CALTRX

Decimal to Fraction

Convert any decimal number into a simplified fraction or mixed number. Shows the step-by-step simplification process for learning.

Enter your values above to see the results.

Tips & Notes

  • ✓Count decimal places: 0.75 has 2 places → denominator is 10² = 100. Then simplify.
  • ✓Repeating single digit: denominator is 9. 0.777...=7/9. 0.444...=4/9.
  • ✓Repeating two digits: denominator is 99. 0.363636...=36/99=4/11.
  • ✓Always simplify using GCF. 75/100: GCF=25 → 3/4. Fraction in lowest terms.
  • ✓Mixed decimal 2.375: convert 0.375=3/8, then add 2 = 2 and 3/8 = 19/8.

Common Mistakes

  • ✗Forgetting to simplify: 75/100 is correct but 3/4 is the simplified form.
  • ✗Wrong denominator power: 0.75 needs 10² (two decimal places), not 10.
  • ✗Repeating decimal: multiply by wrong power. One repeating digit → ×10. Two → ×100.
  • ✗Not subtracting correctly in repeating method: 10x−x=9x, not 10x−x=9 directly.
  • ✗Mixed numbers: convert decimal part only, then add to whole number as fractions.

Decimal to Fraction Overview

A decimal and a fraction are two different notations for the same quantity. Converting between them reveals the exact value behind any approximation and is essential for arithmetic, algebra, and understanding how numbers are stored in computers. This calculator handles terminating decimals, repeating decimals, and mixed numbers, always outputting the fully simplified fraction in lowest terms.

Terminating decimals — those that end after a finite number of digits — convert directly by placing the digits over the appropriate power of 10:

EX: 0.75 → 2 decimal places → 75/100 → GCF(75, 100) = 25 → simplified: 3/4
EX: 0.125 → 3 decimal places → 125/1000 → GCF = 125 → simplified: 1/8
Repeating decimals require an algebraic approach. Set x equal to the repeating decimal, multiply by the power of 10 that shifts exactly one full repeat cycle left, then subtract to cancel the repeating tail:
EX: x = 0.333... → 10x = 3.333... → 10x − x = 3 → 9x = 3 → x = 1/3
EX: x = 0.272727... → 100x = 27.2727... → 99x = 27 → x = 27/99 = 3/11
Converting between decimals and fractions is exact for terminating decimals and algebraically exact for repeating decimals — the fraction captures the full precision that a truncated decimal cannot. Working with fractions throughout a multi-step calculation and converting to decimal only at the final step avoids accumulated rounding error that can distort results in sensitive calculations. Some decimals cannot be expressed as fractions of integers — these are irrational numbers like π, √2, and e, whose decimal representations neither terminate nor repeat. The fraction 22/7 is an approximation of π accurate to 2 decimal places, not an exact representation. For practical applications, knowing whether a decimal is exactly representable as a fraction determines whether exact arithmetic or approximation is appropriate.

Frequently Asked Questions

For a terminating decimal, count the decimal places and use that as the denominator power of 10. 0.75 → 75/100. Simplify by dividing by GCF: GCF(75, 100) = 25 → 3/4. 0.625 → 625/1000. GCF(625, 1000) = 125 → 5/8. 0.3 → 3/10 (already simplified, GCF = 1). Verify by dividing: 3÷4 = 0.75 ✓. The number of decimal places equals the number of zeros in the denominator.

For repeating decimals, use algebra to eliminate the repeating part. Let x = 0.333... Multiply by 10: 10x = 3.333... Subtract: 10x − x = 3.333... − 0.333... → 9x = 3 → x = 3/9 = 1/3. For 0.181818...: let x = 0.181818... Multiply by 100: 100x = 18.1818... Subtract: 99x = 18 → x = 18/99 = 2/11. The multiplier is 10^n where n is the length of the repeating block.

Not every decimal converts to a simple fraction that terminates. Terminating decimals have only 2s and 5s as prime factors in their denominator (in lowest terms). 1/4 = 0.25 (denominator 4 = 2²), 1/8 = 0.125 (denominator 8 = 2³), 1/5 = 0.2. Non-terminating repeating decimals have other prime factors: 1/3 = 0.333..., 1/7 = 0.142857142857..., 1/6 = 0.1666... Irrational numbers like π and √2 are non-terminating and non-repeating — they cannot be expressed as fractions at all.

A mixed number has a whole number part and a proper fraction part. 2.75 → whole part = 2, decimal part = 0.75 = 3/4 → mixed number = 2 and 3/4 (written 2¾). To convert to an improper fraction: 2×4 + 3 = 11 → 11/4. To convert back: 11÷4 = 2 remainder 3 → 2 and 3/4. Mixed numbers are more readable for everyday use; improper fractions are easier for arithmetic operations like multiplication and division.

Engineering and construction use fractional inches extensively: 1.375 inches → 375/1000 → 3/8 (after dividing by GCF 125). Woodworking measurements: 0.0625 inches = 1/16 inch (the smallest common graduation on most tape measures). 0.03125 inches = 1/32 inch. Recipes: 0.25 cups = 1/4 cup, 0.333 cups ≈ 1/3 cup. Financial: $0.25 = 1/4 dollar (quarter), $0.125 = 1/8 dollar. Converting decimals to fractions makes these quantities match the marked intervals on physical measuring tools.

After converting, always verify by dividing the fraction: numerator ÷ denominator should return the original decimal. 7/8: 7 ÷ 8 = 0.875 ✓. Also verify the fraction is in lowest terms by confirming GCF(numerator, denominator) = 1. GCF(7, 8) = 1 ✓ (7 is prime, 8 = 2³, no common factors). If GCF > 1, divide both parts again. Example: converting 0.6 → 6/10 → GCF(6,10)=2 → 3/5. GCF(3,5)=1 ✓. Final check: 3÷5 = 0.6 ✓.