--- title: "Recurring Decimals as Exact Fractions" description: "A decimal that repeats forever equals an exact fraction. Multiplying by a power of ten and then subtracting lets you find it. This gives you a way to use the claim about rational numbers and repeating" canonical: https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd source: https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Recurring Decimals as Exact Fractions A decimal that repeats forever equals an exact fraction. Multiplying by a power of ten and then subtracting lets you find it. This gives you a way to use the claim about rational numbers and repeating decimals instead of only accepting it. Subject: Mathematics · Area: Number Representation & Place Value · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd ## Ready when they can - Converts a recurring decimal such as 0.4545... to a fraction by subtracting a shifted copy of it - Handles a decimal with a non-recurring part before the repeat, such as 0.16333... - Predicts whether a fraction gives a terminating or non-terminating recurring decimal by reducing it to lowest terms and checking the denominator's prime factors ## Lesson: Endless decimals, exact fractions You have seen that rational numbers have decimals that stop or repeat. Each rational number equals a fraction of two integers with a nonzero denominator. A terminating decimal stops. A recurring decimal has a block that repeats forever. Here, 0.1666... means that only the 6 repeats forever. Without dots or a stated repeat, 0.1666 is a finite decimal. Reduce a fraction to lowest terms before testing its denominator. A positive denominator with only 2 and 5 as prime factors gives a terminating decimal. Any other prime factor gives a non-terminating recurring decimal. A denominator of 1 gives an integer. For example, 14/35 reduces to 2/5, so it terminates even though 35 contains a factor of 7. **Example.** To convert a terminating fraction to a decimal, make an equivalent fraction with denominator 10, 100, 1000 or another power of ten. Multiply the numerator and denominator by the same number. For 9/40, multiply both by 25: 9/40 = 225/1000 = 0.225. The denominator 1000 tells you to read the numerator as thousandths. Let x stand for the entire recurring decimal. Here, 10x means 10 times x. If the repeat starts just after the point, multiply by 10 for a one-digit block, 100 for two digits, or 1000 for three. For x = 0.444..., 10x = 4.444... . Subtract the original equation from the shifted equation. The equal decimal tails cancel, leaving 10x - x = 4. Since 10x - x = 9x, divide both sides by 9: x = 4/9. Reduce the fraction if needed. *(drawing: The block 72 has two digits. Multiply by 100, subtract x, then divide by 99 and reduce.)* If digits come before the repeat, first move those digits left of the point. Count any zeros too. Then shift one full repeating block further. Subtract these two equations, whose tails now match. For x = 0.23555..., only 5 repeats. Use 100x = 23.555... and 1000x = 235.555... . Subtract: 900x = 212. Divide by 900 and reduce: x = 212/900 = 53/225. The dots name the entire infinite decimal. They do not describe a finite calculation that eventually stops. Subtracting equal tails gives an exact equation, so the resulting fraction is exact. A finite decimal and an endless repeat can share starting digits without being equal. For example, 0.312 is less than 0.312312..., where 312 repeats forever. **Recap.** Name it x, shift by the repeat, subtract the tail away, and simplify what remains. *Lesson, questions and flashcards written for this dataset.* ## Practice 15 questions on this page, each with its working shown. ## Needs first - [Number Sets & Infinity](https://lightmysky.com/learn/mathematics/number-sets-and-infinity-mt_b4lbTOJYwI) - [Decimals and fractions (age 11+)](https://lightmysky.com/learn/mathematics/decimals-and-fractions-age-11-mt_SBkTGjiZjZ)