Percentage Increases & Decreases

Percentage Increases & Decreases - Comprehensive Notes

Percentage Increases & Decreases: Comprehensive Notes

Welcome to our detailed guide on Percentage Increases & Decreases. Whether you're a student navigating through mathematical concepts or someone aiming to enhance numerical literacy, this guide offers thorough explanations, properties, and a wide range of examples to help you master the fundamentals of calculating and understanding percentage increases and decreases.

Introduction

Percentages are a vital part of everyday mathematics, representing proportions relative to a whole. Understanding how to calculate percentage increases and decreases is essential for various real-life applications, including finance, economics, statistics, and more. This guide provides a comprehensive overview of percentage increases and decreases, their properties, methods, and common pitfalls to ensure a solid mathematical foundation.

Basic Concepts of Percentage Increases & Decreases

Before diving into calculations, it's important to grasp the foundational concepts that underpin percentage increases and decreases.

Understanding Percentage Increase

A percentage increase occurs when a quantity becomes larger by a certain percentage of its original value.

Formula: Percentage Increase (%) = [(New Value - Original Value) / Original Value] × 100

Example: If the price of a book increases from $20 to $25, the percentage increase is:

Percentage Increase = [(25 - 20) / 20] × 100 = (5 / 20) × 100 = 25%

Understanding Percentage Decrease

A percentage decrease occurs when a quantity becomes smaller by a certain percentage of its original value.

Formula: Percentage Decrease (%) = [(Original Value - New Value) / Original Value] × 100

Example: If the price of a shirt decreases from $50 to $40, the percentage decrease is:

Percentage Decrease = [(50 - 40) / 50] × 100 = (10 / 50) × 100 = 20%

Key Terms

Original Value: The initial quantity before the increase or decrease.
New Value: The quantity after the increase or decrease.
Difference: The amount by which the original value has increased or decreased.

Properties of Percentage Increases & Decreases

Understanding the properties of percentage changes is crucial for performing accurate calculations and interpreting results correctly.

Non-commutativity of Percentage Increase and Decrease

Applying a percentage increase followed by the same percentage decrease does not return the original value, and vice versa.

Example: Increase $100 by 10% and then decrease the new amount by 10%:


        Step 1: Increase by 10%
        New Amount = 100 + (0.10 × 100) = 110

        Step 2: Decrease by 10%
        Final Amount = 110 - (0.10 × 110) = 110 - 11 = 99

The final amount is $99, not the original $100.

Percentage Increase and Decrease Relative to Different Bases

Percentage increases and decreases are always relative to the original value, not the new value.

Example: If a population increases from 1,000 to 1,200, the percentage increase is 20%. If it then decreases to 1,100, the percentage decrease is approximately 8.33%, not 20%.

Calculations with Percentage Increases & Decreases

Performing calculations involving percentage increases and decreases involves specific steps to ensure accurate results.

Calculating Percentage Increase

To calculate a percentage increase:

Determine the difference between the new value and the original value.
Divide the difference by the original value.
Multiply the result by 100 to convert it to a percentage.

Formula: [(New Value - Original Value) / Original Value] × 100

Example: Increase from 80 to 100:


        Difference = 100 - 80 = 20
        Percentage Increase = (20 / 80) × 100 = 25%

Calculating Percentage Decrease

To calculate a percentage decrease:

Determine the difference between the original value and the new value.
Divide the difference by the original value.
Multiply the result by 100 to convert it to a percentage.

Formula: [(Original Value - New Value) / Original Value] × 100

Example: Decrease from 150 to 120:


        Difference = 150 - 120 = 30
        Percentage Decrease = (30 / 150) × 100 = 20%

Examples of Percentage Increases & Decreases

Understanding through examples is key to mastering percentage increases and decreases. Below are a variety of problems ranging from easy to hard, each with detailed solutions.

Example 1: Calculating a Percentage Increase

Problem: The price of a laptop increases from $800 to $900. What is the percentage increase?

Solution:


        Difference = 900 - 800 = 100
        Percentage Increase = (100 / 800) × 100 = 12.5%

Therefore, the price of the laptop increased by 12.5%.

Example 2: Calculating a Percentage Decrease

Problem: A smartphone was originally priced at $600 but is now sold for $450. What is the percentage decrease?

Solution:


        Difference = 600 - 450 = 150
        Percentage Decrease = (150 / 600) × 100 = 25%

Therefore, the smartphone's price decreased by 25%.

Example 3: Multiple Percentage Changes

Problem: A company's revenue was $1,200,000 last year. This year, it increased by 10%, and next year it is expected to decrease by 5%. What will be the revenue next year?

Solution:


        Year 1 Revenue = $1,200,000

        Year 2 Revenue after 10% increase:
        Increase = 0.10 × 1,200,000 = 120,000
        New Revenue = 1,200,000 + 120,000 = $1,320,000

        Year 3 Revenue after 5% decrease:
        Decrease = 0.05 × 1,320,000 = 66,000
        Final Revenue = 1,320,000 - 66,000 = $1,254,000

Therefore, the company's revenue next year will be $1,254,000.

Example 4: Finding the Original Value from a Percentage Increase

Problem: After a 15% increase, the price of a bicycle is $230. What was the original price?

Solution:


        Let Original Price = X
        15% of X = 0.15 × X
        X + 0.15X = 1.15X = 230
        X = 230 / 1.15 = 200

Therefore, the original price of the bicycle was $200.

Example 5: Finding the New Value After a Percentage Decrease

Problem: A jacket costs $120. It is on sale for 20% off. What is the sale price?

Solution:


        Discount = 20% of 120 = 0.20 × 120 = 24
        Sale Price = 120 - 24 = $96

Therefore, the sale price of the jacket is $96.

Word Problems: Application of Percentage Increases & Decreases

Applying percentage increases and decreases to real-life scenarios enhances understanding and demonstrates their practical utility. Here are several word problems that incorporate these concepts, along with their solutions.

Example 1: Salary Raise

Problem: Maria's current salary is $45,000 per year. She receives a 10% raise. What is her new salary?

Solution:


        Raise = 10% of 45,000 = 0.10 × 45,000 = 4,500
        New Salary = 45,000 + 4,500 = $49,500

Therefore, Maria's new salary is $49,500 per year.

Example 2: Price Reduction

Problem: A car originally costs $25,000. Due to a promotion, its price is reduced by 12%. What is the new price of the car?

Solution:


        Reduction = 12% of 25,000 = 0.12 × 25,000 = 3,000
        New Price = 25,000 - 3,000 = $22,000

Therefore, the new price of the car is $22,000.

Example 3: Population Decline

Problem: A town has a population of 15,000. Over a year, the population decreases by 5%. What is the current population?

Solution:


        Decrease = 5% of 15,000 = 0.05 × 15,000 = 750
        Current Population = 15,000 - 750 = 14,250

Therefore, the current population of the town is 14,250.

Example 4: Exam Score Improvement

Problem: John scored 70% on his first exam. He aims to improve his score by 20% in his next exam. What score should he aim for?

Solution:


        Improvement = 20% of 70 = 0.20 × 70 = 14
        Target Score = 70 + 14 = 84%

Therefore, John should aim for an 84% score on his next exam.

Example 5: Investment Growth

Problem: Sarah invests $5,000 in a savings account that offers a 6% annual interest rate. What will be the value of her investment after one year?

Solution:


        Interest = 6% of 5,000 = 0.06 × 5,000 = 300
        Value after one year = 5,000 + 300 = $5,300

Therefore, Sarah's investment will be worth $5,300 after one year.

Strategies and Tips for Working with Percentage Increases & Decreases

Enhancing your skills in calculating percentage increases and decreases involves employing effective strategies and consistent practice. Here are some tips to help you improve:

1. Master the Fundamental Percentage Formula

Understand and memorize the core formula for calculating percentage increases and decreases:

Percentage Increase: [(New - Original) / Original] × 100
Percentage Decrease: [(Original - New) / Original] × 100

Example: Calculate a 10% increase from 50.


        Percentage Increase = [(55 - 50) / 50] × 100 = (5 / 50) × 100 = 10%

2. Convert Percentages to Decimals

Always convert percentages to decimals by dividing by 100 before performing calculations.

Example: 25% = 0.25

3. Use Proportional Reasoning

Think of percentages as parts of a whole (100) to simplify problem-solving.

Example: 50% is half of the whole.

4. Practice Converting Between Forms

Regularly practice converting between percentages, decimals, and fractions to build fluency.

Example: Convert 0.75 to a percentage and a fraction.


        0.75 = 75%
        0.75 = 3/4

5. Break Down Complex Problems

For complex percentage problems, break them down into smaller, more manageable parts.

Example: To find a 15% increase followed by a 10% decrease, calculate each step separately.

6. Use Visual Aids

Employ visual tools like pie charts, bar graphs, and number lines to better understand and visualize percentage changes.

Example: A pie chart can help illustrate how a 20% increase affects the total value.

7. Double-Check Your Work

Always review your calculations to catch and correct any mistakes.

Example: After calculating a 25% increase from 80 to 100, verify by checking if 80 + 20 = 100.

8. Apply Real-Life Scenarios

Use real-life situations to practice percentage calculations, making the concepts more relatable and easier to understand.

Example: Calculate discounts while shopping or interest on savings.

9. Memorize Common Percentage Values

Knowing key percentages (10%, 25%, 50%, 75%, etc.) and their decimal and fractional equivalents can speed up calculations.

Example: 10% = 0.10 = 1/10

10. Teach Others

Explaining percentage increase and decrease concepts to someone else can reinforce your understanding and highlight any areas needing improvement.

Common Mistakes in Percentage Increases & Decreases and How to Avoid Them

Being aware of common errors can help you avoid them and improve your calculation accuracy.

1. Confusing Percentage Increase with Percentage of

Mistake: Misunderstanding the difference between calculating a percentage increase and finding a percentage of a number.

Solution: Remember that a percentage increase involves adding to the original value, while finding a percentage of a number involves multiplication.


        Example:
        Incorrect: 20% increase of 50 = 0.20 × 50 = 10
        Correct: 20% increase of 50 = 50 + 10 = 60

2. Forgetting to Convert Percentages to Decimals

Mistake: Performing calculations without converting percentages to decimals first.

Solution: Always convert the percentage to a decimal by dividing by 100 before multiplying or dividing.


        Example:
        Incorrect: 15% of 200 = 15 × 200 = 3,000
        Correct: 15% of 200 = 0.15 × 200 = 30

3. Misapplying the Percentage Change Formula

Mistake: Using the wrong formula or incorrect values in the formula when calculating percentage increases or decreases.

Solution: Use the correct formulas:

Percentage Increase = [(New - Original) / Original] × 100
Percentage Decrease = [(Original - New) / Original] × 100


        Example:
        Incorrect: [(Original - New) / Original] × 100 for percentage increase
        Correct: [(New - Original) / Original] × 100 for percentage increase

4. Incorrectly Calculating the Difference

Mistake: Miscalculating the difference between the new value and the original value.

Solution: Carefully subtract the original value from the new value (for increases) or the new value from the original value (for decreases).


        Example:
        Incorrect: Difference = Original + New
        Correct: Difference = New - Original (for increases) or Original - New (for decreases)

5. Not Simplifying Percentages

Mistake: Leaving percentages unsimplified, leading to unnecessarily complex answers.

Solution: Simplify the percentage results to the nearest appropriate decimal place or fraction.


        Example:
        Incorrect: 12.3456% rounded to 12.3456%
        Correct: 12.3456% rounded to 12.35% (depending on required precision)

6. Ignoring the Base Value in Percentage Calculations

Mistake: Using the wrong base value (original or new value) when performing calculations.

Solution: Always ensure you are using the original value as the base for calculating percentage increases and decreases.


        Example:
        Incorrect: [(New - Original) / New] × 100 for percentage increase
        Correct: [(New - Original) / Original] × 100 for percentage increase

7. Rushing Through Calculations

Mistake: Performing percentage calculations too quickly without ensuring each step is accurate.

Solution: Take your time to follow each step carefully, especially when dealing with complex percentages.

8. Misinterpreting "Of" in Percentage Problems

Mistake: Misunderstanding what the "of" signifies in percentage problems, leading to incorrect calculations.

Solution: Recognize that "of" indicates multiplication in percentage problems.


        Example:
        Incorrect: 20% of 50 = 20 + 50 = 70
        Correct: 20% of 50 = 0.20 × 50 = 10

9. Overcomplicating Simple Problems

Mistake: Adding unnecessary steps or complexity to straightforward percentage problems.

Solution: Simplify your approach and follow the fundamental steps for each operation.


        Example:
        Incorrect: To find 10% increase from 100, calculate 10 + 100 = 110%
        Correct: 10% increase from 100 = 100 + (0.10 × 100) = 110

10. Mixing Up Percentage Increase and Percentage Decrease

Mistake: Confusing percentage increase with percentage decrease, leading to incorrect results.

Solution: Clearly distinguish between percentage increases (which add to the original value) and percentage decreases (which subtract from the original value).


        Example:
        Incorrect: 20% increase from 50 = 50 × 0.80 = 40
        Correct: 20% increase from 50 = 50 + (0.20 × 50) = 60

Practice Questions: Test Your Percentage Increases & Decreases Skills

Practicing with a variety of problems is key to mastering percentage increases and decreases. Below are practice questions categorized by difficulty level, along with their solutions.

Level 1: Easy

A shirt costs $40 and is increased by 10%. What is the new price?
Find the percentage increase from 80 to 100.
A book was priced at $25 and is now sold for $20. What is the percentage decrease?
Calculate a 15% increase on $200.
What is a 5% decrease of $300?

Solutions:

Solution:
Increase = 10% of 40 = 0.10 × 40 = 4
New Price = 40 + 4 = $44
Solution:
Percentage Increase = [(100 - 80) / 80] × 100 = (20 / 80) × 100 = 25%
Solution:
Percentage Decrease = [(25 - 20) / 25] × 100 = (5 / 25) × 100 = 20%
Solution:
Increase = 15% of 200 = 0.15 × 200 = 30
New Amount = 200 + 30 = $230
Solution:
Decrease = 5% of 300 = 0.05 × 300 = 15
New Amount = 300 - 15 = $285

Level 2: Medium

A laptop originally priced at $1,200 is now sold for $1,440. What is the percentage increase?
Find the original price if $80 is a 25% decrease.
Calculate a 12% decrease on $450.
What is the percentage decrease from 150 to 120?
A company's revenue increased from $500,000 to $600,000. What is the percentage increase?

Solutions:

Solution:
Percentage Increase = [(1,440 - 1,200) / 1,200] × 100 = (240 / 1,200) × 100 = 20%
Solution:
Let Original Price = X
25% decrease = 0.25 × X = 0.25X
New Price = X - 0.25X = 0.75X = 80
X = 80 / 0.75 = $106.67
Solution:
Percentage Decrease = 12% of 450 = 0.12 × 450 = 54
New Amount = 450 - 54 = $396
Solution:
Percentage Decrease = [(150 - 120) / 150] × 100 = (30 / 150) × 100 = 20%
Solution:
Percentage Increase = [(600,000 - 500,000) / 500,000] × 100 = (100,000 / 500,000) × 100 = 20%

Level 3: Hard

A car's value depreciates from $25,000 to $20,000. What is the percentage decrease?
Find the new price after a 18% increase on $350.
If a population decreases by 6% from 50,000 to 47,000, what is the percentage decrease?
Calculate the original price if after a 30% increase, the price becomes $130.
A store reduces the price of a TV by 25%, making it $450. What was the original price?

Solutions:

Solution:
Percentage Decrease = [(25,000 - 20,000) / 25,000] × 100 = (5,000 / 25,000) × 100 = 20%
Solution:
Increase = 18% of 350 = 0.18 × 350 = 63
New Price = 350 + 63 = $413
Solution:
Percentage Decrease = [(50,000 - 47,000) / 50,000] × 100 = (3,000 / 50,000) × 100 = 6%
Solution:
Let Original Price = X
30% increase = 0.30 × X = 0.30X
New Price = X + 0.30X = 1.30X = 130
X = 130 / 1.30 = $100
Solution:
Let Original Price = X
25% decrease = 0.25 × X = 0.25X
New Price = X - 0.25X = 0.75X = 450
X = 450 / 0.75 = $600

Combined Exercises: Examples and Solutions

Many mathematical problems require the use of percentage increases and decreases in conjunction with other operations. Below are examples that incorporate these concepts alongside logical reasoning and application to real-world scenarios.

Example 1: Salary Adjustment and Taxation

Problem: Emma's annual salary is $50,000. She receives a 10% raise. After the raise, she pays 15% of her new salary in taxes. What is Emma's salary after the raise and after taxes?

Solution:


        Step 1: Calculate the raise
        Raise = 10% of 50,000 = 0.10 × 50,000 = 5,000
        New Salary = 50,000 + 5,000 = $55,000

        Step 2: Calculate taxes
        Taxes = 15% of 55,000 = 0.15 × 55,000 = 8,250

        Step 3: Salary after taxes
        Salary after Taxes = 55,000 - 8,250 = $46,750

Therefore, Emma's salary after the raise and taxes is $46,750.

Example 2: Investment Growth and Withdrawal

Problem: John invests $10,000 in a fund that grows by 8% in the first year. In the second year, the fund decreases by 5%. What is the value of John's investment after two years?

Solution:


        Year 1: Increase by 8%
        Increase = 0.08 × 10,000 = 800
        Value after Year 1 = 10,000 + 800 = $10,800

        Year 2: Decrease by 5%
        Decrease = 0.05 × 10,800 = 540
        Value after Year 2 = 10,800 - 540 = $10,260

Therefore, the value of John's investment after two years is $10,260.

Example 3: Price Fluctuation

Problem: A gadget's price increased by 12% this year and is expected to decrease by 8% next year. If the current price is $560, what will be the price after these changes?

Solution:


        Step 1: Increase by 12%
        Increase = 0.12 × 560 = 67.20
        New Price = 560 + 67.20 = $627.20

        Step 2: Decrease by 8%
        Decrease = 0.08 × 627.20 = 50.176
        Final Price = 627.20 - 50.176 ≈ $577.02

Therefore, the price after these changes will be approximately $577.02.

Example 4: Discount and Tax Application

Problem: A laptop is priced at $1,500. It is offered at a 20% discount. After applying the discount, sales tax of 8% is added. What is the final price of the laptop?

Solution:


        Step 1: Apply 20% discount
        Discount = 0.20 × 1,500 = 300
        Price after Discount = 1,500 - 300 = $1,200

        Step 2: Apply 8% sales tax
        Tax = 0.08 × 1,200 = 96
        Final Price = 1,200 + 96 = $1,296

Therefore, the final price of the laptop is $1,296.

Example 5: Population Change

Problem: A city's population was 80,000 last year. This year, it increased by 5%. Next year, it is projected to decrease by 3%. What will be the population next year?

Solution:


        Year 1: Increase by 5%
        Increase = 0.05 × 80,000 = 4,000
        New Population = 80,000 + 4,000 = 84,000

        Year 2: Decrease by 3%
        Decrease = 0.03 × 84,000 = 2,520
        Final Population = 84,000 - 2,520 = 81,480

Therefore, the population next year will be 81,480.

Practice Questions: Test Your Percentage Increases & Decreases Skills

Practicing with a variety of problems is key to mastering percentage increases and decreases. Below are practice questions categorized by difficulty level, along with their solutions.

Level 1: Easy

A bike costs $150 and its price increases by 10%. What is the new price?
Find the percentage decrease from 80 to 60.
Calculate a 5% increase on $200.
What is a 20% decrease of $500?
The population of a town increased from 1,000 to 1,100. What is the percentage increase?

Solutions:

Solution:
Percentage Increase = [(165 - 150) / 150] × 100 = (15 / 150) × 100 = 10%
New Price = 150 + 15 = $165
Solution:
Percentage Decrease = [(80 - 60) / 80] × 100 = (20 / 80) × 100 = 25%
Solution:
Percentage Increase = 5% of 200 = 0.05 × 200 = 10
New Amount = 200 + 10 = $210
Solution:
Percentage Decrease = 20% of 500 = 0.20 × 500 = 100
New Amount = 500 - 100 = $400
Solution:
Percentage Increase = [(1,100 - 1,000) / 1,000] × 100 = (100 / 1,000) × 100 = 10%

Level 2: Medium

A laptop originally priced at $1,200 is now sold for $1,320. What is the percentage increase?
Find the original price if $72 is a 20% decrease.
Calculate a 15% decrease on $400.
What is the percentage decrease from 250 to 200?
A company's revenue increased from $750,000 to $825,000. What is the percentage increase?

Solutions:

Solution:
Percentage Increase = [(1,320 - 1,200) / 1,200] × 100 = (120 / 1,200) × 100 = 10%
Solution:
Let Original Price = X
20% decrease = 0.20 × X = 0.20X
New Price = X - 0.20X = 0.80X = 72
X = 72 / 0.80 = $90
Solution:
Percentage Decrease = 15% of 400 = 0.15 × 400 = 60
New Amount = 400 - 60 = $340
Solution:
Percentage Decrease = [(250 - 200) / 250] × 100 = (50 / 250) × 100 = 20%
Solution:
Percentage Increase = [(825,000 - 750,000) / 750,000] × 100 = (75,000 / 750,000) × 100 = 10%

Level 3: Hard

A car's value depreciates from $30,000 to $24,000. What is the percentage decrease?
Find the new price after a 25% increase on $320.
If a population decreases by 7% from 85,000 to 79,550, what is the percentage decrease?
Calculate the original price if after a 40% increase, the price becomes $140.
A store offers a 30% discount on a TV, making it $350. What was the original price?

Solutions:

Solution:
Percentage Decrease = [(30,000 - 24,000) / 30,000] × 100 = (6,000 / 30,000) × 100 = 20%
Solution:
Percentage Increase = 25% of 320 = 0.25 × 320 = 80
New Price = 320 + 80 = $400
Solution:
Percentage Decrease = [(85,000 - 79,550) / 85,000] × 100 = (5,450 / 85,000) × 100 = 6.41%
Solution:
Let Original Price = X
40% increase = 0.40 × X = 0.40X
New Price = X + 0.40X = 1.40X = 140
X = 140 / 1.40 = $100
Solution:
Let Original Price = X
30% discount = 0.30 × X = 0.30X
Sale Price = X - 0.30X = 0.70X = 350
X = 350 / 0.70 = $500

Summary

Calculating percentage increases and decreases are essential mathematical skills that enable precise evaluations of changes in various contexts, such as finance, population studies, and pricing strategies. By understanding the fundamental concepts, mastering the relevant formulas, and practicing consistently, you can perform these calculations accurately and confidently.

Remember to:

Understand and apply the fundamental formulas for percentage increases and decreases.
Convert percentages to decimals to simplify calculations.
Use proportional reasoning to tackle percentage problems effectively.
Memorize common percentage values and their equivalents for quicker calculations.
Break down complex percentage problems into simpler, manageable parts.
Utilize visual aids like pie charts and number lines to enhance comprehension.
Double-check your work to ensure accuracy and catch any mistakes.
Apply percentages to real-life scenarios to reinforce and contextualize the concepts.
Practice regularly with a variety of problems to build confidence and proficiency.
Teach others to reinforce your understanding and identify any areas needing improvement.

With dedication and consistent practice, calculating percentage increases and decreases will become a fundamental skill in your mathematical toolkit, enhancing your analytical and problem-solving abilities.