Welcome to the World of Percentages!
Hi there! Today, we are going to explore a very special way of looking at numbers: Percentages. You see percentages everywhere—when a toy is "50% off" at a shop, when your phone battery is at "20%", or when your teacher gives you a grade.
By the end of these notes, you will be a "Conversion Master," able to switch between Percentages, Decimals, and Fractions with ease. Don’t worry if this seems tricky at first; we will take it one small step at a time!
Section 1: What is a Percentage?
The word "Percent" comes from two words: "Per" (which means "for every") and "Cent" (which means "one hundred," just like there are 100 cents in a dollar). So, Percentage literally means "out of 100."
Analogy: Imagine a huge chocolate bar with 100 tiny squares. If you eat 25 squares, you have eaten 25 out of 100, or 25% of the bar!
Quick Review:
- The symbol for percentage is %.
- \( 100\% \) represents one whole (the entire chocolate bar).
- \( 50\% \) represents half of the whole.
Did you know? The "cent" in percent is the same root word used in "century" (100 years) and "centimetre" (100 in a metre)!
Section 2: Interconversion between Percentages and Fractions
Fractions and percentages are like best friends; they tell the same story in different ways. Since a percentage is always "out of 100," we can easily turn it into a fraction.
A. Converting a Percentage to a Fraction
To change a Percentage to a Fraction, follow these simple steps:
1. Take the number and put it over a denominator of 100.
2. Remove the % sign.
3. Simplify the fraction to its lowest terms.
Example: Convert \( 40\% \) to a fraction.
Step 1 & 2: Write it as \( \frac{40}{100} \).
Step 3: Simplify by dividing the top and bottom by 20. \( \frac{40 \div 20}{100 \div 20} = \frac{2}{5} \).
So, \( 40\% = \frac{2}{5} \).
B. Converting a Fraction to a Percentage
There are two main ways to turn a Fraction into a Percentage:
Method 1: The "Make it 100" Trick
If the denominator (the bottom number) can easily be multiplied to become 100, do that!
Example: Convert \( \frac{3}{5} \) to a percentage.
We know \( 5 \times 20 = 100 \).
Multiply the top and bottom by 20: \( \frac{3 \times 20}{5 \times 20} = \frac{60}{100} \).
Since it is now "out of 100," it is 60%!
Method 2: Multiply by 100%
This works for any fraction! Just multiply the fraction by \( 100\% \).
Example: Convert \( \frac{1}{4} \) to a percentage.
\( \frac{1}{4} \times 100\% = \frac{100}{4}\% = 25\% \).
Common Mistake to Avoid: When converting a percentage to a fraction, always remember to simplify! Teachers love seeing fractions in their simplest form.
Key Takeaway: A percentage is just a fraction with a denominator of 100. \( x\% = \frac{x}{100} \).
Section 3: Interconversion between Percentages and Decimals
Decimals are another way to show parts of a whole. Moving between percentages and decimals is like moving a "sliding door" (the decimal point).
A. Converting a Percentage to a Decimal
To change a Percentage to a Decimal, you divide by 100. This is the same as moving the decimal point two places to the left.
Example: Convert \( 85\% \) to a decimal.
Imagine the decimal point is at the end: \( 85.0\% \).
Move it two places left: \( .85 \).
Answer: 0.85
Example: Convert \( 7\% \) to a decimal.
Move it two places left from \( 7.0 \). You will need to add a zero as a placeholder!
Answer: 0.07
B. Converting a Decimal to a Percentage
To change a Decimal to a Percentage, you multiply by 100. This is the same as moving the decimal point two places to the right and adding the % sign.
Example: Convert \( 0.42 \) to a percentage.
Move the point two places right: \( 42 \).
Answer: 42%
Example: Convert \( 0.6 \) to a percentage.
Move the point two places right. Add a zero to fill the empty "jump" space!
Answer: 60%
Memory Aid: Dr. Pepper (D.P.)
Think of Decimal and Percentage in alphabetical order:
- To go from D to P (Decimal to Percentage), move the point to the Right.
- To go from P to D (Percentage to Decimal), move the point to the Left.
Key Takeaway: Moving the decimal point two spots is the magic trick for decimals and percentages!
Summary Table of Common Conversions
It helps to memorize these common "VIP" numbers:
Fraction | Decimal | Percentage
\( \frac{1}{2} \) | 0.5 | 50%
\( \frac{1}{4} \) | 0.25 | 25%
\( \frac{3}{4} \) | 0.75 | 75%
\( \frac{1}{5} \) | 0.2 | 20%
\( \frac{1}{10} \) | 0.1 | 10%
Final Encouragement
You’ve done a great job! Percentages are just another way of writing fractions and decimals. Whether you are moving decimal points or simplifying fractions, the most important thing to remember is that percentage means "out of 100." Keep practicing, and soon you'll be doing these conversions in your sleep!