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Understanding fractions is key to mastering many mathematical concepts. This overview covers the basics of fraction notation, including the roles of numerators and denominators, and provides step-by-step procedures for multiplying and dividing fractions. Simplification strategies for complex operations and interactions with whole numbers are also discussed, along with methods for converting and calculating with mixed numbers and algebraic fractions.

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## Definition of Fractions

### Numerator and Denominator

A fraction is made up of a numerator and a denominator, representing the number of equal parts and the total number of equal parts, respectively

### Division by Zero

The denominator of a fraction must never be zero, as division by zero is undefined

### Examples of Fractions

Common examples of fractions include \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), and \(\dfrac{3}{4}\)

## Multiplying Fractions

### Process of Multiplication

To multiply fractions, multiply the numerators and denominators separately and simplify the result

### Simplifying Fractions

The resulting fraction should be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD)

### Examples of Multiplication

When multiplying \(\dfrac{4}{9}\) by \(\dfrac{3}{5}\), the product is \(\dfrac{12}{45}\), which simplifies to \(\dfrac{4}{15}\) after dividing by their GCD, which is 3

## Dividing Fractions

### Process of Division

To divide fractions, take the reciprocal of the divisor and multiply it by the dividend

### Examples of Division

To divide \(\dfrac{4}{5}\) by \(\dfrac{7}{10}\), first find the reciprocal of \(\dfrac{7}{10}\), which is \(\dfrac{10}{7}\), and then multiply to get \(\dfrac{40}{35}\), which simplifies to \(\dfrac{8}{7}\) or \(1\dfrac{1}{7}\)

### Multiplying or Dividing by a Whole Number

When multiplying or dividing a fraction by a whole number, convert the whole number to a fraction with a denominator of one and apply the standard rules for fractions

## Simplifying Fraction Operations

### Canceling Common Factors

When multiplying or dividing several fractions, it is efficient to simplify the expression by canceling out common factors between numerators and denominators

### Converting Mixed Numbers

Mixed numbers must be converted to improper fractions before multiplication or division

### Algebraic Fractions

Algebraic fractions follow the same rules for multiplication and division as numerical fractions

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