Percents Unit Study Guide Answer Key

Percent Unit Study Guide Answer Key: Mastering Percentages

This comprehensive study guide provides answers and explanations to common percentage problems, equipping you with the skills to confidently tackle any percentage-related question. We'll cover fundamental concepts, advanced techniques, and real-world applications, ensuring a thorough understanding of this essential mathematical topic.

Understanding the Fundamentals of Percentages

Before diving into complex problems, let's solidify our understanding of the basics. A percentage represents a fraction of 100. The term "percent" literally means "per hundred." This means that 50% is equivalent to 50/100, which simplifies to 1/2 or 0.5.

Key Terms and Concepts:

• Percentage: A portion or fraction of 100. Represented by the symbol "%".
• Base: The whole amount or the original value from which a percentage is calculated.
• Rate: The percentage itself, expressed as a number.
• Portion: The part of the base that represents the percentage.

Formula: The Core of Percentage Calculations

The fundamental formula governing percentage calculations is:

Portion = Rate × Base

This formula can be rearranged to solve for any of the three variables:

• To find the Portion: Multiply the Rate (as a decimal) by the Base.
• To find the Rate: Divide the Portion by the Base, then multiply by 100 to express it as a percentage.
• To find the Base: Divide the Portion by the Rate (as a decimal).

Types of Percentage Problems and Their Solutions

Let's explore various types of percentage problems and demonstrate their solutions using the fundamental formula and its variations.

1. Finding the Percentage of a Number

Problem: What is 30% of 80?

Solution:

Here, the Base is 80, and the Rate is 30% (or 0.30 as a decimal). We need to find the Portion.

Portion = Rate × Base = 0.30 × 80 = 24

Answer: 30% of 80 is 24.

2. Finding the Percentage One Number is of Another

Problem: What percentage of 60 is 15?

Solution:

Here, the Portion is 15, and the Base is 60. We need to find the Rate.

Rate = (Portion / Base) × 100 = (15 / 60) × 100 = 0.25 × 100 = 25%

Answer: 15 is 25% of 60.

3. Finding the Original Number (Base) Given a Percentage and a Portion

Problem: 20% of what number is 12?

Solution:

Here, the Portion is 12, and the Rate is 20% (or 0.20). We need to find the Base.

Base = Portion / Rate = 12 / 0.20 = 60

Answer: 20% of 60 is 12.

4. Percentage Increase and Decrease

These problems involve calculating the change in a value expressed as a percentage.

Problem (Increase): A shirt's price increased from $25 to $30. What is the percentage increase?

Solution:

1. Find the increase: $30 - $25 = $5
2. Calculate the percentage increase: ($5 / $25) × 100 = 20%

Answer: The shirt's price increased by 20%.

Problem (Decrease): A store is having a 15% off sale. If an item originally cost $50, what is the sale price?

Solution:

1. Calculate the discount: 15% of $50 = 0.15 × $50 = $7.50
2. Subtract the discount from the original price: $50 - $7.50 = $42.50

Answer: The sale price is $42.50.

5. Compound Percentage Increase/Decrease

Compound percentages involve calculating percentages on already adjusted values.

Problem: A population increases by 10% annually. If the initial population is 1000, what will the population be after two years?

Solution:

• Year 1: 1000 + (10% of 1000) = 1000 + 100 = 1100
• Year 2: 1100 + (10% of 1100) = 1100 + 110 = 1210

Answer: The population will be 1210 after two years. Note that this is not simply a 20% increase (which would be 1200).

Advanced Percentage Applications and Problem-Solving Techniques

Let's delve into more complex scenarios where percentage calculations are crucial.

1. Simple Interest Calculations

Simple interest is calculated only on the principal amount.

Formula: Simple Interest = Principal × Rate × Time

Problem: Calculate the simple interest on a $1000 loan at a 5% annual interest rate over 3 years.

Solution: Simple Interest = $1000 × 0.05 × 3 = $150

Answer: The simple interest is $150.

2. Compound Interest Calculations

Compound interest is calculated on both the principal and accumulated interest.

Formula: A = P (1 + r/n)^(nt) Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (decimal)
• n = the number of times that interest is compounded per year
• t = the number of years the money is invested or borrowed for

Problem: Calculate the compound interest on a $1000 investment at a 5% annual interest rate, compounded annually, over 3 years.

Solution: A = 1000 (1 + 0.05/1)^(1*3) = 1000 (1.05)^3 ≈ $1157.63

Answer: The compound interest is approximately $157.63 ($1157.63 - $1000).

3. Profit and Loss Percentages

Problem: A shopkeeper bought an item for $50 and sold it for $60. What is the profit percentage?

Solution:

1. Profit: $60 - $50 = $10
2. Profit Percentage: ($10 / $50) × 100 = 20%

Answer: The profit percentage is 20%.

4. Discount and Sale Price Calculations

We've already touched upon this, but let's solidify it with another example.

Problem: A jacket is on sale for 25% off its original price of $80. What is the sale price?

Solution:

1. Discount amount: 25% of $80 = 0.25 × $80 = $20
2. Sale Price: $80 - $20 = $60

Answer: The sale price is $60.

5. Tax Calculations

Problem: A product costs $100, and the sales tax is 6%. What is the total cost?

Solution:

1. Tax amount: 6% of $100 = 0.06 × $100 = $6
2. Total cost: $100 + $6 = $106

Answer: The total cost is $106.

Tips and Tricks for Mastering Percentages

• Practice Regularly: Consistent practice is key to mastering percentages. Work through a variety of problems, focusing on different types and complexities.
• Visual Aids: Diagrams and charts can be helpful in visualizing percentage problems, especially those involving increase or decrease.
• Use a Calculator: Don't hesitate to use a calculator for complex calculations. Focus on understanding the concepts and applying the formulas correctly.
• Check Your Work: Always double-check your answers. A simple mistake in calculation can lead to an incorrect result.
• Real-World Applications: Relate percentage problems to real-world situations to improve comprehension and retention. Consider budgeting, sales, discounts, and interest rates.

Conclusion: Becoming a Percentage Pro

Mastering percentages isn't about memorizing formulas; it's about understanding the underlying concepts and applying them effectively. By practicing consistently and exploring various problem types, you'll build the confidence and skills needed to tackle any percentage-related challenge. Remember the fundamental formula, practice regularly, and you'll become a percentage pro in no time! Remember to break down complex problems into smaller, manageable steps, and always double-check your calculations. Good luck, and happy calculating!