Converting Decimals to Fractions

Every terminating or repeating decimal can be converted back to a ratio of two integers p/q. In this quick tutorial, we will show you how to do both.

Part 1: Terminating Decimals (The Easy Way)

To convert a decimal that ends cleanly:

  1. Count the number of decimal places (let this be d).
  2. Place the decimal digits over a denominator of 10 to the power of d.
  3. Simplify the fraction by dividing the numerator and denominator by their Greatest Common Divisor (GCD).

Example: Convert 0.375 to a fraction

  • There are 3 decimal places, so the denominator is 10^3 = 1000.
  • Write as a fraction: 375 / 1000
  • Find the GCD of 375 and 1000, which is 125.
  • Divide both parts by 125: (375 / 125) / (1000 / 125) = 3 / 8

Part 2: Repeating Decimals (The Algebraic Way)

To convert a decimal that repeats forever:

  1. Let x equal the decimal (e.g. x = 0.777…).
  2. Multiply x by 10^k, where k is the number of digits in the repeating sequence (e.g., 10x = 7.777…).
  3. Subtract the original equation (x) from the new equation (10x) to eliminate the repeating decimal part.
  4. Solve for x and reduce to lowest terms.

Example: Convert 0.777… to a fraction

  • x = 0.777…
  • Since 1 digit repeats, multiply by 10^1 = 10: 10x = 7.777…
  • Subtract the first equation from the second: (10x - x) = (7.777… - 0.777…) 9x = 7
  • Solve for x: x = 7 / 9

Try It Yourself!

If you don’t want to calculate this manually, our Rational Number Calculator does the heavy lifting for you instantly, generating full step-by-step conversion steps!