Basic Percentages | Free Learning Resources

Basic Percentages - Comprehensive Notes

Basic Percentages: Comprehensive Notes

Welcome to our detailed guide on Basic Percentages. 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 percentages.

Introduction

Percentages are a fundamental part of mathematics, representing a fraction of 100. They are widely used in various real-life applications, including finance, statistics, discounts, and more. Understanding how to calculate and manipulate percentages is essential for making informed decisions in everyday life. This guide provides a comprehensive overview of basic percentages, their properties, methods, and common pitfalls to ensure a solid mathematical foundation.

Basic Concepts of Percentages

Before diving into operations with percentages, it's important to understand the foundational concepts that make these operations possible.

Understanding Percentages

A percentage represents a part of a whole expressed as a fraction of 100. It is denoted by the symbol "%".

Formula: Percentage (%) = (Part / Whole) × 100

Example: If you have 25 apples out of 100, the percentage is:

Percentage = (25 / 100) × 100 = 25%

Converting Between Fractions, Decimals, and Percentages

Fraction to Percentage: Multiply the fraction by 100.
Decimal to Percentage: Multiply the decimal by 100.
Percentage to Fraction: Divide the percentage by 100.
Percentage to Decimal: Divide the percentage by 100.

Example: Convert 3/4 to a percentage:

3/4 × 100 = 75%

Properties of Percentages

Understanding the properties of percentages is crucial for performing accurate calculations.

Percentage Increase and Decrease

Increase: To increase a number by a percentage, multiply the number by the percentage and add it to the original number.

Decrease: To decrease a number by a percentage, multiply the number by the percentage and subtract it from the original number.

Example: Increase $200 by 10%:


        Increase = 10% of 200 = 0.10 × 200 = 20
        New amount = 200 + 20 = $220

Percentage of a Percentage

When finding a percentage of a percentage, convert both percentages to decimals and multiply.

Example: Find 20% of 30%:


        20% of 30% = 0.20 × 0.30 = 0.06 = 6%

Calculations with Percentages

Working with percentages involves various types of calculations, including finding percentages, percentage increases/decreases, and more.

Finding the Percentage of a Number

To find a percentage of a number, convert the percentage to a decimal and multiply by the number.

Formula: Percentage of Number = (Percentage / 100) × Number

Example: Find 25% of 80:


        25% of 80 = 0.25 × 80 = 20

Finding What Percentage One Number is of Another

To determine what percentage one number is of another, divide the first number by the second and multiply by 100.

Formula: (Part / Whole) × 100

Example: What percentage is 45 of 60?


        (45 / 60) × 100 = 0.75 × 100 = 75%

Calculating Percentage Increase

To calculate a percentage increase, determine the difference between the new and original values, divide by the original value, and multiply by 100.

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

Example: Increase $150 by 20%:


        Increase = 20% of 150 = 0.20 × 150 = 30
        New amount = 150 + 30 = $180

Calculating Percentage Decrease

To calculate a percentage decrease, determine the difference between the original and new values, divide by the original value, and multiply by 100.

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

Example: Decrease $200 by 15%:


        Decrease = 15% of 200 = 0.15 × 200 = 30
        New amount = 200 - 30 = $170

Examples of Basic Percentages

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

Example 1: Finding the Percentage of a Number

Problem: Find 30% of 50.

Solution:


        30% of 50 = 0.30 × 50 = 15

Therefore, 30% of 50 is 15.

Example 2: What Percentage One Number is of Another

Problem: What percentage is 25 of 200?

Solution:


        (25 / 200) × 100 = 0.125 × 100 = 12.5%

Therefore, 25 is 12.5% of 200.

Example 3: Calculating Percentage Increase

Problem: A shirt originally costs $40. It is now priced at $50. What is the percentage increase?

Solution:


        Increase = 50 - 40 = 10
        Percentage Increase = (10 / 40) × 100 = 25%

Therefore, the shirt's price increased by 25%.

Example 4: Calculating Percentage Decrease

Problem: A laptop was priced at $800 but is now sold for $600. What is the percentage decrease?

Solution:


        Decrease = 800 - 600 = 200
        Percentage Decrease = (200 / 800) × 100 = 25%

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

Example 5: Finding the Original Price After a Discount

Problem: After a 20% discount, a jacket costs $80. What was the original price?

Solution:


        Let Original Price = X
        80 = X - (20% of X)
        80 = X - 0.20X
        80 = 0.80X
        X = 80 / 0.80 = 100

Therefore, the original price was $100.

Example 6: Percentage Change in Population

Problem: A town had a population of 15,000 last year. This year, the population increased to 18,000. What is the percentage increase in population?

Solution:


        Increase = 18,000 - 15,000 = 3,000
        Percentage Increase = (3,000 / 15,000) × 100 = 20%

Therefore, the population increased by 20%.

Word Problems: Application of Basic Percentages

Applying percentages 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: Shopping Discount

Problem: A store is offering a 15% discount on all items. If a pair of shoes costs $120 before the discount, how much will they cost after the discount?

Solution:


        Discount = 15% of 120 = 0.15 × 120 = 18
        Price after discount = 120 - 18 = $102

Therefore, the shoes will cost $102 after the discount.

Example 2: Salary Increase

Problem: Jane's salary was increased by 10%. If her original salary was $50,000, what is her new salary?

Solution:


        Increase = 10% of 50,000 = 0.10 × 50,000 = 5,000
        New Salary = 50,000 + 5,000 = $55,000

Therefore, Jane's new salary is $55,000.

Example 3: Exam Scores

Problem: Tom scored 85% on his math exam. If the exam was out of 200 points, how many points did he earn?

Solution:


        Points Earned = 85% of 200 = 0.85 × 200 = 170 points

Therefore, Tom earned 170 points.

Example 4: Population Growth

Problem: A city's population grows by 5% each year. If the current population is 100,000, what will the population be after one year?

Solution:


        Increase = 5% of 100,000 = 0.05 × 100,000 = 5,000
        New Population = 100,000 + 5,000 = 105,000

Therefore, the population after one year will be 105,000.

Example 5: Tax Calculation

Problem: An item costs $80 before tax. If the sales tax rate is 8%, what is the total cost after tax?

Solution:


        Tax = 8% of 80 = 0.08 × 80 = 6.40
        Total Cost = 80 + 6.40 = $86.40

Therefore, the total cost after tax is $86.40.

Strategies and Tips for Working with Percentages

Enhancing your skills in working with percentages involves employing effective strategies and consistent practice. Here are some tips to help you improve:

1. Understand the Relationship Between Fractions, Decimals, and Percentages

Recognize that percentages are simply fractions out of 100 and can be easily converted to and from decimals and fractions.

Example: 50% = 50/100 = 0.50

2. Use Proportional Reasoning

When dealing with percentages, think in terms of proportions and ratios to simplify calculations.

Example: To find 25% of 80, think of it as 25/100 × 80 = 1/4 × 80 = 20

3. 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

4. Break Down Complex Percentages

For complex percentages, break them down into simpler parts that are easier to calculate.

Example: To find 35% of a number, calculate 30% plus 5% separately and then add the results.

5. 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.

6. Use Visual Aids

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

Example: A pie chart can help illustrate how 25%, 50%, and 75% relate to a whole.

7. Double-Check Calculations

Always review your calculations to catch and correct any mistakes.

Example: After calculating 20% of 150 as 30, verify by dividing 150 by 5.

8. Apply Percentages to 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. Use the Percent Formula

Familiarize yourself with the fundamental percentage formulas to handle various types of percentage problems.

Formulas:

Percentage of a Number: (Percentage / 100) × Number
Percentage Change: [(New - Original) / Original] × 100
Finding the Original Number: Part / (Percentage / 100)

10. Teach Others

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

Common Mistakes in Working with Percentages and How to Avoid Them

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

1. Confusing Percentages with Fractions or Decimals

Mistake: Treating percentages as whole numbers or mixing up their relationship with fractions and decimals.

Solution: Always remember that percentages are out of 100 and convert them to fractions or decimals as needed.


        Example:
        Incorrect: 25% of 80 = 25 × 80 = 2000
        Correct: 25% of 80 = 0.25 × 80 = 20

2. Forgetting to Convert Percentages to Decimals Before Multiplying

Mistake: Multiplying a percentage directly without converting it to a decimal first.

Solution: Always convert the percentage to a decimal by dividing by 100 before performing multiplication.


        Example:
        Incorrect: 20% of 50 = 20 × 50 = 1000
        Correct: 20% of 50 = 0.20 × 50 = 10

3. Misapplying the Percentage Change Formula

Mistake: Incorrectly calculating percentage increases or decreases by not using the correct formula.

Solution: Use the correct formula: [(New - Original) / Original] × 100


        Example:
        Incorrect: Increase of $50 to $70 = (70 + 50) / 50 × 100 = 240%
        Correct: [(70 - 50) / 50] × 100 = 40%

4. Incorrectly Calculating the Original Amount After a Percentage Change

Mistake: Miscalculating the original amount when given the new amount and the percentage change.

Solution: Use the formula: Original = New / (1 ± Percentage Change)


        Example:
        Incorrect: If $80 is after a 20% discount, original price = 80 × 1.20 = 96
        Correct: Original price = 80 / 0.80 = 100

5. Overlooking the Need to Simplify Percentages

Mistake: Leaving percentages unsimplified, making the answer unnecessarily complex.

Solution: Simplify percentages to their lowest terms or appropriate decimal places.


        Example:
        Incorrect: 50% of 30 = 0.50 × 30 = 15.00
        Correct: 50% of 30 = 0.50 × 30 = 15

6. 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.

7. 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

8. Not Practicing Enough

Mistake: Lack of practice can result in slower calculations and increased errors.

Solution: Engage in regular practice through exercises, quizzes, and real-life applications to build speed and accuracy.

9. Ignoring the Context of Percentage Problems

Mistake: Misapplying percentage operations without considering the real-life context, leading to incorrect conclusions.

Solution: Always interpret percentage problems within their real-life context to ensure logical and accurate solutions.

10. Mixing Up Percentage Increase and Percentage of

Mistake: Confusing calculating a percentage increase with finding a percentage of a number.

Solution: Understand the distinction between percentage increase (which involves addition) and finding a percentage of a number (which involves multiplication).


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

Practice Questions: Test Your Basic Percentages Skills

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

Level 1: Easy

Find 20% of 50.
What is 25% of 200?
Calculate 10% of 80.
Find 50% of 120.
What is 5% of 300?

Solutions:

Solution:
20% of 50 = 0.20 × 50 = 10
Solution:
25% of 200 = 0.25 × 200 = 50
Solution:
10% of 80 = 0.10 × 80 = 8
Solution:
50% of 120 = 0.50 × 120 = 60
Solution:
5% of 300 = 0.05 × 300 = 15

Level 2: Medium

Find 15% of 200.
What percentage of 250 is 50?
Calculate 30% of 150.
Find the original price if $80 is 20% of it.
Increase $400 by 12%.

Solutions:

Solution:
15% of 200 = 0.15 × 200 = 30
Solution:
(50 / 250) × 100 = 0.20 × 100 = 20%
Solution:
30% of 150 = 0.30 × 150 = 45
Solution:
Let Original Price = X
20% of X = 80 → 0.20 × X = 80 → X = 80 / 0.20 = 400
Solution:
Increase = 12% of 400 = 0.12 × 400 = 48
New Amount = 400 + 48 = 448

Level 3: Hard

Find 18% of 350.
What is 40% of 275?
Calculate 22.5% of 160.
Decrease $500 by 25%.
If 30% of a number is 45, what is the number?

Solutions:

Solution:
18% of 350 = 0.18 × 350 = 63
Solution:
40% of 275 = 0.40 × 275 = 110
Solution:
22.5% of 160 = 0.225 × 160 = 36
Solution:
Decrease = 25% of 500 = 0.25 × 500 = 125
New Amount = 500 - 125 = 375
Solution:
Let Number = X
30% of X = 45 → 0.30 × X = 45 → X = 45 / 0.30 = 150

Combined Exercises: Examples and Solutions

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

Example 1: Shopping with Multiple Discounts

Problem: A jacket is priced at $120. It is first discounted by 10%, and then the discounted price is further reduced by 5%. What is the final price of the jacket?

Solution:


        First Discount:
        10% of 120 = 0.10 × 120 = 12
        Price after first discount = 120 - 12 = $108

        Second Discount:
        5% of 108 = 0.05 × 108 = 5.40
        Final Price = 108 - 5.40 = $102.60

Therefore, the final price of the jacket is $102.60.

Example 2: Salary Adjustment

Problem: Mark's current salary is $50,000. He receives a 12% raise and then has to pay 8% tax on his new salary. What is Mark's salary after the raise and tax?

Solution:


        Salary after Raise:
        12% of 50,000 = 0.12 × 50,000 = 6,000
        New Salary = 50,000 + 6,000 = $56,000

        Tax:
        8% of 56,000 = 0.08 × 56,000 = 4,480
        Salary after Tax = 56,000 - 4,480 = $51,520

Therefore, Mark's salary after the raise and tax is $51,520.

Example 3: Population Growth

Problem: A town has a population of 20,000. Over a year, the population grows by 5%. What will be the population after the growth?

Solution:


        Population Growth = 5% of 20,000 = 0.05 × 20,000 = 1,000
        New Population = 20,000 + 1,000 = 21,000

Therefore, the population after the growth will be 21,000.

Example 4: Exam Score Improvement

Problem: Lisa scored 70% on her first exam. She aims to improve her score by 15% in her next exam. What score should she aim for?

Solution:


        Improvement = 15% of 70 = 0.15 × 70 = 10.5
        Target Score = 70 + 10.5 = 80.5%

Therefore, Lisa should aim for an 80.5% score on her next exam.

Example 5: Budget Allocation

Problem: You have a monthly budget of $2,500. You decide to allocate 30% to rent, 20% to groceries, and 10% to entertainment. How much money is allocated to each category, and how much remains unallocated?

Solution:


        Rent: 30% of 2,500 = 0.30 × 2,500 = $750
        Groceries: 20% of 2,500 = 0.20 × 2,500 = $500
        Entertainment: 10% of 2,500 = 0.10 × 2,500 = $250
        Total Allocated = 750 + 500 + 250 = $1,500
        Unallocated = 2,500 - 1,500 = $1,000

Therefore, $750 is allocated to rent, $500 to groceries, $250 to entertainment, and $1,000 remains unallocated.

Practice Questions: Test Your Basic Percentages Skills

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

Level 1: Easy

Find 20% of 50.
What is 25% of 200?
Calculate 10% of 80.
Find 50% of 120.
What is 5% of 300?

Solutions:

Solution:
20% of 50 = 0.20 × 50 = 10
Solution:
25% of 200 = 0.25 × 200 = 50
Solution:
10% of 80 = 0.10 × 80 = 8
Solution:
50% of 120 = 0.50 × 120 = 60
Solution:
5% of 300 = 0.05 × 300 = 15

Level 2: Medium

Find 15% of 200.
What percentage of 250 is 50?
Calculate 30% of 150.
Find the original price if $80 is 20% of it.
Increase $400 by 12%.

Solutions:

Solution:
15% of 200 = 0.15 × 200 = 30
Solution:
(50 / 250) × 100 = 0.20 × 100 = 20%
Solution:
30% of 150 = 0.30 × 150 = 45
Solution:
Let Original Price = X
20% of X = 80 → 0.20 × X = 80 → X = 80 / 0.20 = 400
Solution:
Increase = 12% of 400 = 0.12 × 400 = 48
New Amount = 400 + 48 = 448

Level 3: Hard

Find 18% of 350.
What is 40% of 275?
Calculate 22.5% of 160.
Decrease $500 by 25%.
If 30% of a number is 45, what is the number?

Solutions:

Solution:
18% of 350 = 0.18 × 350 = 63
Solution:
40% of 275 = 0.40 × 275 = 110
Solution:
22.5% of 160 = 0.225 × 160 = 36
Solution:
Decrease = 25% of 500 = 0.25 × 500 = 125
New Amount = 500 - 125 = 375
Solution:
Let Number = X
30% of X = 45 → 0.30 × X = 45 → X = 45 / 0.30 = 150

Summary

Understanding and working with percentages are essential mathematical skills that enable precise calculations in various contexts. By grasping the fundamental concepts, practicing conversions between fractions, decimals, and percentages, and applying percentage formulas, you can confidently handle percentage-related problems.

Remember to:

Understand the relationship between fractions, decimals, and percentages.
Convert between different forms to simplify calculations.
Use proportional reasoning to tackle percentage problems.
Memorize common percentage values and their equivalents.
Break down complex percentages into simpler parts for easier computation.
Utilize visual aids like pie charts and number lines to enhance understanding.
Double-check your work to ensure accuracy.
Apply percentages to real-life scenarios to reinforce concepts.
Familiarize yourself with key percentage formulas.
Teach others to reinforce your own understanding and identify any gaps.

With dedication and consistent practice, working with percentages will become a fundamental skill in your mathematical toolkit, enhancing your analytical and problem-solving abilities.