Understanding Fractions and Decimals - Complete Guide

Learn how fractions and decimals work, how to convert between them, and how to perform arithmetic operations. Includes simplification, comparison, and practical tips.

What Is a Fraction?

A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number), separated by a fraction bar. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have. For example, 3/4 means three out of four equal parts. Fractions can also represent division: 3/4 is equivalent to 3 divided by 4, which equals 0.75. Fractions are used everywhere from cooking (half a cup) to construction (three-eighths of an inch) to probability (one in six chance).

Types of Fractions

Fractions come in several varieties. A proper fraction has a numerator smaller than its denominator (like 2/5), meaning it represents less than one whole. An improper fraction has a numerator greater than or equal to its denominator (like 7/3), meaning it represents one or more wholes. A mixed number combines a whole number with a proper fraction (like 2 1/3). To convert an improper fraction to a mixed number, divide the numerator by the denominator: the quotient is the whole part, the remainder is the new numerator, and the denominator stays the same. So 7/3 = 2 remainder 1, which is 2 1/3.

Simplifying Fractions

A fraction is in simplest form (or lowest terms) when the numerator and denominator have no common factor other than 1. To simplify a fraction, find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For example, 12/18 has a GCF of 6, so 12/18 simplifies to 2/3. You can find the GCF by listing factors, using prime factorization, or applying the Euclidean algorithm. Simplifying fractions makes them easier to work with and easier to compare. Always present final answers in simplest form unless the context requires a specific denominator.

Adding and Subtracting Fractions

To add or subtract fractions, they must have the same denominator (a common denominator). If they already do, simply add or subtract the numerators and keep the denominator. For example, 2/7 + 3/7 = 5/7. If the denominators differ, find the least common denominator (LCD), convert each fraction, then operate. For example, 1/3 + 1/4 requires LCD = 12: convert to 4/12 + 3/12 = 7/12. For mixed numbers, you can either convert to improper fractions first or handle the whole-number and fractional parts separately (being careful with borrowing when subtracting).

Multiplying and Dividing Fractions

Multiplying fractions is simpler than adding them: multiply the numerators together and multiply the denominators together. For example, 2/3 x 4/5 = 8/15. You can simplify before multiplying by cross-canceling common factors. To divide fractions, multiply by the reciprocal of the divisor. For example, 2/3 divided by 4/5 equals 2/3 x 5/4 = 10/12 = 5/6. The phrase "invert and multiply" summarizes this rule. These operations extend naturally to mixed numbers: convert to improper fractions first, then multiply or divide as usual.

Converting Fractions to Decimals

To convert a fraction to a decimal, simply divide the numerator by the denominator. For example, 3/8 = 3 divided by 8 = 0.375. Some fractions produce terminating decimals (like 1/4 = 0.25), while others produce repeating decimals (like 1/3 = 0.333...). A fraction in lowest terms produces a terminating decimal if and only if the denominator has no prime factors other than 2 and 5. Understanding this distinction helps you anticipate whether a conversion will be exact or approximate. For repeating decimals, a bar notation is used: 0.333... is written as 0.3 with a bar over the 3.

Converting Decimals to Fractions

To convert a terminating decimal to a fraction, write the digits after the decimal point as the numerator, and the appropriate power of 10 as the denominator, then simplify. For example, 0.625 = 625/1000 = 5/8 after dividing both by 125. For repeating decimals, use an algebraic method: let x = 0.666..., then 10x = 6.666..., so 10x - x = 6, giving 9x = 6, and x = 6/9 = 2/3. This technique works for any repeating pattern, though longer repeating blocks require multiplying by larger powers of 10. Being comfortable with both directions of conversion is essential for flexible mathematical work.

Comparing Fractions and Decimals

To compare fractions, you can convert them to a common denominator or to decimals. For example, which is larger: 3/7 or 5/12? Converting to decimals gives approximately 0.4286 and 0.4167, so 3/7 is larger. Alternatively, cross-multiply: 3 x 12 = 36 and 5 x 7 = 35; since 36 > 35, 3/7 > 5/12. When comparing decimals, align the decimal points and compare digit by digit from left to right. A useful benchmark strategy is to compare fractions to 1/2: since 3/7 > 1/2 (because 3 x 2 > 7) and 5/12 < 1/2 (because 5 x 2 < 12), 3/7 is the larger fraction without further calculation.

Try These Calculators

Put what you learned into practice with these free calculators.

Fraction Calculator Fraction to Decimal Calculator Decimal to Fraction Calculator Greatest Common Factor Calculator