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{{T}}
[[Category:English]]
[[Category:Percent Increase and Decrease]]

{{BR}}
= 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.

[[File:Percent sign icon.svg|250px|frameless|center]]

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.

{{#ev:youtube|https://www.youtube.com/watch?v=ZL2ILv98IZU|500|center}}

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.

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= Understanding Percent Change =

{{BR}}
== Percent Means Per Hundred ==

The word [[English:Percentage|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.

[[File:Percentage Pie chart.png|500px|frameless|center]]

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.

{{BR}}
== Original Value, New Value, and Change ==

Every basic percent-change problem begins with two values:

# [[English:Original value|Original value]]: The starting amount before the change.
# [[English:New value|New value]]: The amount after the change.
# [[English:Amount of 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.

{{#ev:youtube|https://www.youtube.com/watch?v=n21E7OzXOwI|500|center}}

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.

{{BR}}
== 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.

{{BR}}
= 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%'''.

{{#ev:youtube|https://www.youtube.com/watch?v=7c8eR_CCx-o|500|center}}

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

{{BR}}
== 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'''.

{{#ev:youtube|https://www.youtube.com/watch?v=lwctkmKBDxA|500|center}}

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

{{BR}}
== Reading a Percentage-Increase Graphic ==

[[File:WPWP Percentage Increase in Contents.png|500px|frameless|center]]

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.

{{BR}}
= 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%'''.

[[File:Sale sign.jpg|400px|frameless|center]]

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.

{{BR}}
== 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'''.

[[File:Shoeby tot 50% sale sign, Hoogezand (2019) 01.jpg|350px|frameless|center]]

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.

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= Real-Life Applications =

{{BR}}
== Discounts and Shopping ==

[[File:Prozente im Verkauf.jpg|500px|frameless|center]]

Discounts connect percent decrease with [[English:Financial literacy|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.

{{BR}}
== 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.

[[File:Price Tag.png|400px|frameless|center]]

{{BR}}
== 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%.

{{BR}}
= 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.

{{BR}}
= 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.

{{BR}}
= 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 [[English:Compound growth|compound change]]: each new percent change acts on the result of the previous step.

{{BR}}
= 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.

{{BR}}
= A Problem-Solving Strategy =

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

# [[English:Identify the baseline|Identify the baseline]]: Find the original value.
# [[English:Find the change|Find the change]]: Subtract the smaller relevant value from the larger one.
# [[English:Compare with the original|Compare with the original]]: Divide the amount of change by the original value.
# [[English:Convert to a percent|Convert to a percent]]: Multiply the decimal by 100%.
# [[English:Interpret the result|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.

{{BR}}
== Common Mistakes ==

{| class="wikitable"
! 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
|}

{{BR}}
= Worked Examples =

{{BR}}
== 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%.'''

{{BR}}
== 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%.'''

{{BR}}
== 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.'''

{{BR}}
== 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%.'''

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| 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
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! 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
|}
{{E}}

<br>
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| 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?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Percent+Increase+and+Decrease </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
Percent change compares a change with the { original value }. An increase occurs when the new value is { greater } than the starting value. A decrease occurs when the new value is { smaller } than the starting value. To calculate a percent increase, divide the amount of increase by the { original value }. A 20 percent increase can be represented by the multiplier { 1.20 }. A 20 percent decrease can be represented by the multiplier { 0.80 }. Equal percent increases and decreases do not usually cancel because the second change uses a { different baseline }. A rise from 40 percent to 50 percent is a change of { 10 percentage points }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Sale Price Poster|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.
# [[English:Percent Change Diary|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.
# [[English:Visual Percent Model|Visual Percent Model]]: Draw or digitally create a bar, grid, or circle model that shows an original amount and a 25% increase or decrease.
# [[English:Explain the Baseline|Explain the Baseline]]: Write a short explanation for a younger learner showing why percent change is divided by the original value.

{{BR}}
=== Standard ===
# [[English:Shopping Investigation|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.
# [[English:Interview About Percentages|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.
# [[English:Data Change Report|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.
# [[English:Percent Change Video|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.

{{BR}}
=== Advanced ===
# [[English:Successive Changes Experiment|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.
# [[English:Reverse Percent Challenge|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.
# [[English:Percentage Point Investigation|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.
# [[English:Consumer Decision Project|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.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Method Selection|Method Selection]]: Given four different percent-change situations, choose an efficient method for each and justify why the method fits the information provided.
# [[English:Error Analysis|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.
# [[English:Multi-Step Shopping Problem|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.
# [[English:Compare Two Changes|Compare Two Changes]]: Compare two situations with the same absolute change but different original values and decide which has the greater relative change.
# [[English:Reverse Reasoning|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.
# [[English:Communicate a Data Claim|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.

<br>
<br>

{{BR}}
= Evidence of Learning =

Strong evidence of learning includes the following:

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

<br>
<br>

{{BR}}
= OERs on the Topic =

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

<iframe> https://en.m.wikipedia.org/wiki/Percentage </iframe>

<br>

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

{{BR}}
= 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.

{| align=center
{{:D-Tab}}
'''[[English:Percent Increase and Decrease|Percent Increase and Decrease]]'''
# [[English:Percentages|Percentages]]
# [[English:Ratios|Ratios]]
# [[English:Proportional reasoning|Proportional reasoning]]
# [[English:Decimals|Decimals]]
# [[English:Financial literacy|Financial literacy]]
# [[English:Data analysis|Data analysis]]
# [[English:Compound growth|Compound growth]]
|}

[[Category:English]]
[[Category:Mathematics]]
[[Category:Percentages]]
[[Category:Arithmetic]]
[[Category:Financial literacy]]
[[Category:Grades 7-8]]

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Percent Increase and Decrease]]
[[Category:Mathematics]]
[[Category:Percentages]]
[[Category:Arithmetic]]
[[Category:Financial literacy]]
[[Category:Grades 7-8]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Percent Increase and Decrease - Versionsgeschichte

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