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Egyptian Fraction Calculator with Step-by-Step Greedy Expansion

Turn any proper fraction into a sum of distinct unit fractions with the Fibonacci-Sylvester greedy algorithm, with the steps shown, browser-only

  • Runs locally
  • Category Calculator
  • Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Ancient Egyptians wrote every fraction as a sum of distinct unit fractions (numerator 1). The greedy algorithm reproduces that here.

Try:
5/6 = Egyptian fraction
1/2 + 1/3
2 unit fractions

Greedy steps

Each step takes the largest unit fraction that fits, then continues on the remainder.

  • 5/6→ take1/2, leaves1/3
  • 1/3→ take1/3, leaves0

What this tool does

A free Egyptian fraction calculator that rewrites a fraction as a sum of distinct unit fractions, the way Egyptian scribes did 4000 years ago. A unit fraction has 1 on top, so 5/6 becomes 1/2 + 1/3 and 4/5 becomes 1/2 + 1/4 + 1/20. Type a numerator and a denominator and the tool runs the Fibonacci-Sylvester greedy method: at each step it takes the largest unit fraction that still fits, subtracts it, and repeats on what is left until nothing remains. Every step is laid out so you can see why the decomposition works, not just the final answer. Improper fractions are handled by splitting off the whole part first, the fraction is reduced before the expansion, and a term cap guards against pathological inputs. One click copies the result as plain text such as 1/2 + 1/3, and the numerator and denominator live in the URL so a shared link reopens the same problem. Everything runs locally in your browser with no upload and no server round trip.

Tool details

Input
Files + Numbers
The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output
Live result + Copy
The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy
Browser-side processing
The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share
Shareable URL state
Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget
Initial JS <= 9 KB
No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit
Calculator · Student
Category and role tags drive related tools, internal links, and quick fit checks.

How to use

  1. 1. Input

    Paste or drop your content into the tool panel.

  2. 2. Process

    Click the button. All processing is local in your browser.

  3. 3. Copy / Download

    Copy the result or download to disk in one click.

How Egyptian Fraction Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

  • Getting a realistic range before a purchase, plan, workout, or schedule decision.
  • Comparing scenarios by changing one input at a time.
  • Turning rough assumptions into a number you can discuss.

Calculation checks

  • Double-check units, dates, rates, and rounding assumptions.
  • Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
  • Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

  1. 1 Fraction Calculator Add, subtract, multiply, divide fractions — auto-reduced, with steps, mixed ⇄ improper ⇄ decimal — browser-only Open
  2. 2 GCD & LCM Calculator GCD + LCM of any list of integers — Euclidean steps, prime factorization, prime-factor table — exact BigInt math, browser-only Open
  3. 3 Decimal to Fraction Converter Turn 0.75 into 3/4, type 0.(3) to get 1/3, or flip a fraction back to its decimal with the repeating block marked Open

Real-world use cases

  • Teach the Fibonacci-Sylvester algorithm in class

    A teacher introducing Egyptian fractions can put 5/6 on the board, show that 1/2 is the biggest unit fraction that fits, and let the tool reveal the remaining 1/3 step by step. Students see the greedy choice and the subtraction at every stage instead of memorizing a final answer. Change the numbers live and the whole class watches the expansion update, which makes the why obvious rather than abstract.

  • Check homework on unit fraction decomposition

    When a worksheet asks for the Egyptian fraction form of 4/5, a student can compare their work against 1/2 + 1/4 + 1/20 and trace where they went wrong. Because the tool reduces the fraction first and shows each greedy step, it is easy to spot a missed reduction or a wrong ceiling. The copy button makes it painless to paste the checked answer back into a digital assignment.

  • Explore the history of ancient Egyptian mathematics

    A history-of-math enthusiast reading about the Rhind Papyrus can try the same fractions the scribes recorded and see how the greedy method compares with their 2/n table entries. Punching in values like 2/7 or 7/15 turns a museum-case curiosity into something you can experiment with, and the step view connects the modern algorithm to the doubling arithmetic the Egyptians actually used.

  • Generate examples for a math blog post or video

    A content creator writing about number theory needs clean, verifiable examples. Enter a fraction, copy the expansion as text, and drop 3/7 = 1/3 + 1/11 + 1/231 straight into a script or caption. The shareable URL lets a reader open the exact same decomposition, so your audience can replay the result instead of trusting a screenshot.

  • Sanity-check a coding solution to the greedy problem

    A programmer implementing the greedy Egyptian fraction algorithm for a coding challenge can diff their output against this tool across many inputs. Edge cases like an already-unit fraction, an improper fraction, or a value that reduces first are exactly where bugs hide, and seeing the reference expansion plus each step makes it quick to confirm the implementation matches.

Common pitfalls

  • Expecting the greedy answer to be the shortest. The greedy method always works but is not always minimal. Take 4/5, which greedily gives 1/2 + 1/4 + 1/20, while 1/2 + 1/5 + 1/10 also works with smaller denominators. The tool reports the greedy expansion on purpose because it is deterministic.

  • Forgetting to reduce the fraction first. 10/12 and 5/6 are the same value, so they have the same Egyptian fraction 1/2 + 1/3. If you decompose the unreduced form by hand you can get a different and longer answer; this tool reduces before expanding to avoid that.

  • Treating an improper fraction as if it had no whole part. 7/3 is not a single unit-fraction sum below 1, it is 2 + 1/3. The classical problem only covers the part under 1, so the whole number 2 is shown separately rather than absorbed into the units.

Privacy

Every part of this calculator runs as plain JavaScript inside your browser tab: the greedy decomposition, the reduction, the step table and the formatted output. Nothing you type is uploaded, logged or sent to a server. The one thing worth knowing is that the numerator and denominator are encoded in the page URL so a shared link can reopen the same problem, which means a link you paste into chat will record those two numbers in the recipient server access log. For math problems that is almost never sensitive, but if it matters use the copy button and paste the text instead of sharing the URL.

FAQ

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Made by Toolora · 100% client-side · Updated 2026-05-30