Hello Maths Explorers! Let's Uncover the Secrets of Percentages!
Welcome to your study notes for Percentages! Ever seen a sign in a shop that says "50% off"? Or checked your phone and seen it has "20% battery left"? That's percentages in action! They are a super useful way to understand parts of a whole, and we use them every single day.
In this chapter, you'll learn:
- What a percentage really is.
- How to change percentages into fractions and decimals (and back again!).
- How to solve fun, real-life problems using percentages.
Don't worry if this seems tricky at first. We'll go step-by-step, and you'll be a percentage pro in no time!
Part 1: What is a Percentage?
The word "percent" sounds a bit like "per centipede," but it's not about insects with 100 legs! It's much simpler.
"Per cent" means "for every 100" or "out of 100".
Imagine you have a giant chocolate bar with 100 little squares. If you eat 25 squares, you've eaten 25 out of 100 squares. As a percentage, we say you've eaten 25% of the chocolate bar!
The symbol for percent is %.
Real-World Examples:
- If you get 85 questions right out of 100 on a test, your score is 85%. Well done!
- If your tablet is fully charged, it's at 100%.
- If it's half charged, it's at 50%.
Key Takeaway
A percentage is just a special type of fraction where the bottom number (the denominator) is always 100.
25% is the same as $$ \frac{25}{100} $$
70% is the same as $$ \frac{70}{100} $$
Part 2: The Magic Conversions! (Percentages, Fractions & Decimals)
Percentages, fractions, and decimals are like best friends—they all represent parts of a whole, just in different ways. Learning how to change one into another is a maths superpower!
1. From Percentage to Fraction
This is the easiest conversion! Just remember what "percent" means.
Step 1: Write the percentage number over 100.
Step 2: Simplify the fraction if you can (find the simplest form).
Example: Convert 50% to a fraction.
Step 1: Write it over 100. $$ 50\% = \frac{50}{100} $$
Step 2: Simplify. We can divide both the top and bottom by 50. $$ \frac{50 \div 50}{100 \div 50} = \frac{1}{2} $$
So, 50% is the same as $$ \frac{1}{2} $$. That makes sense, 50 is half of 100!
2. From Fraction to Percentage
To turn a fraction into a percentage, we just do the opposite.
The Rule: Multiply the fraction by 100%.
Example: Convert $$ \frac{1}{4} $$ to a percentage.
We calculate: $$ \frac{1}{4} \times 100\% $$
This is the same as asking, "What is one-quarter of 100?"
$$ 100 \div 4 = 25 $$
So, $$ \frac{1}{4} $$ is the same as 25%.
3. From Percentage to Decimal
Think of the % sign as a secret code that means "divide by 100".
The Rule: Divide the percentage by 100. (A simple trick is to move the decimal point two places to the left!)
Example: Convert 75% to a decimal.
Divide by 100: $$ 75 \div 100 = 0.75 $$
Simple Trick: Imagine 75 is 75.0. Move the decimal point two spots to the left: 75.0 -> 7.50 -> .750 or 0.75.
So, 75% is the same as 0.75.
4. From Decimal to Percentage
To go back, we just do the opposite of dividing!
The Rule: Multiply the decimal by 100. (A simple trick is to move the decimal point two places to the right!)
Example: Convert 0.25 to a percentage.
Multiply by 100: $$ 0.25 \times 100 = 25 $$
Simple Trick: Move the decimal point two spots to the right: 0.25 -> 2.5 -> 25.0 or 25.
So, 0.25 is the same as 25%.
Key Takeaway & Quick Review
Remember these common conversions! They will help you a lot.
- 50% = $$ \frac{1}{2} $$ = 0.5 (Half)
- 25% = $$ \frac{1}{4} $$ = 0.25 (A quarter)
- 75% = $$ \frac{3}{4} $$ = 0.75 (Three quarters)
- 10% = $$ \frac{1}{10} $$ = 0.1 (A tenth)
- 100% = $$ \frac{1}{1} $$ = 1.0 (The whole thing!)
Part 3: Solving Percentage Problems
Now let's use our new skills to solve some real problems. There are a few different types of questions you'll see.
Problem Type 1: Finding a percentage OF a number
This is the most common type of problem. It asks "What is a certain percent of a total amount?"
Example Question: There are 30 students in a class. 20% of them have blue eyes. How many students have blue eyes?
The question is asking: What is 20% of 30?
Step-by-step solution:
Step 1: Convert the percentage to a fraction or a decimal. Let's use a fraction first.
$$ 20\% = \frac{20}{100} $$ Simplified, this is $$ \frac{1}{5} $$
Step 2: In maths, the word "of" usually means multiply (×).
So, the problem becomes: $$ \frac{1}{5} \times 30 $$
Step 3: Solve it. $$ 30 \div 5 = 6 $$
Answer: 6 students have blue eyes.
You could also use the decimal method: 20% = 0.20. Then, 0.20 × 30 = 6. Both ways give the same answer!
Problem Type 2: What percentage is it?
This type of problem gives you the 'part' and the 'whole' and asks you to find the percentage.
Example Question: You scored 15 out of 20 on a spelling test. What is your score as a percentage?
The question is asking: What percentage of 20 is 15?
Step-by-step solution:
Step 1: Write the numbers as a fraction. The 'part' goes on top, and the 'whole' goes on the bottom.
Fraction = $$ \frac{\text{Part}}{\text{Whole}} = \frac{15}{20} $$
Step 2: Convert the fraction to a percentage by multiplying by 100%.
$$ \frac{15}{20} \times 100\% $$
Step 3: Solve it. You can simplify first: $$ \frac{15}{20} $$ is the same as $$ \frac{3}{4} $$.
So, $$ \frac{3}{4} \times 100\% = 75\% $$
Answer: Your score was 75%.
Memory Aid: A simple way to remember the fraction is "is over of". For "What percentage of 20 is 15?", the fraction is $$ \frac{15}{20} $$.
Problem Type 3: Increasing a number by a percentage
This is useful for things like when a price goes up.
Example Question: A video game costs $50. The price is increased by 10%. What is the new price?
Step-by-step solution:
Step 1: Find the amount of the increase. We need to find 10% of $50.
$$ 10\% \text{ of } 50 = \frac{10}{100} \times 50 = \frac{1}{10} \times 50 = 5 $$
The price increase is $5.
Step 2: Add the increase to the original price.
$$ \text{Original Price} + \text{Increase} = \text{New Price} $$
$$ $50 + $5 = $55 $$
Answer: The new price of the game is $55.
Common Mistake: Don't stop at Step 1! The question asks for the new price, not just the increase.
Problem Type 4: Decreasing a number by a percentage
This is what happens during a sale!
Example Question: A T-shirt costs $50, but it's in a sale with 10% off. What is the new price?
Step-by-step solution:
Step 1: Find the amount of the decrease (the discount). We need to find 10% of $50.
$$ 10\% \text{ of } 50 = 5 $$
The discount is $5.
Step 2: Subtract the decrease from the original price.
$$ \text{Original Price} - \text{Decrease} = \text{New Price} $$
$$ $50 - $5 = $45 $$
Answer: The new price of the T-shirt is $45.
Key Takeaway
To solve percentage problems:
- Read the question carefully to see what it's asking.
- Decide if you need to find a part, the percentage, an increase, or a decrease.
- Remember that "of" means multiply.
- For increases, ADD. For decreases, SUBTRACT.
Did you know?
The "%" sign is believed to have evolved from an Italian symbol used in the 15th century. It started as "per 100" which was written as "p c" with a little circle over the 'c', and over time it morphed into the symbol we use today!
You've done an amazing job! Keep practicing, and you'll see percentages everywhere. They are a great tool for understanding the world around you.