Number to Fraction Converter

About Number to Fraction Converter

The Number to Fraction Converter transforms any decimal number, repeating decimal, mixed number, percentage, or unsimplified fraction into its simplest fractional form. It shows a complete step-by-step breakdown of how the conversion works, visual fraction bars, and a long division verification that proves the result is correct.

How to Use the Number to Fraction Converter

1. Enter your number in the input field. You can type a decimal like 0.75, a repeating decimal like 0.(3), a mixed number like 2 3/4, a percentage like 75%, or a fraction like 6/8.
2. Click "Convert" or press Enter to perform the conversion.
3. View the simplified fraction displayed prominently with the decimal equivalent and mixed number form (if applicable).
4. Study the step-by-step solution to understand the math behind the conversion, including GCD simplification.
5. Check the verification: an animated long division converts the fraction back to a decimal, proving the result matches your input.

Converting Terminating Decimals to Fractions

A terminating decimal is one that ends after a finite number of digits, like 0.75 or 3.14. To convert:

1. Count the decimal places (0.75 has 2).
2. Write the number over \(10^n\) where \(n\) is the number of decimal places: \(\frac{75}{100}\).
3. Simplify by dividing both by the GCD: \(\gcd(75, 100) = 25\), so \(\frac{75}{100} = \frac{3}{4}\).

Converting Repeating Decimals to Fractions

A repeating decimal has a block of digits that repeats forever, like \(0.333... = 0.\overline{3}\). The algebraic method works like this:

1. Let \(x = 0.\overline{3}\)
2. Multiply both sides by 10: \(10x = 3.\overline{3}\)
3. Subtract: \(10x - x = 3\), so \(9x = 3\)
4. Solve: \(x = \frac{3}{9} = \frac{1}{3}\)

For decimals with both non-repeating and repeating parts like \(0.1\overline{6}\), use two multiplications to align the repeating blocks before subtracting.

Repeating Decimal Notation

Use parentheses to indicate the repeating block:

• 0.(3) means 0.333... (1/3)
• 0.(6) means 0.666... (2/3)
• 0.1(6) means 0.1666... (1/6)
• 0.(142857) means 0.142857142857... (1/7)
• 0.(09) means 0.090909... (1/11)

Common Decimal to Fraction Conversions

• 0.5 = 1/2
• 0.25 = 1/4
• 0.75 = 3/4
• 0.125 = 1/8
• 0.2 = 1/5
• 0.(3) = 1/3
• 0.(6) = 2/3
• 0.1(6) = 1/6
• 0.(142857) = 1/7

Why GCD (Greatest Common Divisor) Matters

The GCD of two numbers is the largest number that divides both evenly. When converting decimals to fractions, we divide both the numerator and denominator by their GCD to get the simplest form. For example, \(\frac{75}{100}\): \(\gcd(75, 100) = 25\), so \(\frac{75 \div 25}{100 \div 25} = \frac{3}{4}\).

FAQ

How do I convert a decimal to a fraction?

Write the decimal over 1, multiply both numerator and denominator by 10 for each decimal place, then simplify by dividing both by their greatest common divisor (GCD). For example, 0.75 = 75/100 = 3/4 after dividing by GCD of 25.

How do I convert a repeating decimal to a fraction?

Set the repeating decimal equal to x. Multiply x by a power of 10 to shift the repeating block, then subtract to eliminate the repeating part. For example, x = 0.333... so 10x = 3.333..., subtract to get 9x = 3, so x = 3/9 = 1/3.

What is the fraction form of 0.333...?

The repeating decimal 0.333... equals exactly 1/3. This can be proven by letting x = 0.333..., multiplying by 10 to get 10x = 3.333..., and subtracting: 9x = 3, so x = 1/3.

How do I enter a repeating decimal?

Use parentheses around the repeating digits. For example, type 0.(3) for 0.333..., 0.1(6) for 0.1666..., or 1.2(345) for 1.2345345345...

Can every decimal be written as a fraction?

Every terminating decimal and every repeating decimal can be written as a fraction of two integers. However, irrational numbers like pi (3.14159...) and the square root of 2 cannot be expressed as exact fractions because their decimal expansions never terminate or repeat.

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