A fraction is a mathematical way of expressing a part of a whole. It consists of two integers separated by a horizontal bar, written in the form \( \frac{a}{b} \), where:
Addition (Different Denominators):
\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \]
Subtraction (Different Denominators):
\[ \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \]
Multiplication:
\[ \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \]
Division:
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc} \]
Mixed Number to Improper Fraction:
If \( a \frac{b}{c} \) is a mixed number, then:
\[ a \frac{b}{c} = \frac{ac + b}{c} \]
Reducing to Lowest Terms:
Divide the numerator and denominator by their greatest common divisor (GCD):
\[ \frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \]
Comparing Fractions:
Use common denominators or decimal conversion.
\[ \frac{2}{5} = 0.4,\quad \frac{3}{8} = 0.375 \Rightarrow \frac{2}{5} > \frac{3}{8} \]
Mastering fractions is fundamental in mathematics and daily life. Whether you're analyzing data, managing money, or cooking a recipe, fractions help interpret and solve real-world problems accurately.