A fraction represents a part of a whole and consists of a numerator (top number) and a denominator (bottom number). The fraction \(\frac{3}{4}\) is read as "three-fourths" or "three over four," meaning three parts out of four equal parts. The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts you have.

To add fractions with different denominators, first find the least common multiple (LCM) of the denominators to create a common denominator. Convert each fraction to an equivalent fraction with this common denominator by multiplying both numerator and denominator by the appropriate number. Then add the numerators while keeping the common denominator the same. Finally, simplify the result if possible. For example, \(\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).

A proper fraction has a numerator smaller than its denominator (like \(\frac{3}{4}\) or \(\frac{2}{5}\)), and its value is always less than 1. An improper fraction has a numerator equal to or greater than its denominator (like \(\frac{7}{4}\) or \(\frac{5}{5}\)), and its value is equal to or greater than 1. Improper fractions can be converted to mixed numbers, which combine a whole number with a proper fraction.

To multiply fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. No common denominator is needed. The formula is \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\). For example, \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\). Always simplify the final answer by dividing both numerator and denominator by their greatest common divisor.

To divide fractions, use the "Keep, Change, Flip" method: keep the first fraction unchanged, change the division sign to multiplication, and flip (take the reciprocal of) the second fraction by swapping its numerator and denominator. Then multiply the fractions normally. For example, \(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\).

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, to convert \(2\frac{3}{4}\): multiply \(2 \times 4 = 8\), add the numerator \(8 + 3 = 11\), so the improper fraction is \(\frac{11}{4}\). The formula is \(a\frac{b}{c} = \frac{(a \times c) + b}{c}\).

Equivalent fractions are different fractions that represent the same value. To find equivalent fractions, multiply or divide both the numerator and denominator by the same non-zero number. For example, \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}\) are all equivalent. To check if two fractions are equivalent, cross-multiply: if \(a \times d = b \times c\), then \(\frac{a}{b} = \frac{c}{d}\).

To simplify a fraction to lowest terms, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by this GCD. A fraction is in simplest form when the only common factor between numerator and denominator is 1. For example, to simplify \(\frac{24}{36}\): find GCD(24, 36) = 12, then divide both: \(\frac{24 \div 12}{36 \div 12} = \frac{2}{3}\).

To convert a fraction to a decimal, divide the numerator by the denominator using long division or a calculator. For example, \(\frac{3}{4} = 3 \div 4 = 0.75\). Some fractions result in terminating decimals (like 0.75), while others produce repeating decimals (like \(\frac{1}{3} = 0.333...\)). For mixed numbers, convert the fraction part to a decimal and add it to the whole number.

To convert a decimal to a fraction, count the number of decimal places, write the digits as the numerator, and use a power of 10 as the denominator (10 for one decimal place, 100 for two, etc.). Then simplify. For example, 0.625 has three decimal places, so write \(\frac{625}{1000}\). Simplify by dividing both by GCD(625, 1000) = 125 to get \(\frac{5}{8}\). For mixed decimals like 2.5, separate the whole number: \(2\frac{5}{10} = 2\frac{1}{2}\).

To compare fractions, convert them to equivalent fractions with a common denominator (using the LCM of the denominators), then compare the numerators. The fraction with the larger numerator is larger. Alternatively, convert both fractions to decimals and compare. For example, to compare \(\frac{2}{3}\) and \(\frac{3}{4}\): convert to \(\frac{8}{12}\) and \(\frac{9}{12}\). Since 9 > 8, we have \(\frac{3}{4} > \frac{2}{3}\). You can also cross-multiply: if \(a \times d > b \times c\), then \(\frac{a}{b} > \frac{c}{d}\).

Fractions, decimals, and percentages are three different ways to represent parts of a whole. A fraction uses numerator and denominator (\(\frac{1}{4}\)), a decimal uses place value (0.25), and a percentage uses parts per hundred (25%). They are interchangeable: \(\frac{1}{4} = 0.25 = 25\%\). Each representation has advantages in different contexts—fractions for exact values, decimals for calculations, and percentages for comparisons and proportions.