# Fraction, Percentage, and Decimal Equivalency Calculator Convert between fractions, percentages, and decimals instantly, with clear step-by-step proofs shown for repeating decimals too. --- - **Canonical URL:** https://dothecalculation.com/calculators/fraction-percentage-decimal-calculator - **Category:** Everyday utilities - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology - **Reviewed by:** Sheharyar Shahid, Data Analyst & Advanced Excel Specialist (DAE) (https://dothecalculation.com/about/team/sheharyar-shahid) --- ## Fraction, Percentage & Decimal Equivalency Calculator Convert between fractions, percentages, and decimals, and view step-by-step algebraic proofs for repeating decimals. - Simplifies fractions to their lowest terms using GCD reduction - Converts repeating decimals to fractions using algebraic proofs - Generates visual division proofs and decimal scale factor steps ## The Algebraic Proof for Converting Repeating Decimals to Fractions Converting a repeating decimal (like \(0.333...\)) to a fraction requires an algebraic proof that isolates the repeating decimal period. Let \(x\) equal the repeating decimal: $$x = 0.333...$$. Since the repeating pattern is 1 digit long, we multiply both sides of the equation by $$10^1 = 10$$: $$10x = 3.333...$$ Next, we subtract the original equation from the multiplied equation: $$10x - x = 3.333... - 0.333... \implies 9x = 3$$. Solving for \(x\) gives: $$x = 3/9$$, which simplifies to \(1/3\). This algebraic method works for any repeating pattern length by multiplying by \(10^n\), where \(n\) is the number of repeating digits. ## Decimal Equivalency: Terminating vs. Repeating Fractional Series A fraction is represented as \(a/b\), where \(a\) is the numerator and \(b\) is the denominator. When converted to a decimal, a fraction will either terminate (end) or repeat infinitely. This behavior is determined entirely by the prime factorization of the denominator when the fraction is in its simplest form. If the prime factors of the simplified denominator consist only of 2s and/or 5s, the decimal will terminate. This is because our base-10 number system is based on the factors of 10 ($$2 \times 5$$). If the denominator contains any other prime factors (such as 3, 7, or 11), the resulting decimal division will repeat infinitely. ## Simplifying Fractions: Greatest Common Divisor (GCD) Reduction A fraction is in its simplest form when its numerator and denominator are coprime, meaning they share no common factors other than 1. To simplify a fraction, we calculate the Greatest Common Divisor (GCD) of the numerator and denominator using the Euclidean algorithm. The Euclidean algorithm operates on the principle that the GCD of two numbers also divides their difference: $$\text{gcd}(a, b) = \text{gcd}(b, a \pmod b)$$. Once the GCD is found, we divide both the numerator and the denominator by this divisor to reduce the fraction to its lowest terms: $$a_{\text{simple}} = a / \text{gcd}(a, b)$$ and $$b_{\text{simple}} = b / \text{gcd}(a, b)$$. ## Percentages in Financial and Scientific Calculations: Standardizing Ratios A percentage is simply a fraction with a fixed denominator of 100, representing "per hundred". To convert a percentage to a decimal, we divide by 100, which shifts the decimal point two places to the left: $$P_{\text{decimal}} = P_{\%}/100$$. To convert to a fraction, we place the percentage value over 100 and reduce it to its lowest terms. For advanced division and math operations, you can use our [long division calculator](/calculators/long-division-calculator) to see step-by-step long division workouts with remainders, or use the [decimal to fraction calculator](/calculators/decimal-to-fraction-calculator) to focus exclusively on decimal conversions. ## How to Use This Equivalency Calculator Select your input type — Fraction, Decimal, or Percentage — then enter the value. For a repeating decimal like 0.333..., check the "Is this a repeating decimal?" box and enter the repeating digit block (e.g. "3"). The calculator returns all three equivalent representations at once, plus a step-by-step algebraic proof showing exactly how the conversion and simplification were performed. ## Worked Example: Converting 3/4 and the Repeating Decimal 0.333... The default input, the fraction 3/4, converts directly to the decimal 0.75 and the percentage 75%, with no simplification needed since 3 and 4 share no common factors. For the repeating decimal 0.333... (pattern "3"), the algebraic proof sets x = 0.333..., multiplies by 10 to get 10x = 3.333..., then subtracts to get 9x = 3, giving x = 3/9, which simplifies via GCD (3) to the fraction 1/3 — confirmed as 0.(3) and 33.3333%. ## Related Calculators This calculator is part of an Academic Tools cluster. For grade-related percentage math, see the [GPA Calculator](/calculators/gpa-calculator) and [Grade Calculator](/calculators/grade-calculator). For other everyday conversions, try the [Unit Converter](/calculators/unit-converter). ## Frequently asked questions ### What is a repeating decimal? A repeating decimal is a decimal representation of a number whose digits repeat periodically at regular intervals, such as 0.333... or 0.142857... ### How do you convert a fraction to a decimal? Divide the numerator by the denominator using long division. If the division ends, it is a terminating decimal; if it repeats, it is a repeating decimal. ### Why do some fractions result in repeating decimals? If the denominator of a simplified fraction contains prime factors other than 2 or 5, it cannot be divided evenly into a base-10 number, resulting in a repeating decimal. ### What is the Euclidean algorithm for GCD? The Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers by repeatedly taking the remainder of their division. ### How do you convert a percentage to a fraction? Place the percentage value over 100 (e.g., 45% = 45/100) and then simplify the fraction by dividing the numerator and denominator by their greatest common divisor (e.g., 45/100 = 9/20). ### What is a terminating decimal? A terminating decimal is a decimal number that has a finite number of digits after the decimal point, such as 0.25 or 0.875. ### Is 0.999... equal to 1? Yes, mathematically 0.999... is exactly equal to 1. This can be proven algebraically: let x = 0.999..., then 10x = 9.999...; subtracting the equations yields 9x = 9, so x = 1. ### How do you simplify a fraction? Find the greatest common divisor (GCD) of the numerator and the denominator, and then divide both the numerator and the denominator by that number. ### What is a repeating pattern block? The repeating pattern block (or repetend) is the sequence of digits that repeat infinitely in a repeating decimal, often written with a bar over it or in parentheses. ### How do you convert a decimal to a percentage? Multiply the decimal number by 100 and append a percentage sign (e.g., 0.125 * 100 = 12.5%). ## Related concepts - **Euclidean Algorithm** — An algorithmic method for calculating the greatest common divisor of two integers. - **Repeating Decimal (Repetend)** — A decimal number containing a repeating sequence of digits that extends infinitely. - **Prime Factorization** — The process of factoring a composite number into a product of prime numbers. ## Related guides - [Fraction Operations Guide: Add, Subtract, Multiply, Divide](https://dothecalculation.com/blog/math/fraction-operations-guide) — Learn the rules for adding, subtracting, multiplying, and dividing fractions, with simplification steps and live DTC calculator workflows. ## Related calculators - [Fraction to Decimal Calculator](https://dothecalculation.com/calculators/fraction-to-decimal-calculator) — Convert fractions to decimals with step-by-step long division and repeating decimal bar notation shown clearly and instantly. - [Decimal to Fraction Calculator](https://dothecalculation.com/calculators/decimal-to-fraction-calculator) — Use our free decimal to fraction calculator to convert any decimal to a fraction. Shows step-by-step work for terminating and repeating decimals. - [Mixed Number Fraction Calculator](https://dothecalculation.com/calculators/mixed-number-calculator) — Add, subtract, multiply, divide, and simplify mixed number fractions with clear step-by-step conversions and instant results. - [Circle Equation & Geometry Solver](https://dothecalculation.com/calculators/circle-solver) — Convert between standard and general circle equation forms, find centers and radii, and compute areas with this free geometry solver. - [Extended Euclidean GCD Solver](https://dothecalculation.com/calculators/euclidean-algorithm-calculator) — Compute the GCD of two numbers and find Bezout coefficients using the Extended Euclidean algorithm with clear step-by-step results. - [Modular Arithmetic & Congruence Solver](https://dothecalculation.com/calculators/modular-arithmetic-calculator) — Perform modular addition, multiplication, and exponentiation, and solve linear modular congruence equations with clear steps. --- _This calculator is for general educational and reference purposes only. Results are estimates and should not be used as the sole basis for critical decisions._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/fraction-percentage-decimal-calculator). Quote freely with attribution and a link to this page._