Fraction & Ratio Calculator
Add, subtract, multiply, and divide fractions with automatic simplification, convert between fractions, decimals, and percents, and scale ratios to any term.
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Disclaimer: This free tool is provided “as is,” without warranties of any kind, and is for general informational purposes only — not professional, legal, financial, medical, tax, or engineering advice. Results may contain errors; verify anything important independently and use at your own risk. We accept no liability for any loss or damage arising from its use. See our Terms of Use for details.
Step-by-Step Guide
Choose a mode: Arithmetic for fraction operations, Convert for decimal-fraction-percent conversion, or Scale Ratio.
In Arithmetic mode, enter the numerator and denominator for each fraction. Select the operation (+, −, ×, ÷) and the result is shown as an improper fraction in lowest terms, as a mixed number, and as a decimal. Division by zero is flagged.
In Convert mode, enter either a decimal (e.g. 0.625) to see it as a simplified fraction and percentage, or a fraction to see its decimal and percentage equivalents.
In Scale Ratio mode, enter two ratio terms (e.g. 3:4) and one of the target terms. The calculator finds the unknown term that preserves the ratio. For example, 3:4 scaled so the second term is 12 gives 9:12.
Arithmetic: 1/2 + 1/3 → find a common denominator of 6 → 3/6 + 2/6 = 5/6 (0.8333..., not a whole number so no mixed number form). Convert: 0.75 → 3/4 (75%). Arithmetic: 2/3 × 3/4 → multiply numerators (2×3=6) and denominators (3×4=12) → 6/12 → simplify by GCD 6 → 1/2. Scale: 3:4 with target second term 12 → 3/4 = ?/12 → ? = 9 → result 9:12.
Who it's for
Students, teachers, cooks, woodworkers, and anyone working with fractions or ratios.
Core Features
- Add, subtract, multiply, and divide two fractions with automatic lowest-terms simplification.
- Show the answer as an improper fraction, a mixed number, and a decimal.
- Convert a decimal into a simplified fraction and a percentage.
- Scale a ratio so one term hits a target value (e.g. 3:4 → ?:12).
🛡️ No tracking — your inputs, keys, and details never leave this client sandbox.
How do you add fractions?
To add fractions, convert them to a common denominator (the least common multiple of the two denominators), then add the numerators. For 1/3 + 1/4: the LCM of 3 and 4 is 12. Convert: 4/12 + 3/12 = 7/12. The tool does this automatically and simplifies the result to lowest terms.
How do you multiply and divide fractions?
Multiplication: multiply numerators together and denominators together, then simplify. 2/3 × 3/4 = 6/12 = 1/2. Division: flip the second fraction (find its reciprocal) and multiply. 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9. Division is always easier with fractions than with decimals because no long division is needed — just flip and multiply.
How do you convert a decimal to a fraction?
Count the decimal places. For 0.625 (3 decimal places), write it as 625/1000, then simplify by dividing both by their GCD (125): 5/8. For a repeating decimal like 0.333..., it is exactly 1/3. The converter handles terminating decimals directly; very long repeating decimals may have precision limits due to floating-point arithmetic.
What is lowest terms and how is it computed?
A fraction is in lowest terms (also called simplified or reduced form) when the numerator and denominator share no common factors other than 1. To simplify, divide both by their greatest common divisor (GCD). The GCD can be computed using the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is 0 — the last non-zero remainder is the GCD.
Adding fractions vs. multiplying fractions: two operations with opposite difficulty profiles
Fraction addition and multiplication look like they should have the same complexity. In practice, they are almost reversed. Addition seems simple — you are combining two quantities — but requires finding a common denominator first, which involves computing the LCM and converting both fractions. Multiplication seems more complex — there are more numbers involved — but requires no common denominator at all: just multiply the numerators together and the denominators together, then simplify. Division is even simpler structurally: flip the second fraction and multiply. Understanding why the difficulty is inverted helps you remember the procedures when you need them.
Why addition needs a common denominator and multiplication doesn't
Fractions represent proportions of a whole. Adding 1/3 and 1/4 asks: 'how much of a whole do we have when we combine a third and a quarter?' Those thirds and quarters are different-sized pieces that cannot be counted together without first cutting them into the same size — that is what a common denominator does. Multiplying 1/3 by 1/4 asks a different question: 'what is a third of a quarter?' You are applying one fraction as a proportion to the other, not combining same-sized pieces. No common denominator is needed because you are not adding — you are scaling. Multiplying numerators scales the part; multiplying denominators scales the whole.
The Euclidean algorithm: why simplification is fast
Before displaying a result, the tool reduces fractions to lowest terms using the Euclidean algorithm — one of the oldest algorithms in mathematics and still the most efficient way to find the greatest common divisor. For two numbers a and b, repeatedly replace the larger with the remainder of dividing it by the smaller until the remainder is zero. The last non-zero remainder is the GCD. For 48 and 36: 48 mod 36 = 12; 36 mod 12 = 0 → GCD = 12. So 48/36 = 4/3. This runs in logarithmic time — it is fast even for very large numerators and denominators.
From fractions to ratios: the same math, different notation
A ratio like 3:4 is the same relationship as the fraction 3/4. Proportional scaling — finding the unknown in 3:4 = ?:12 — is the same as solving 3/4 = x/12, giving x = 3 × (12/4) = 9. This appears in recipe scaling (double a recipe by scaling every ingredient ratio), map reading (1:50,000 scale means 1 cm on the map is 50,000 cm in reality), mixing ratios in chemistry, and aspect ratio conversions in design.