Common fraction

Common Fraction

A common fraction (or simple fraction) is a way of expressing a number as the quotient of two integers, where the numerator is divided by the denominator. Common fractions are a fundamental concept in mathematics and are used to represent parts of a whole.

Definition

A common fraction is written in the form \( \frac{a}{b} \), where:

- \( a \) is the numerator, representing the number of equal parts being considered. - \( b \) is the denominator, representing the total number of equal parts in the whole.

For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator, indicating that three parts out of a total of four equal parts are being considered.

Types of Common Fractions

Proper Fractions

A proper fraction is a fraction where the absolute value of the numerator is less than the absolute value of the denominator (\( |a| < |b| \)). For example, \( \frac{3}{4} \) is a proper fraction.

Improper Fractions

An improper fraction is a fraction where the absolute value of the numerator is greater than or equal to the absolute value of the denominator (\( |a| \geq |b| \)). For example, \( \frac{5}{3} \) is an improper fraction.

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. For example, \( 1\frac{1}{2} \) is a mixed number equivalent to the improper fraction \( \frac{3}{2} \).

Operations with Common Fractions

Addition and Subtraction

To add or subtract fractions, they must have a common denominator. For example:

\[ \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]

Multiplication

To multiply fractions, multiply the numerators together and the denominators together:

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

Division

To divide fractions, multiply by the reciprocal of the divisor:

\[ \frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9} \]

Simplifying Fractions

A fraction is simplified by dividing the numerator and the denominator by their greatest common divisor (GCD). For example, the fraction \( \frac{8}{12} \) can be simplified to \( \frac{2}{3} \) by dividing both the numerator and the denominator by 4.

Applications

Common fractions are used in various fields such as engineering, science, and everyday life to represent ratios, proportions, and probabilities. They are essential in calculations involving measurements, cooking, and financial transactions.

Also see

• Decimal fraction
• Mixed number
• Improper fraction
• Rational number
• Greatest common divisor

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