Fractions

1. What is a Fraction?

A fraction represents a part of a whole.

It is written in the form:

$$\frac{\text{Numerator}}{\text{Denominator}}$$

• Numerator: The top number — tells how many parts you have.

• Denominator: The bottom number — tells how many equal parts the whole is divided into.

Example:

$$\frac{3}{4} \text{ means 3 parts out of 4.}$$

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2. Types of Fractions

a) Proper Fractions

• Numerator < Denominator

• Value is less than 1

Example:

$$\frac{3}{5}, \frac{7}{9}$$

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b) Improper Fractions

• Numerator ≥ Denominator

• Value is equal to or more than 1

Example:

$$\frac{5}{4}, \frac{8}{8}, \frac{9}{6}$$

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c) Mixed Fractions

• A combination of a whole number and a proper fraction

Example:

$$2 \frac{1}{3} \text{ (means } 2 + \frac{1}{3})$$

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3. Equivalent Fractions

Fractions that look different but have the same value.

To get an equivalent fraction, multiply or divide both the numerator and the denominator by the same number.

Examples:

$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$$ $$\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$$

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4. Comparison of Fractions

To compare fractions:

• Same denominator: compare numerators directly

• Different denominators: find common denominators or convert to decimals

Example 1 (same denominators):

$$\frac{3}{7} < \frac{5}{7}$$

Example 2 (different denominators):

Compare $\frac{2}{3}$ and $\frac{3}{4}$
→ Common denominator = 12

$$\frac{2}{3} = \frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12} \Rightarrow \frac{2}{3} < \frac{3}{4}$$

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5. Operations on Fractions

a) Addition

Same denominators:

$$\frac{2}{5} + \frac{1}{5} = \frac{3}{5}$$

Different denominators:

$$\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$$

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b) Subtraction

Example:

$$\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$$

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c) Multiplication

Multiply numerators and denominators:

$$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$$

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d) Division

Keep the first fraction, flip the second, then multiply:

$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1 \frac{7}{8}$$

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Bonus: Converting Between Improper and Mixed Fractions

Improper to Mixed:

$$\frac{7}{4} = 1 \frac{3}{4}$$

Mixed to Improper:

$$2 \frac{1}{3} = \frac{7}{3}$$

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Summary:

Concept	Example
Proper Fraction	$\frac{3}{5}$
Improper Fraction	$\frac{9}{6}$
Mixed Fraction	$2 \frac{1}{3}$
Equivalent	$\frac{1}{2} = \frac{2}{4}$
Addition	$\frac{1}{4} + \frac{1}{2} = \frac{3}{4}$
Multiplication	$\frac{2}{3} \times \frac{3}{5} = \frac{6}{15}$

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