Working with Fractions

Arithmetic Operations with Fractions Fractions are fundamental to mathematics, and working with them requires understanding several key operations. This guide covers the essential techniques you'll need to add, subtract, multiply, and divide fractions, as well as convert between fractions and decimals. Working with Equivalent Fractions and Reducing Equivalent fractions are different representations of the same value. The key insight is that when you multiply both the numerator and denominator by the same non-zero number, you create an equivalent fraction. For example, $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$ are all equivalent because: $$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$$ This principle is powerful because it allows you to rewrite fractions in forms that are easier to work with. The reverse process is reducing (or simplifying) a fraction. This means dividing both the numerator and denominator by their greatest common divisor (GCD)—the largest number that divides evenly into both. When you reduce a fraction, you get it into lowest terms, its simplest equivalent form. For instance, to reduce $\frac{12}{18}$, you find that the GCD of 12 and 18 is 6. Then: $$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$$ Why this matters: Always reduce your final answers to lowest terms. This makes them easier to interpret and is expected in most mathematical contexts. Comparing Fractions To determine which fraction is larger, you need to consider whether they share a common denominator. When denominators are the same If two fractions have the same denominator, simply compare the numerators. The fraction with the larger numerator is larger. For example, $\frac{7}{10} > \frac{3}{10}$ because 7 > 3. When denominators are different When denominators differ, you can use cross-multiplication to compare. To determine whether $\frac{a}{b}$ is greater than $\frac{c}{d}$, multiply the numerator of the first fraction by the denominator of the second, and multiply the numerator of the second by the denominator of the first. If the first product is larger, the first fraction is larger. Mathematically: $\frac{a}{b} > \frac{c}{d}$ if and only if $ad > bc$. Example: Compare $\frac{3}{4}$ and $\frac{5}{7}$. Cross-multiply: $3 \times 7 = 21$ and $5 \times 4 = 20$. Since $21 > 20$, we know $\frac{3}{4} > \frac{5}{7}$. Adding and Subtracting Fractions To add fractions, you cannot simply add the numerators and denominators separately. Instead, you must first ensure both fractions share the same denominator. Finding a common denominator Convert each fraction to an equivalent fraction with a shared denominator. The least common multiple (LCM) of the denominators is often the most efficient choice, though any common denominator works. Example: Add $\frac{1}{3} + \frac{1}{4}$. The LCM of 3 and 4 is 12. Convert each fraction: $$\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}$$ Now add the numerators: $$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$$ Subtracting fractions Subtraction follows the identical process: find a common denominator, then subtract the numerators. Example: $\frac{5}{6} - \frac{1}{4}$. The LCM of 6 and 4 is 12: $$\frac{5}{6} = \frac{10}{12}, \quad \frac{1}{4} = \frac{3}{12}$$ $$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$$ Multiplying Fractions Multiplying fractions is simpler than adding them—no common denominator is needed. Simply multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$ Example: $\frac{2}{3} \times \frac{5}{7} = \frac{10}{21}$. Cancellation: A shortcut to keep answers simple Before multiplying, you can cancel common factors between any numerator and any denominator. This reduces your numbers before multiplication, making calculations simpler and keeping your answer already in lowest terms. Example: Simplify $\frac{3}{4} \times \frac{8}{9}$. Notice that 3 and 9 share a common factor of 3, and 4 and 8 share a common factor of 4: $$\frac{3}{4} \times \frac{8}{9} = \frac{\cancel{3}^1}{\cancel{4}^1} \times \frac{\cancel{8}^2}{\cancel{9}^3} = \frac{1 \times 2}{1 \times 3} = \frac{2}{3}$$ Multiplying by whole numbers A whole number can be treated as a fraction with denominator 1. So $\frac{2}{5} \times 3 = \frac{2}{5} \times \frac{3}{1} = \frac{6}{5}$. Multiplying mixed numbers A mixed number like $2\frac{1}{3}$ combines a whole number and a fraction. To multiply mixed numbers, first convert each to an improper fraction (where the numerator is larger than the denominator), then multiply using the standard rule. Example: Multiply $2\frac{1}{3} \times 1\frac{1}{2}$. Convert to improper fractions: $2\frac{1}{3} = \frac{7}{3}$ and $1\frac{1}{2} = \frac{3}{2}$. Multiply: $\frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}$. Dividing Fractions Division by a fraction is equivalent to multiplication by its reciprocal. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$—simply flip the numerator and denominator. Key rule: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$. Dividing by a whole number When dividing a fraction by a whole number, you have two equivalent approaches: Divide the numerator by the whole number (if it divides evenly): $\frac{8}{9} \div 2 = \frac{4}{9}$ Multiply the denominator by the whole number: $\frac{8}{9} \div 2 = \frac{8}{18} = \frac{4}{9}$ Converting Between Fractions and Decimals Fraction to decimal To convert a fraction to a decimal, perform long division: divide the numerator by the denominator. The result is the decimal representation. Example: $\frac{3}{4} = 3 \div 4 = 0.75$ Terminating decimal to fraction A terminating decimal has a finite number of decimal places. To convert it to a fraction: Remove the decimal point to get the numerator Use $10^n$ as the denominator, where $n$ is the number of decimal places Example: Convert 0.35 to a fraction. There are 2 decimal places, so: $0.35 = \frac{35}{100} = \frac{7}{20}$ (after reducing). Repeating decimals to fractions A repeating decimal has one or more digits that repeat infinitely. There's an elegant pattern for converting these. For a purely repeating decimal (the repetition starts right after the decimal point), place the repeating block of digits over that many nines: $$0.\overline{37} = \frac{37}{99}$$ For mixed repeating decimals (some non-repeating digits come first), use this formula: subtract the non-repeating part from the entire number (repeating part included), then divide by the appropriate denominator. Example: Convert $0.12\overline{3}$ to a fraction. The non-repeating part is 12 (2 digits), and the repeating part is 3 (1 digit). Use: $$0.12\overline{3} = \frac{123 - 12}{990} = \frac{111}{990} = \frac{37}{330}$$ The denominator has 2 nines (for the repeating digit) and 1 zero (for the non-repeating digit). <extrainfo> Algebraic approach to repeating decimals An alternative method uses algebra. Let $x$ equal the repeating decimal, multiply by an appropriate power of 10 to shift the decimal point, subtract the original equation, and solve for $x$. Example: Find the fraction for $x = 0.3\overline{45}$ (where 45 repeats). $$x = 0.34545...$$ $$100x = 34.545...$$ $$10000x = 3454.545...$$ Subtracting the first from the third equation: $$9900x = 3420$$ $$x = \frac{3420}{9900} = \frac{19}{55}$$ </extrainfo> Complex Fractions A complex fraction (also called a compound fraction) is a fraction that contains another fraction in its numerator, denominator, or both. They look complicated but are simplified using standard fraction operations. Example: $\frac{\frac{2}{3}}{\frac{5}{7}}$ To simplify, treat the main fraction bar as a division operation: $$\frac{\frac{2}{3}}{\frac{5}{7}} = \frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} = \frac{14}{15}$$ The key is recognizing that a complex fraction is simply division in disguise, so apply the division rule for fractions.