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 (including discounts, GST, and interest).
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 by dividing 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 → 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.
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 skills to solve real-world problems. Here are the key types of questions you will meet.
Problem Type 1: Finding a percentage OF a number (Finding the part)
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 1: Convert the percentage to a fraction: \( 20\% = \frac{20}{100} = \frac{1}{5} \).
Step 2: In maths, "of" means multiply (\( \times \)):
\( \frac{1}{5} \times 30 = 6 \)
Answer: 6 students have blue eyes.
Problem Type 2: Finding what percentage one number is of another
Example Question: You scored 15 out of 20 on a spelling test. What is your score as a percentage?
Step 1: Write the part over the whole: \( \frac{\text{Part}}{\text{Whole}} = \frac{15}{20} \).
Step 2: Multiply by 100%: \( \frac{15}{20} \times 100\% = \frac{3}{4} \times 100\% = 75\% \).
Answer: Your score was 75%.
Problem Type 3: Finding the whole given a percentage part
Example Question: 40% of the books on a shelf are storybooks. There are 20 storybooks. How many total books are on the shelf?
Method: The Unitary Method (Find 1% first)
Step 1: Find what 1% represents:
\( 40\% \rightarrow 20 \text{ books} \)
\( 1\% \rightarrow 20 \div 40 = 0.5 \text{ books} \)
Step 2: Find the whole (100%):
\( 100\% \rightarrow 0.5 \times 100 = 50 \text{ books} \)
Answer: There are 50 books in total on the shelf.
Problem Type 4: Percentage Increase (Goods and Services Tax / GST)
An increase means adding to the original 100%.
Example Question: A board game costs $50 before tax. A 9% GST is added to the price. What is the total amount to pay?
Step 1: Find the GST amount.
\( 9\% \text{ of } \$50 = \frac{9}{100} \times \$50 = \$4.50 \)
Step 2: Add the tax to the original price.
\( \$50 + \$4.50 = \$54.50 \)
Alternative Method: Total percentage = \( 100\% + 9\% = 109\% \). Then calculate \( \frac{109}{100} \times \$50 = \$54.50 \).
Answer: The total price including GST is $54.50.
Problem Type 5: Percentage Decrease (Discounts)
A decrease or discount means subtracting from the original 100%.
Example Question: A T-shirt costs $50, but it is on sale with a 10% discount. What is the discounted price?
Step 1: Find the discount amount.
\( 10\% \text{ of } \$50 = \frac{10}{100} \times \$50 = \$5 \)
Step 2: Subtract the discount from the original price.
\( \$50 - \$5 = \$45 \)
Answer: The discounted price of the T-shirt is $45.
Problem Type 6: Annual Interest
When you deposit money in a bank, the bank pays you extra money called interest over a period of time (usually 1 year).
Example Question: Mei Ling deposits $1000 into a savings account that pays an annual interest rate of 2%. How much interest does she earn at the end of 1 year?
\( \text{Annual Interest} = 2\% \text{ of } \$1000 = \frac{2}{100} \times \$1000 = \$20 \)
Answer: Mei Ling earns $20 in interest after 1 year (making her total savings $1020).
Key Takeaway
To solve percentage problems:
- Read the question carefully to identify the 100% (the base/whole).
- Decide if you are finding a part, the whole, an increase (add), or a decrease (subtract).
- Use the unitary method (finding 1%) whenever you need to work backwards from a part to 100%.
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.