Let's Explore Percentages!

Hello, Super Mathematician! Get ready to learn about a super useful part of maths called percentages. It might sound like a big word, but don't worry, it's really simple once you get the hang of it.

You see percentages everywhere:

- In shops during a sale (like "50% off!")
- On your test scores (like "You got 90%!")
- On food labels (like "Contains 10% of your daily vitamins")

In these notes, we'll learn what percentages are, and how they are best friends with fractions and decimals. Let's begin our adventure!


What is a Percentage?

The Secret is in the Name!

The word "percent" comes from two smaller words: "per" and "cent".

- "Per" means "out of".
- "Cent" means "100" (think of a century having 100 years, or a dollar having 100 cents!).

So, percent means "out of 100". That's the most important thing to remember! A percentage is just a special type of fraction where the bottom number (the denominator) is always 100.

We use this symbol for percent: %

Analogy Time: The Pizza Party!

Imagine a giant pizza cut into 100 equal slices.

- If you eat 50 slices, you have eaten 50 out of 100 slices. That's 50% of the pizza!
- If you eat 25 slices, you have eaten 25 out of 100 slices. That's 25% of the pizza!
- If you eat all 100 slices (wow!), you have eaten 100 out of 100 slices. That's 100%, or the whole pizza!

Key Takeaway

A percentage is a part of a whole, where the whole is always thought of as 100 parts.


The Magic Trio: Percentages, Fractions, and Decimals

Percentages, fractions, and decimals are like three best friends who look different but are actually saying the same thing. They are just different ways to show a part of a whole.

For example, if a glass is half full, you can say:
- It is 50% full. (Percentage)
- It is $$ \frac{1}{2} $$ full. (Fraction)
- It is 0.5 full. (Decimal)

See? They all mean the exact same amount! Since they are related, we can learn to change them from one form to another. This is called interconversion.


Changing Between Percentages, Fractions, and Decimals

1. From Percentage (%) to Fraction (/)

This is the easiest one! Just remember what "percent" means: "out of 100".

Step-by-step:

1. Take the number in front of the % sign.
2. Write it as the top number (numerator) of a fraction.
3. Write 100 as the bottom number (denominator).
4. Simplify the fraction if you can!

Let's try some examples:

- Change 40% to a fraction: $$40\% = \frac{40}{100}$$ Now, we simplify it. We can divide the top and bottom by 20. $$\frac{40 \div 20}{100 \div 20} = \frac{2}{5}$$ So, 40% is the same as $$ \frac{2}{5} $$.

- Change 75% to a fraction: $$75\% = \frac{75}{100}$$ Simplify by dividing the top and bottom by 25. $$\frac{75 \div 25}{100 \div 25} = \frac{3}{4}$$ So, 75% is the same as $$ \frac{3}{4} $$.

Key Takeaway

To change a percentage to a fraction, put the number over 100 and simplify.


2. From Fraction (/) to Percentage (%)

To go the other way, we just need to do the opposite. We want to find out how many parts it would be if the whole was 100.

Step-by-step:

1. Take your fraction.
2. Multiply it by 100.
3. Add the % sign. That's it!

Let's try some examples:

- Change $$ \frac{1}{2} $$ to a percentage: $$\frac{1}{2} \times 100\% = 50\%$$

- Change $$ \frac{1}{4} $$ to a percentage: $$\frac{1}{4} \times 100\% = 25\%$$

Key Takeaway

To change a fraction to a percentage, multiply the fraction by 100%.


3. From Percentage (%) to Decimal (.)

This one has a super easy trick! Remember, a percentage is a number out of 100. Dividing by 100 is the key.

Step-by-step:

1. Take the percentage number (and imagine it has a decimal point at the end, even if you can't see it, like 50 is 50.0).
2. Move the decimal point two places to the LEFT.
3. Remove the % sign.

Let's try some examples:

- Change 65% to a decimal:
The number is 65. The decimal point is 65.
Move it two places left: .65 or 0.65.

- Change 9% to a decimal:
The number is 9. The decimal point is 9.
Move it two places left. We need to add a zero to hold the place! .09 or 0.09.

Key Takeaway

To change a percentage to a decimal, divide by 100 (or just move the decimal point two places left).


4. From Decimal (.) to Percentage (%)

You guessed it! We just do the opposite of what we just did.

Step-by-step:

1. Take your decimal number.
2. Move the decimal point two places to the RIGHT.
3. Add the % sign at the end.

Let's try some examples:

- Change 0.85 to a percentage:
Move the decimal two places right: 85.
Add the % sign: 85%.

- Change 0.4 to a percentage:
Move the decimal two places right. We need to add a zero! 40.
Add the % sign: 40%.

Key Takeaway

To change a decimal to a percentage, multiply by 100 (or just move the decimal point two places right) and add the % sign.


Putting Your Percentage Skills to Work!

Now that we know how to change between our three friends, let's use them to solve some real-life problems. Don't worry if this seems tricky at first, we'll go step-by-step.

Finding a Percentage of a Number

This is the most common type of percentage problem. It's like finding a piece of a whole amount.

Question: What is 60% of 50?

Step-by-step:

1. Change the percentage into a fraction or a decimal. Let's use a decimal because it's often faster. 60% becomes 0.60.
2. The word "of" in maths almost always means "multiply" (×).
3. Now, multiply your decimal by the number: $$0.60 \times 50 = 30$$

Answer: 60% of 50 is 30.

Another example: A T-shirt costs $20. It's on sale for 25% off. How much money do you save?
We need to find 25% of $20.
$$25\% = 0.25$$ $$0.25 \times 20 = 5$$ You save $5!


Finding "What Percentage is...?"

This is like getting a score on a test and figuring out your percentage.

Question: What percentage of 50 is 30?

Step-by-step:

1. Make a fraction. The "part" goes on top and the "whole" goes on the bottom. Here, 30 is the part and 50 is the whole. So our fraction is $$ \frac{30}{50} $$.
2. Change this fraction into a percentage. Remember how? We multiply by 100%!

$$\frac{30}{50} \times 100\% = 60\%$$

Answer: 30 is 60% of 50. (It's like getting 30 out of 50 on a test... you scored 60%!)


Percentage Increase and Decrease

Sometimes, an amount gets bigger (increase) or smaller (decrease) by a certain percentage.

1. Percentage Increase

Question: What is the result when 50 is increased by 10%?

Step-by-step (in two parts):

1. Find the increase amount: First, find out what 10% of 50 is.
$$10\% \text{ of } 50 = 0.10 \times 50 = 5$$ 2. Add it on: Now, add this amount to the original number.
$$50 + 5 = 55$$

Answer: When 50 is increased by 10%, the result is 55.

2. Percentage Decrease

Question: What is the result when 50 is decreased by 10%?

Step-by-step (in two parts):

1. Find the decrease amount: First, find out what 10% of 50 is.
$$10\% \text{ of } 50 = 0.10 \times 50 = 5$$ 2. Take it away: Now, subtract this amount from the original number.
$$50 - 5 = 45$$

Answer: When 50 is decreased by 10%, the result is 45.

Did You Know?

The "%" symbol is over 500 years old! It started as the letters "pc" (for per cento), which then became a little circle over a line, and finally turned into the symbol we use today.

You've done an amazing job! Percentages are a key part of maths, and with a little practice, you'll be a percentages expert in no time. Keep practicing and stay curious!