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Introduction:
Fractions are a fundamental concept in mathematics that represent a part of a whole. They are used to describe and compare quantities that are not whole numbers. In this lesson, we will explore the basic concepts of fractions and learn hncepts:**
ow to manipulate and work with them.
Key Concepts:
1. Definition of Fractions:
- A fraction consists of two parts: the numerator and the denominator.
- The numerator represents the part of the whole, while the denominator represents the total number of equal parts the whole is divided into.
Example: In the fraction \( \frac{3}{5} \), 3 is the numerator, and 5 is the denominator.
2. Types of Fractions:
- **Proper Fractions:** The numerator is smaller than the denominator (e.g., \( \frac{2}{3} \)).
- **Improper Fractions:** The numerator is equal to or greater than the denominator (e.g., \( \frac{5}{4} \)).
- **Mixed Numbers:** A whole number combined with a proper fraction (e.g., \( 1 \frac{1}{2} \)).
3. Equivalent Fractions:
- Equivalent fractions represent the same part of a whole.
- They are obtained by multiplying or dividing both the numerator and denominator by the same non-zero number.
Example: \( \frac{2}{3} \) is equivalent to \( \frac{4}{6} \) because both represent two-thirds of a whole.
4. Comparing Fractions:
- To compare fractions, find a common denominator.
- If the denominators are the same, compare the numerators.
Example: Comparing \( \frac{2}{5} \) and \( \frac{3}{5} \), since the denominators are the same, \( \frac{3}{5} \) is greater.
5. Adding and Subtracting Fractions:
- To add or subtract fractions, find a common denominator.
- Add or subtract the numerators while keeping the denominator the same.
Example: \( \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \)
6. Multiplying and Dividing Fractions:
- To multiply fractions, multiply the numerators together and the denominators together.
- To divide fractions, multiply by the reciprocal of the divisor.
Example: \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)
Important Notes:
- Zero in the Denominator: Division by zero is undefined, so fractions with a denominator of zero are not valid.
- Simplifying Fractions: Always simplify fractions to their lowest terms by dividing both the numerator and denominator by their greatest common factor.
- Real-world Applications: Fractions are commonly used in everyday situations, such as cooking, measurements, and financial calculations.
