Math Formula for Fractions

Math Formula for Fractions

Introduction to Fractions

A fraction represents a part of a whole and is written in the form \( \frac{a}{b} \), where:

• \( a \) is the numerator (the top number).
• \( b \) is the denominator (the bottom number).

For example, \( \frac{3}{4} \) means three parts out of four.

Basic Operations with Fractions

Addition of Fractions

When adding fractions with the same denominator:

\[ \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} \]

Example:

\[ \frac{2}{5} + \frac{3}{5} = \frac{2+3}{5} = \frac{5}{5} = 1 \]

For different denominators, find the least common denominator (LCD):

\[ \frac{a}{b} + \frac{c}{d} = \frac{a \times d + c \times b}{b \times d} \]

Example:

\[ \frac{1}{3} + \frac{1}{4} = \frac{1 \times 4 + 1 \times 3}{3 \times 4} = \frac{4+3}{12} = \frac{7}{12} \]

Subtraction of Fractions

For the same denominator:

\[ \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} \]

Example:

\[ \frac{5}{7} - \frac{2}{7} = \frac{5-2}{7} = \frac{3}{7} \]

For different denominators:

\[ \frac{a}{b} - \frac{c}{d} = \frac{a \times d - c \times b}{b \times d} \]

Example:

\[ \frac{5}{6} - \frac{1}{4} = \frac{5 \times 4 - 1 \times 6}{6 \times 4} = \frac{20-6}{24} = \frac{14}{24} = \frac{7}{12} \]

Multiplication of Fractions

Multiply the numerators and the denominators:

\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]

Example:

\[ \frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20} = \frac{3}{10} \]

Division of Fractions

Multiply by the reciprocal:

\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \]

Example:

\[ \frac{3}{7} \div \frac{2}{5} = \frac{3}{7} \times \frac{5}{2} = \frac{3 \times 5}{7 \times 2} = \frac{15}{14} \]

Advanced Concepts in Fractions

Simplifying Fractions

A fraction is simplified when there are no common factors between the numerator and denominator. Use the Greatest Common Divisor (GCD) to simplify.

Example:

\[ \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \]

Converting Improper Fractions to Mixed Numbers

Divide the numerator by the denominator:

Example:

\[ \frac{11}{4} = 2 \frac{3}{4} \]

Converting Mixed Numbers to Improper Fractions

Use the formula:

\[ a \frac{b}{c} = \frac{a \times c + b}{c} \]

Example:

\[ 3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15+2}{5} = \frac{17}{5} \]

Real-Life Applications of Fractions

Fractions are used in measurements, cooking, financial calculations, and engineering. Understanding how to manipulate fractions is essential for daily life and professional applications.

Conclusion

Understanding fraction formulas is essential for solving mathematical problems. By mastering addition, subtraction, multiplication, and division, you can handle fractions confidently in real-life applications.

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