Fractions are mathematical expressions that represent a part of a whole or a division of one quantity by another. They consist of a numerator and a denominator, separated by a horizontal line called a fraction bar or a solidus. The numerator represents the number of parts we have, while the denominator represents the total number of equal parts that make up a whole. For example, \frac{2}{5} means 2 part out of 5

## Types of Fractions

### Proper Fraction

A proper fraction is a fraction where the numerator is smaller than the denominator. For example, \frac{2}{5}, \frac{3}{8}, \frac{7}{9} are all proper fractions. Proper fractions represent values less than one.

### Improper Fraction

An improper fraction is a fraction where the numerator is equal to or greater than the denominator. For example, \frac{5}{5}, \frac{7}{3}, \frac{11}{2} are all improper fractions. Improper fractions represent values equal to or greater than one.

### Mixed Number

A mixed number is a combination of a whole number and a proper fraction. It is expressed as a whole number followed by a proper fraction. For example, 3\frac{1}{2}, 2\frac{3}{4}, 5\frac{2}{7} are all mixed numbers.

### Equivalent Fractions

Equivalent fractions are fractions that represent the same value or proportion, but their numerators and denominators may differ. To find equivalent fractions, we multiply or divide the numerator and denominator by the same non-zero number.

**For example, ** \frac {2}{5} = \frac {2}{5} \times \frac {3}{3} = \frac{6}{15}

## How to determine equivalent fractions

### Multiplying or Dividing

Multiply or divide both the numerator and denominator of a fraction by the same non-zero number. For example, multiplying 2/3 by 2 gives an equivalent fraction of 4/6. Dividing 4/8 by 2 results in an equivalent fraction of 2/4.

### Simplifying

To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. For example, the fraction 4/8 can be simplified by dividing both the numerator and denominator by 4, resulting in 1/2.

### Cross-multiplication

Cross-multiplication is used to determine equivalent fractions when comparing two fractions. For example, to compare 2/3 and 4/6, we multiply the numerator of the first fraction (2) by the denominator of the second fraction (6) and compare it with the product of the denominator of the first fraction (3) and the numerator of the second fraction (4). If the products are equal, the fractions are equivalent.

### Common Factors

Sometimes, equivalent fractions can be found by recognizing common factors between the numerator and denominator. For instance, the fraction 8/12 can be simplified by recognizing that both 8 and 12 are divisible by 4. Dividing both by 4 gives an equivalent fraction of 2/3.

Understanding fractions and their equivalents is essential for various mathematical operations, such as addition, subtraction, multiplication, and division involving fractions. Equivalent fractions allow us to manipulate fractions while preserving their underlying values and proportions.

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