utility calculator

Fraction, Percentage, and Decimal Equivalency Calculator

Convert between fractions, percentages, and decimals instantly, with clear step-by-step proofs shown for repeating decimals too.

Formula Verified by Dr. Arthur Pendelton, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Inputs

Input Value Type

Input Value

Results

Fraction Equivalency

3/4

Decimal Equivalency0.75

Percentage Equivalency75%

Step-by-Step Conversion Proof

Input is fraction: \(\frac{3}{4}\).

Tip

Save this page to quickly convert values, solve math homework, and verify fraction-percentage-decimal equivalencies instantly.

Formula

Decimal-Fraction Equivalency Formulas

Fraction to Decimal = Numerator ÷ Denominator 
Decimal to Fraction = Decimal × 10^n / 10^n 
Repeating Decimal Proof: 10^n × x - x = Integer Difference

Leverages GCD greatest common divisor algorithms to simplify fractions, and solves repeating decimals via linear algebra equations.

Step by step

How the calculation works

1Select your input type: Fraction (e.g. 5/8), Decimal (e.g. 0.125), or Percentage (e.g. 75%).
2Enter the numerical value to convert.
3If entering a repeating decimal, check the Repeating option and provide the repeating digit sequence.
4Read the equivalent representations and review the step-by-step algebraic proof.

Related guides

Learn more about this calculation

Guide

Fraction Operations Guide: Add, Subtract, Multiply, Divide

Learn the rules for adding, subtracting, multiplying, and dividing fractions, with simplification steps and live DTC calculator workflows.

Site Guide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Learn the fastest reliable workflow for using Do The Calculation calculators, reading results, checking formulas, and using save, print, share, and export actions correctly.

How to understand and use this calculator

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... ⇒ 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 × 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: gcd(a, b) = 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_(simple) = a / gcd(a, b) and b_(simple) = b / 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_(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 to see step-by-step long division workouts with remainders, or use the 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 and Grade Calculator. For other everyday conversions, try the Unit Converter.

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.

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%).

How the calculation works

1Select your input type: Fraction (e.g. 5/8), Decimal (e.g. 0.125), or Percentage (e.g. 75%).

2Enter the numerical value to convert.

3If entering a repeating decimal, check the Repeating option and provide the repeating digit sequence.

4Read the equivalent representations and review the step-by-step algebraic proof.

How to understand and use this calculator

The Algebraic Proof for Converting Repeating Decimals to Fractions

10x = 3.333...

Decimal Equivalency: Terminating vs. Repeating Fractional Series

Simplifying Fractions: Greatest Common Divisor (GCD) Reduction

Percentages in Financial and Scientific Calculations: Standardizing Ratios

How to Use This Equivalency Calculator

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.

Related Calculators

This calculator is part of an Academic Tools cluster. For grade-related percentage math, see the GPA Calculator and Grade Calculator. For other everyday conversions, try the Unit Converter.

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.

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%).

Fraction, Percentage, and Decimal Equivalency Calculator

Inputs

Results

Step-by-Step Conversion Proof

Decimal-Fraction Equivalency Formulas

How the calculation works

Learn more about this calculation

Fraction Operations Guide: Add, Subtract, Multiply, Divide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Fraction, Percentage, and Decimal Equivalency Calculator

Inputs

Results

Step-by-Step Conversion Proof

Key output metrics

Fraction as Part of Whole

How to understand and use this calculator

The Algebraic Proof for Converting Repeating Decimals to Fractions

Decimal Equivalency: Terminating vs. Repeating Fractional Series

Simplifying Fractions: Greatest Common Divisor (GCD) Reduction

Percentages in Financial and Scientific Calculations: Standardizing Ratios

How to Use This Equivalency Calculator

Worked Example: Converting 3/4 and the Repeating Decimal 0.333...

Related Calculators

Related concepts

Frequently Asked Questions

Decimal-Fraction Equivalency Formulas

How the calculation works

Learn more about this calculation

Fraction Operations Guide: Add, Subtract, Multiply, Divide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Key output metrics

Fraction as Part of Whole

How to understand and use this calculator

The Algebraic Proof for Converting Repeating Decimals to Fractions

Decimal Equivalency: Terminating vs. Repeating Fractional Series

Simplifying Fractions: Greatest Common Divisor (GCD) Reduction

Percentages in Financial and Scientific Calculations: Standardizing Ratios

How to Use This Equivalency Calculator

Worked Example: Converting 3/4 and the Repeating Decimal 0.333...

Related Calculators

Related concepts

Frequently Asked Questions