Fractions

Welcome to the World of Fractions!

In this chapter, we are going to explore fractions. You might think of fractions as just numbers on a page, but they are actually everywhere! Whether you are splitting a pizza with friends, measuring ingredients for a cake, or calculating a discount at your favorite shop, you are using fractions. By the end of these notes, you’ll be a pro at adding, subtracting, and even converting them into decimals.

Don’t worry if fractions seem tricky at first. We’ll take it one slice at a time!

1. The Basics: What is a Fraction?

A fraction represents a part of a whole. It consists of two main numbers:
1. The Numerator (the top number): Tells us how many parts we have.
2. The Denominator (the bottom number): Tells us how many equal parts the whole is divided into.

Analogy: Imagine a pizza cut into 8 equal slices. If you eat 3 slices, you have eaten \( \frac{3}{8} \) of the pizza. The "8" is the total slices available, and the "3" is what you took!

Did you know? The line between the numerator and denominator is called a vinculum. It actually means "to divide"!

Key Takeaway:

The denominator is the "down" number (D for Down), and it shows the total parts.

2. Equivalent Fractions and Simplifying

Equivalent fractions are different fractions that actually represent the same amount. For example, eating \( \frac{1}{2} \) of a cake is the same as eating \( \frac{2}{4} \) or \( \frac{4}{8} \).

The Golden Rule: Whatever you do to the top, you must do to the bottom. To find an equivalent fraction, multiply or divide the numerator and denominator by the same number.

Simplifying Fractions: This means writing the fraction using the smallest possible numbers. We do this by dividing both the top and bottom by their Highest Common Factor (HCF).
Example: Simplify \( \frac{10}{15} \).
1. Both 10 and 15 can be divided by 5.
2. \( 10 \div 5 = 2 \)
3. \( 15 \div 5 = 3 \)
4. So, \( \frac{10}{15} = \frac{2}{3} \).

Quick Review:

• To simplify, divide until you can't divide anymore.
• To find equivalents, multiply top and bottom by the same number.

3. Mixed Numbers and Improper Fractions

Sometimes we have more than one "whole."
• Proper Fraction: The numerator is smaller than the denominator (e.g., \( \frac{3}{4} \)).
• Improper Fraction: The numerator is larger than or equal to the denominator (e.g., \( \frac{7}{4} \)). These are sometimes called "top-heavy" fractions.
• Mixed Number: A mix of a whole number and a fraction (e.g., \( 1 \frac{3}{4} \)).

Converting Improper to Mixed:
Divide the top by the bottom. The answer is your whole number, and the remainder is your new numerator.
Example: \( \frac{11}{4} \)
11 divided by 4 is 2 with a remainder of 3. So, \( \frac{11}{4} = 2 \frac{3}{4} \).

Converting Mixed to Improper:
1. Multiply the whole number by the denominator.
2. Add the numerator.
3. Put that total over the original denominator.
Example: \( 3 \frac{1}{2} \)
\( (3 \times 2) + 1 = 7 \). So, the fraction is \( \frac{7}{2} \).

Memory Aid (The MAD Method):

Multiply the whole by the bottom.
Add the top.
Denominator stays the same!

4. Adding and Subtracting Fractions

Scenario A: Same Denominators
If the bottom numbers are the same, just add or subtract the top numbers and keep the bottom number the same.
Example: \( \frac{2}{7} + \frac{3}{7} = \frac{5}{7} \)

Scenario B: Different Denominators
You cannot add them until the denominators are the same! You must find a Common Denominator (usually the Lowest Common Multiple).
Example: \( \frac{1}{4} + \frac{2}{5} \)
1. Find a number both 4 and 5 go into: 20.
2. Change \( \frac{1}{4} \): \( (1 \times 5) / (4 \times 5) = \frac{5}{20} \).
3. Change \( \frac{2}{5} \): \( (2 \times 4) / (5 \times 4) = \frac{8}{20} \).
4. Add: \( \frac{5}{20} + \frac{8}{20} = \frac{13}{20} \).

Common Mistake to Avoid: Never, ever add the bottom numbers together! \( \frac{1}{2} + \frac{1}{2} \) is \( \frac{2}{2} \) (one whole), not \( \frac{2}{4} \)!

5. Multiplying and Dividing Fractions

Multiplying: This is actually the easiest part! Just multiply the tops and multiply the bottoms.
Example: \( \frac{2}{3} \times \frac{5}{6} = \frac{10}{18} \). (Then simplify to \( \frac{5}{9} \))

Dividing: Use the KCF method!
1. Keep the first fraction as it is.
2. Change the sign from \( \div \) to \( \times \).
3. Flip the second fraction upside down (this is called the reciprocal).
Example: \( \frac{2}{3} \div \frac{1}{4} \)
Keep: \( \frac{2}{3} \)
Change: \( \times \)
Flip: \( \frac{4}{1} \)
Result: \( \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} \) (or \( 2 \frac{2}{3} \)).

Key Takeaway:

Multiplication is "straight across." Division is "flip and multiply."

6. Fractions of a Quantity

To find a fraction of an amount (like "find \( \frac{2}{5} \) of £35"), follow these two steps:
1. Divide the amount by the denominator (bottom number).
2. Multiply the result by the numerator (top number).

Example: Find \( \frac{2}{5} \) of £35.
1. \( 35 \div 5 = 7 \)
2. \( 7 \times 2 = 14 \). Answer: £14.

Expressing one quantity as a fraction of another: Simply put the first number over the second and simplify.
Example: What is 20p as a fraction of £1?
1. Make sure units are the same: 20p and 100p.
2. Write as a fraction: \( \frac{20}{100} \).
3. Simplify: \( \frac{1}{5} \).

7. Converting Fractions and Decimals

To turn any fraction into a decimal, just divide the top by the bottom.
• \( \frac{1}{2} = 1 \div 2 = 0.5 \)
• \( \frac{3}{4} = 3 \div 4 = 0.75 \)
• \( \frac{1}{8} = 1 \div 8 = 0.125 \)

Terminating vs Recurring:
• Terminating: The decimal ends (like \( 0.4 \)).
• Recurring: The decimal goes on forever in a pattern (like \( \frac{1}{3} = 0.333... \)). We show this by putting a dot over the repeating number: \( 0.\dot{3} \).

Higher Tier Tip:

For higher tier, you may need to convert recurring decimals back into fractions. For example, \( 0.\dot{4}\dot{1} = \frac{41}{99} \). Notice how many digits repeat determines how many 9s go on the bottom!

8. Ordering Fractions

If you need to put fractions in order from smallest to largest, the easiest way is to either:
1. Convert them all into decimals using division.
2. Find a common denominator for all of them so you can compare the numerators.

Example: Which is larger, \( \frac{3}{4} \) or \( \frac{2}{3} \)?
Common denominator is 12.
\( \frac{3}{4} = \frac{9}{12} \)
\( \frac{2}{3} = \frac{8}{12} \)
So, \( \frac{3}{4} \) is larger.

Final Quick Review:

• Simplifying: Divide top and bottom by the same number.
• Adding/Subtracting: Get the bottoms the same first!
• Multiplying: Top \( \times \) Top, Bottom \( \times \) Bottom.
• Dividing: Keep, Change, Flip (KCF).
• Fraction of amount: Divide by the bottom, multiply by the top.