Revision of Basic Percentage Concepts

TL;DR

Percentages are a way to express a part of a whole as a fraction of 100, making comparisons easier. You'll learn to convert between fractions, decimals, and percentages, and calculate percentage changes. Mastering these basics will build a strong foundation for more complex math problems.

1. The Mental Model

Think of a percentage as "out of 100." If you have 50% of something, you have 50 out of every 100 units of that thing. It's just a standardized way to talk about parts of a whole.

2. The Core Material

Percentages are incredibly useful for comparing quantities, understanding growth or reduction, and analyzing data. Let's break down the fundamentals.

What is a Percentage?

Photo by Towfiqu barbhuiya on Pexels

A percentage is simply a fraction where the denominator is 100. The symbol "%" means "per hundred" or "out of 100." So, 25% means 25/100.

Converting Between Forms

Photo by Tima Miroshnichenko on Pexels

You'll often need to switch between percentages, decimals, and fractions.

Percentage to Decimal

To change a percentage to a decimal, divide it by 100 (or move the decimal point two places to the left).
* Example: 75% = 75 / 100 = 0.75
* Example: 5% = 5 / 100 = 0.05

Decimal to Percentage

To change a decimal to a percentage, multiply it by 100 (or move the decimal point two places to the right and add the "%" symbol).
* Example: 0.25 = 0.25 * 100 = 25%
* Example: 1.5 = 1.5 * 100 = 150%

Fraction to Percentage

First, convert the fraction to a decimal, then convert the decimal to a percentage.
* Example: 1/4 = 0.25 (as a decimal) = 25%
* Example: 3/5 = 0.6 (as a decimal) = 60%

Percentage to Fraction

Write the percentage as a fraction with a denominator of 100, then simplify if possible.
* Example: 40% = 40/100 = 4/10 = 2/5 (simplified)
* Example: 12.5% = 12.5/100 = 125/1000 = 1/8 (simplified)

Calculating a Percentage of a Quantity

Photo by Nataliya Vaitkevich on Pexels

To find a percentage of a given number, convert the percentage to a decimal (or fraction) and multiply it by the number.

• Formula: Percentage (as decimal) × Quantity = Part

• Example: What is 20% of 80?

  • 20% as a decimal is 0.20
  • 0.20 × 80 = 16
  • So, 20% of 80 is 16.

Finding the Whole When Given a Part and a Percentage

Photo by Towfiqu barbhuiya on Pexels

Sometimes you know a part and what percentage it represents, and you need to find the total (the whole).

• Formula: Part / Percentage (as decimal) = Whole

• Example: If 15 is 30% of a number, what is the number?

  • 30% as a decimal is 0.30
  • 15 / 0.30 = 50
  • So, 15 is 30% of 50.

Percentage Change (Increase or Decrease)

Percentage change tells you how much a quantity has changed relative to its original value.

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

• Example (Increase): A shirt originally cost $20 and now costs $25. What's the percentage increase?

  • Change = $25 - $20 = $5
  • ($5 / $20) × 100% = 0.25 × 100% = 25% increase

• Example (Decrease): A bike originally cost $400 and is now $300. What's the percentage decrease?

  • Change = $400 - $300 = $100
  • ($100 / $400) × 100% = 0.25 × 100% = 25% decrease

Here's a flow of how you'd typically handle percentage problems:

graph TD
    A["Start with problem"] --> B{Is it about conversion?};
    B -- Yes --> C{Fraction <--> Decimal <--> Percentage};
    C --> D["Choose correct conversion rule"];
    D --> E["Perform conversion"];
    B -- No --> F{Is it about finding a part or whole?};
    F -- Yes --> G{Identify 'Part', 'Whole', 'Percentage'};
    G --> H{Use 'Part = Percentage x Whole' (rearrange as needed)};
    F -- No --> I{Is it about percentage change?};
    I -- Yes --> J{Identify 'Original Value' and 'New Value'};
    J --> K["Calculate Change: New - Original"];
    K --> L["Calculate Percentage Change: (Change / Original) x 100%"];
    E --> M["Solution"];
    H --> M;
    L --> M;
    M --> Z["End"];

3. Worked Example

Let's say you're buying a new gadget that costs $150. There's a 10% discount, and then a 5% sales tax is applied to the discounted price. What's the final price?

1. Calculate the discount:

   • Discount amount = 10% of $150
   • 10% as a decimal = 0.10
   • Discount = 0.10 * $150 = $15

2. Calculate the discounted price:

   • Discounted price = Original price - Discount
   • Discounted price = $150 - $15 = $135

3. Calculate the sales tax:

   • Sales tax amount = 5% of $135 (the discounted price)
   • 5% as a decimal = 0.05
   • Sales tax = 0.05 * $135 = $6.75

4. Calculate the final price:

   • Final price = Discounted price + Sales tax
   • Final price = $135 + $6.75 = $141.75

So, the final price you'd pay is $141.75.

4. Key Takeaways

• Percentages are always "out of 100" and provide a standard way to express proportions.
• You can easily convert between fractions, decimals, and percentages.
• To find a percentage of a quantity, convert the percentage to a decimal and multiply.
• To find the original whole, divide the part by the percentage (as a decimal).
• Percentage change is calculated relative to the original value.
• Be careful with sequential percentage changes – they apply to the new amount each time.

Common Mistakes to Avoid:
* Forgetting to divide or multiply by 100 when converting between percentages and decimals.
* Calculating percentage change using the wrong base (e.g., using the new value instead of the original value for the denominator).
* Incorrectly adding/subtracting percentages directly (e.g., 10% discount then 5% tax is NOT a net 5% change).
* Misplacing the decimal point when converting (e.g., 5% to 0.5 instead of 0.05).

5. Now Try It

You earned