Fractions, Decimals and Percentages

Welcome to the World of Fractions, Decimals, and Percentages!

Have you ever shared a pizza with friends, looked at a price tag with 20% off, or seen a sports statistic like 0.75? These are all different ways of talking about the same thing: parts of a whole. In this chapter, we are going to learn how to swap between these three "languages" of math so you can use whichever one is easiest for the job. Don't worry if this seems tricky at first—once you see the patterns, it becomes much easier!

1. Fractions: The "Parts" of a Whole

A fraction tells us how many parts of a whole we have. It looks like this: \( \frac{2}{5} \).

The Top Number (Numerator): This tells us how many parts we actually have. (Memory aid: Numerator is North/Top).
The Bottom Number (Denominator): This tells us how many equal parts the whole is split into. (Memory aid: Denominator is Down/Bottom).

Simplifying Fractions

Sometimes fractions look big and scary, like \( \frac{10}{20} \). We can make them simpler by dividing the top and bottom by the same number. If we divide both by 10, we get \( \frac{1}{2} \). It’s the same amount, just easier to read!

Example: Simplify \( \frac{12}{18} \). Both numbers can be divided by 6. \( 12 \div 6 = 2 \) and \( 18 \div 6 = 3 \). So, the simplest form is \( \frac{2}{3} \).

Adding and Subtracting Fractions

To add or subtract, the Denominators must be the same. Think of it like this: you can’t easily add 3 apples and 2 oranges to get "5 app-oranges." They need to be the same "type" (the same denominator).
1. Find a common denominator (a number both bottom numbers fit into).
2. Change the numerators to match.
3. Add or subtract the top numbers only! Keep the bottom number the same.

Multiplying and Dividing Fractions

Multiplying: This is actually the easiest one! Just multiply the tops and multiply the bottoms.
\( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)

Dividing: Use the KCF trick:
1. Keep the first fraction.
2. Change the sign to multiply.
3. Flip the second fraction upside down.
Then just multiply as usual!

Quick Review:

Key Takeaway: Fractions are just divisions. Always try to simplify your final answer to make it look "cleaner."

2. Decimals: The Power of Ten

Decimals are another way to write fractions, but they always use a "base-10" system. This means every column to the right of the decimal point represents a fraction of 10, 100, or 1,000.

Place Value:
0.1 = One Tenth (\( \frac{1}{10} \))
0.01 = One Hundredth (\( \frac{1}{100} \))
0.001 = One Thousandth (\( \frac{1}{1000} \))

Comparing Decimals

Sometimes it’s hard to tell which decimal is bigger. For example: Is 0.5 bigger than 0.45?
The "Filler Zero" Trick: Add a zero to the end so they have the same number of digits.
0.5 becomes 0.50.
Now compare: Is 0.50 bigger than 0.45? Yes! It’s like comparing 50p to 45p.

Multiplying and Dividing by 10, 100, and 1,000

When we multiply by 10, we move the decimal point one place to the right (making the number bigger).
When we divide by 10, we move the decimal point one place to the left (making the number smaller).
The number of zeros tells you how many places to move!

Did you know? The word "decimal" comes from the Latin word "decimus," which means "tenth." This is why we have 10 digits (0-9) and why our money system uses 100 pence to a pound!

Quick Review:

Common Mistake to Avoid: Don't just "add a zero" at the end when multiplying a decimal by 10. You must move the decimal point! For example, \( 1.5 \times 10 \) is \( 15 \), not \( 1.50 \).

3. Percentages: Out of 100

The word "Percent" literally means "per hundred." So, 70% just means 70 out of every 100.

Finding a Percentage of an Amount

If you need to find a percentage without a calculator, use these "building blocks":
- To find 50%, divide by 2 (half it).
- To find 10%, divide by 10.
- To find 5%, find 10% and then half it.
- To find 1%, divide by 100.

Example: Find 15% of \$40.
10% of 40 = \$4.
5% of 40 = \$2 (half of the 10%).
15% = \$4 + \$2 = \$6.

Quick Review:

Key Takeaway: Percentages are just fractions with a denominator of 100. 25% is exactly the same as \( \frac{25}{100} \).

4. The Conversion Bridge: Swapping Between Them

Being able to switch between fractions, decimals, and percentages is like being a math superhero. Here is how you do it:

From Fraction to Decimal

Divide the top number by the bottom number.
Example: \( \frac{1}{4} \) is \( 1 \div 4 = 0.25 \).

From Decimal to Percentage

Multiply by 100 (move the decimal point two places to the right) and add the % sign.
Example: \( 0.35 = 35\% \).

From Percentage to Fraction

Put the percentage number over 100 and simplify.
Example: \( 60\% = \frac{60}{100} \). This simplifies to \( \frac{6}{10} \), and then to \( \frac{3}{5} \).

Common "Must-Know" Conversions

These are the ones that appear all the time. Try to memorize these!
- \( \frac{1}{2} \) = 0.5 = 50%
- \( \frac{1}{4} \) = 0.25 = 25%
- \( \frac{3}{4} \) = 0.75 = 75%
- \( \frac{1}{10} \) = 0.1 = 10%
- \( \frac{1}{5} \) = 0.2 = 20%

Final Summary:

1. Fractions use a numerator and denominator to show parts.
2. Decimals use place value (tenths, hundredths) to show parts.
3. Percentages always compare parts to the number 100.
4. You can switch between them by dividing, multiplying by 100, or using fractions over 100.

Don't worry if this seems like a lot! The more you practice finding 10% of a number or simplifying a fraction, the more natural it will feel. You use these skills every time you go shopping or check your phone's battery percentage!