Percentages Of An Amount Corbettmaths

Mastering Percentages: A Comprehensive Guide Inspired by Corbettmaths

Understanding percentages is a fundamental skill in mathematics, crucial for navigating everyday life, from calculating discounts and sales tax to interpreting data and statistics. This comprehensive guide, inspired by the clear and concise teaching style of Corbettmaths, will equip you with the knowledge and strategies to confidently tackle any percentage problem. We'll explore various methods, delve into the underlying concepts, and address common misconceptions. By the end, you'll not only be able to calculate percentages of amounts but also understand the why behind the calculations.

Understanding the Basics: What are Percentages?

A percentage is simply a fraction where the denominator is 100. The word "percent" itself means "out of 100" ( per cent). Therefore, 50% means 50 out of 100, which can be written as the fraction 50/100 or the decimal 0.5. Understanding this foundational equivalence is key to mastering percentage calculations.

Method 1: Using the Unitary Method

The unitary method is a powerful and versatile approach for calculating percentages of amounts. It involves finding the value of 1% first, then scaling up to find the required percentage.

Steps:

1. Find 1%: Divide the total amount by 100. This gives you the value of 1%.

2. Multiply to find the desired percentage: Multiply the value of 1% by the percentage you want to calculate.

Example: Find 15% of 200.

1. 1% of 200: 200 ÷ 100 = 2

2. 15% of 200: 2 × 15 = 30

Therefore, 15% of 200 is 30.

Method 2: Using Decimals

This method leverages the decimal equivalent of a percentage. Converting the percentage to a decimal simplifies the calculation.

Steps:

1. Convert the percentage to a decimal: Divide the percentage by 100 (or move the decimal point two places to the left).

2. Multiply by the amount: Multiply the decimal equivalent by the original amount.

Example: Find 30% of 150.

1. Convert 30% to a decimal: 30 ÷ 100 = 0.3

2. Multiply: 0.3 × 150 = 45

Therefore, 30% of 150 is 45.

Method 3: Using Fractions

Many percentages have simple fraction equivalents. Using fractions can be quicker and easier for certain percentages.

Common Percentage-Fraction Equivalents:

• 10% = 1/10
• 20% = 1/5
• 25% = 1/4
• 50% = 1/2
• 75% = 3/4

Example: Find 25% of 80.

1. Use the fraction equivalent: 25% = 1/4

2. Multiply: 80 × (1/4) = 20

Therefore, 25% of 80 is 20.

Finding the Original Amount (Reverse Percentages)

Often, you might know the value after a percentage increase or decrease and need to find the original amount. This involves working backwards.

Example: After a 10% increase, a price is now £110. What was the original price?

Since the final price includes the original price plus 10%, the final price represents 110% of the original price.

1. Find 1%: Divide the final price by 110: £110 ÷ 110 = £1

2. Find 100% (original price): Multiply 1% by 100: £1 × 100 = £100

Therefore, the original price was £100.

Percentage Increase and Decrease

Calculating percentage increases and decreases involves understanding how the percentage change affects the original amount.

Percentage Increase:

• Formula: New Amount = Original Amount × (1 + Percentage Increase/100)

Example: Increase 50 by 20%.

New Amount = 50 × (1 + 20/100) = 50 × 1.2 = 60

Percentage Decrease:

• Formula: New Amount = Original Amount × (1 - Percentage Decrease/100)

Example: Decrease 80 by 15%.

New Amount = 80 × (1 - 15/100) = 80 × 0.85 = 68

Percentage Change

Determining the percentage change between two values helps analyze growth or decline.

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

Example: A value increases from 20 to 25. What is the percentage increase?

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

Compound Interest and Depreciation

Compound interest involves earning interest on both the principal amount and accumulated interest. Depreciation works similarly but represents a decrease in value over time.

Compound 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

Depreciation Formula: Similar to compound interest, but the rate (r) is negative.

Solving More Complex Problems: Proportion and Ratio

Many percentage problems involve proportions and ratios. Understanding these concepts allows you to solve more challenging scenarios.

Example: A recipe calls for 200g of flour to make 12 cakes. If you want to make 18 cakes, how much flour do you need?

This problem can be solved using proportions. Set up a ratio:

200g / 12 cakes = x g / 18 cakes

Cross-multiply and solve for x:

12x = 200 × 18 x = 300g

Applying Percentages to Real-World Scenarios

The applications of percentages are vast and varied. Here are a few examples:

• Sales and Discounts: Calculating discounts offered in shops.
• Tax Calculations: Determining sales tax or VAT amounts.
• Interest Rates: Understanding interest on savings accounts or loans.
• Data Analysis: Interpreting percentages in charts and graphs.
• Profit Margins: Calculating profit as a percentage of revenue in business.

Frequently Asked Questions (FAQ)

Q1: How do I calculate a percentage of a percentage?

To calculate a percentage of a percentage, simply multiply the two percentages together (as decimals) and multiply by the original amount. For example, 20% of 50% of 100 is: (0.20 * 0.50 * 100) = 10.

Q2: What if the percentage is greater than 100%?

Percentages greater than 100% represent values exceeding the original amount. The calculation remains the same; simply multiply the decimal equivalent by the amount. For example, 150% of 50 is (1.50 * 50) = 75.

Q3: How can I improve my speed and accuracy with percentage calculations?

Practice is key! Regularly solve percentage problems of varying difficulty. Memorize common percentage-fraction equivalents. Use mental math strategies wherever possible. Consider using online quizzes and interactive exercises.

Q4: What resources are available for further learning?

Numerous online resources, including Corbettmaths itself, offer further practice exercises, video tutorials, and comprehensive explanations. Textbooks and educational websites also provide valuable support.

Conclusion

Mastering percentages is a journey, not a destination. This guide has provided you with the foundational knowledge and practical methods to confidently tackle percentage calculations. Remember to practice regularly, explore different approaches, and apply your learning to real-world scenarios. By understanding the underlying concepts and utilizing the various techniques presented, you'll become proficient in using percentages and unlock a deeper understanding of their significance in mathematics and beyond. Embrace the challenge, and you will undoubtedly succeed in your journey of mastering this crucial mathematical skill!