷ 50 = 0.20 = 20%.

Now suppose another game grows from 200 players to 210 players. It also gains 10 players, but its percent increase is only 10 ÷ 200 = 0.05 = 5%.

The same numerical change can represent very different percent changes. That is why percent change is useful when comparing quantities with different starting sizes.

Percent Increase

A percent increase occurs when the new value is greater than the original value. You first find how much was added, then compare that increase with the original amount.

Example: A bicycle helmet costs $80 and later costs $92.

The increase is $92 − $80 = $12.
Compare the increase with the original price: 12 ÷ 80 = 0.15.
Convert to a percent: 0.15 × 100% = 15%.

The price increased by 15%.

This video applies percent increase to a price. Notice that the original price, not the new price, is the denominator.

Increasing by a Given Percent

Sometimes you know the original value and the percent increase, and you need the new value. There are two useful methods.

Method 1: Find the increase, then add it.

If a quantity of 240 increases by 15%, first find 15% of 240:

0.15 × 240 = 36

Then add the increase:

240 + 36 = 276

Method 2: Use a multiplier.

A 15% increase means the new value is 115% of the original value. As a decimal, 115% = 1.15.

240 × 1.15 = 276

So an increase of r% can be represented by the multiplier 1 + r/100.

Using equivalent expressions helps you connect “add 15%” with “multiply by 1.15.”

Reading a Percentage-Increase Graphic

Real graphics may report percentage increases for several categories. When you read one, ask: Increase compared with what original amount? A percent increase is meaningful only when you understand its baseline.

Percent Decrease

A percent decrease occurs when the new value is less than the original value. You find the amount removed and compare that decrease with the original value.

Example: A backpack costs $60 before a sale and $45 after the discount.

The decrease is $60 − $45 = $15.
Compare the decrease with the original price: 15 ÷ 60 = 0.25.
Convert to a percent: 0.25 × 100% = 25%.

The price decreased by 25%.

Sale signs are a common place to see percent decreases. A “25% off” sign means 25% of the original price is removed, so you pay 75% of the original price.

Decreasing by a Given Percent

If a quantity decreases by 30%, then 70% remains. You can therefore multiply the original amount by 0.70.

Example: A $90 item is reduced by 30%.

0.30 × 90 = 27, so the discount is $27.

90 − 27 = 63, so the sale price is $63.

Using a multiplier gives the same result:

90 × 0.70 = 63

A decrease of r% can be represented by the multiplier 1 − r/100.

A sign offering “up to 50%” does not mean every item is half price. It means the largest advertised discount may be 50%, so you still need the actual percentage for the item you are buying.

Real-Life Applications

Discounts and Shopping

Discounts connect percent decrease with financial literacy. To check a sale price, identify the original price, calculate the discount amount, and subtract it. You can also multiply by the percentage that remains.

For example, a jacket priced at $120 with a 35% discount leaves 65% of the price to pay:

120 × 0.65 = 78

The sale price is $78.

Markups, Tax, Tips, and Fees

A markup or tax usually acts like a percent increase. If a shop increases a $50 wholesale cost by 40%, the retail amount before any tax is:

50 × 1.40 = 70

If a restaurant bill is $36 and you choose a 20% tip, the tip is:

36 × 0.20 = 7.20

The total is $43.20.

The order of operations matters in multi-step situations. Read carefully to decide which amount each percentage is based on.

Population, Science, and Data

Percent change can compare measurements across time. A town might grow from 8,000 to 8,600 residents, a plant might grow from 20 cm to 26 cm, or an experiment might record a lower energy use after a design change. In each case, the percent change compares the difference with the original measurement.

For the town:

Increase = 8,600 − 8,000 = 600

Percent increase = 600 ÷ 8,000 × 100% = 7.5%

The town's population increased by 7.5%.

Comparing Absolute Change and Percent Change

Absolute change tells you how many units were added or removed. Percent change tells you how large the change is relative to the starting amount.

Consider two school clubs:

Club A grows from 20 members to 30 members. The increase is 10 members, and the percent increase is 10 ÷ 20 × 100% = 50%.

Club B grows from 100 members to 110 members. The increase is also 10 members, but the percent increase is 10 ÷ 100 × 100% = 10%.

Both clubs gained the same number of members, but Club A had the greater relative growth.

Reverse Percent Problems

A reverse percent problem gives you the new value and the percent change, then asks for the original value.

Example: After a 15% increase, a price is $138.

A 15% increase means the new price is 115% of the original, so:

Original × 1.15 = 138

Original = 138 ÷ 1.15 = 120

The original price was $120.

Example: After a 30% decrease, a price is $63.

A 30% decrease means 70% remains, so:

Original × 0.70 = 63

Original = 63 ÷ 0.70 = 90

The original price was $90.

A useful habit is to ask, “What percent of the original remains?” before choosing the multiplier.

Successive Percent Changes

Two equal percent changes in opposite directions do not usually cancel because the second change uses a different starting value.

Suppose a value is 100.

After a 20% increase:

100 × 1.20 = 120

Then after a 20% decrease:

120 × 0.80 = 96

The final value is 96, which is 4% below the original 100.

This happens because 20% of 120 is larger than 20% of 100. In multiplier form, 1.20 × 0.80 = 0.96.

This idea is an introduction to compound change: each new percent change acts on the result of the previous step.

Percent Change and Percentage Points

Percent change and percentage-point change are not the same idea.

Suppose the share of students choosing an activity rises from 40% to 50%.

The increase in the share is 10 percentage points because 50% − 40% = 10 percentage points.

Relative to the original 40%, the percent increase is:

10 ÷ 40 × 100% = 25%

So the same situation can be described as a rise of 10 percentage points or a 25% relative increase. Always check which meaning is intended.

A Problem-Solving Strategy

Use this sequence when you solve a percent-change problem:

Identify the baseline: Find the original value.
Find the change: Subtract the smaller relevant value from the larger one.
Compare with the original: Divide the amount of change by the original value.
Convert to a percent: Multiply the decimal by 100%.
Interpret the result: State clearly whether the result is an increase or a decrease.

For a problem that gives the percent change instead, convert the percent to a decimal and use either the change amount or a multiplier.

Common Mistakes

Mistake	Why it is a problem	Better approach
Dividing by the new value	Percent change is normally measured relative to the starting value	Divide the change by the original value
Treating 20% off as paying 20%	A 20% discount removes 20%, so 80% remains	Multiply the original price by 0.80
Assuming equal increases and decreases cancel	The second percent acts on a different amount	Use successive multipliers
Confusing percentage points with percent change	A change between two percentages can be described in two different ways	State whether you mean percentage points or relative percent change

Worked Examples

Example: Increase

A streaming club grows from 160 members to 200 members.

Increase = 200 − 160 = 40

Percent increase = 40 ÷ 160 × 100% = 25%

Answer: The membership increased by 25%.

Example: Decrease

A water tank level drops from 250 liters to 190 liters.

Decrease = 250 − 190 = 60

Percent decrease = 60 ÷ 250 × 100% = 24%

Answer: The water level decreased by 24%.

Example: Find a New Value

A school orders 320 notebooks and then increases the order by 12.5%.

12.5% = 0.125

Increase = 320 × 0.125 = 40

New order = 320 + 40 = 360

Answer: The new order contains 360 notebooks.

Example: Check a Claim

A game score rises from 400 to 500. Someone claims this is a 100% increase.

Actual increase = 500 − 400 = 100

Percent increase = 100 ÷ 400 × 100% = 25%

Answer: The claim is incorrect. The score increased by 25%, not 100%.

Interactive Tasks

Quiz: Test Your Knowledge

A value rises from 50 to 60. What is the percent increase? (20 percent) (!10 percent) (!50 percent) (!120 percent)

A value of 80 is decreased by 25 percent. What is the new value? (60) (!20) (!55) (!75)

Which value is used as the denominator when calculating percent change? (Original value) (!New value) (!Difference only) (!Largest value)

A price of 200 increases by 15 percent. What is the new price? (230) (!215) (!170) (!300)

A quantity falls from 120 to 90. What is the percent decrease? (25 percent) (!30 percent) (!20 percent) (!75 percent)

A value of 100 increases by 10 percent and then decreases by 10 percent. What is the final value? (99) (!100) (!90) (!101)

An item costs 48 after a 20 percent discount. What was the original price? (60) (!58) (!68) (!40)

A group grows from 40 people to 50 people. What is the percent increase? (25 percent) (!10 percent) (!20 percent) (!50 percent)

Which multiplier represents a 12 percent decrease? (0.88) (!1.12) (!0.12) (!0.98)

A rate rises from 30 percent to 36 percent. How many percentage points did it rise? (6 percentage points) (!20 percentage points) (!36 percentage points) (!66 percentage points)

Memory Game

Original value	Starting amount before a change
New value	Amount after a change
Increase	Amount added when a quantity becomes larger
Decrease	Amount removed when a quantity becomes smaller
Percent change	Change compared with the starting amount and expressed per hundred
Multiplier	Factor used to scale an original amount directly

Drag and Drop

Match the correct terms.	Topic
Percent increase	New value is greater than the original value
Percent decrease	New value is less than the original value
Original value	Baseline used in the denominator
Increase multiplier	Factor greater than one
Decrease multiplier	Factor between zero and one

Crossword Puzzle

Percent	What word means a ratio expressed per hundred?
Original	What word describes the starting value in a percent-change problem?
Increase	What word describes a change to a larger value?
Decrease	What word describes a change to a smaller value?
Multiplier	What factor can be used to calculate a new value directly?
Discount	What price reduction is often expressed as a percent?

LearningApps

Cloze Text

Percent change compares a change with the

. An increase occurs when the new value is

than the starting value. A decrease occurs when the new value is

than the starting value. To calculate a percent increase, divide the amount of increase by the

. A 20 percent increase can be represented by the multiplier

. A 20 percent decrease can be represented by the multiplier

. Equal percent increases and decreases do not usually cancel because the second change uses a

. A rise from 40 percent to 50 percent is a change of

.

Open-Ended Tasks

Easy

Sale Price Poster: Choose three imaginary products, give each an original price and a discount, calculate each sale price, and design a clear poster that shows your calculations.
Percent Change Diary: Record four examples of increases or decreases you notice in everyday life and explain which amount would be the original value in each case.
Visual Percent Model: Draw or digitally create a bar, grid, or circle model that shows an original amount and a 25% increase or decrease.
Explain the Baseline: Write a short explanation for a younger learner showing why percent change is divided by the original value.

Standard

Shopping Investigation: Visit a shop or online store, record at least five advertised discounts, calculate the sale prices yourself, and compare your results with the displayed prices.
Interview About Percentages: Interview an adult about where they use percent increases or decreases at work or in daily life, then summarize one calculation from the interview.
Data Change Report: Collect a small set of data measured at two times, calculate the absolute and percent change for each item, and present your results in a table or chart.
Percent Change Video: Produce a two-minute teaching video that solves one increase problem and one decrease problem using both the difference method and a multiplier.

Advanced

Successive Changes Experiment: Test several pairs of equal percent increases and decreases with a calculator or spreadsheet, identify the pattern, and explain why the final value is below the original.
Reverse Percent Challenge: Create and solve four reverse problems in which the final amount and percent change are known but the original amount is missing.
Percentage Point Investigation: Find a real report containing two percentage rates, calculate both the percentage-point change and relative percent change, and explain why the two results answer different questions.
Consumer Decision Project: Compare two multi-step offers such as a discount followed by tax or two successive discounts, calculate the final costs, and justify which offer is better.

Learning Assessment

Method Selection: Given four different percent-change situations, choose an efficient method for each and justify why the method fits the information provided.
Error Analysis: Analyze a worked solution in which the new value was used as the denominator, identify the error, correct the calculation, and explain the effect of the mistake.
Multi-Step Shopping Problem: Calculate the final cost of an item after a discount and a later percent increase, then explain why the percentages cannot simply be added or subtracted.
Compare Two Changes: Compare two situations with the same absolute change but different original values and decide which has the greater relative change.
Reverse Reasoning: Determine an original value from a known final value and percent decrease, then verify your answer by applying the decrease to the original.
Communicate a Data Claim: Read a short data statement involving percentages and rewrite it so that the baseline, direction of change, and meaning of the percent are unambiguous.

Evidence of Learning

Strong evidence of learning includes the following:

Knowledge: You can explain original value, new value, amount of change, percent increase, percent decrease, multiplier, and percentage point.
Calculation skills: You can accurately calculate percent increases and decreases and can find new or original values.
Reasoning: You can choose the correct baseline, distinguish absolute from relative change, and explain successive percent changes.
Communication: You can show calculations clearly and write a complete interpretation using the words increase or decrease.
Products: You can create tables, posters, charts, written explanations, or videos that use percent change correctly.
Transfer: You can apply percent change to unfamiliar problems involving shopping, data, science, school, or financial decisions.

OERs on the Topic

The English Wikipedia article on Percentage provides background on percentages, percentage increase and decrease, and repeated percentage changes.

Khan Academy also provides Grade 7 learning materials and practice on percent increase, percent decrease, percentage change, and financial percent problems.

Linked Learning Areas

Percent increase and decrease connects arithmetic with proportional reasoning, algebraic thinking, data interpretation, and financial literacy. Understanding these links helps you recognize the same structure in many different problems.

Percent Increase and Decrease

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