Simplifying a fraction with decimals
Move the decimal point the same number of places on the top and on the bottom, then reduce as normal. For 2.5 over 0.75 the longer of the two has two decimal places, so both are multiplied by a hundred to give 250/75. The greatest common divisor of those is 25, so the fraction is 10/3, or 3 and 1/3. Multiplying the top and the bottom by the same number never changes what a fraction is worth, which is why the step is allowed at all.
Euclid's algorithm, step by step
| 1 | 250 | 75 | 3 | 25 |
| 2 | 75 | 25 | 3 | 0 |
Results are provided as-is, with no warranty of accuracy. The method and its sources are published below so you can check the working.
Both numbers must be scaled by the same power of ten. Scaling only the top changes the value rather than the way it is written.
Use the larger of the two decimal place counts. Two point five over nought point seven five needs a hundred, not a ten, or the bottom stays fractional.
A recurring value cannot be typed in exactly. 0.333 is three hundred and thirty-three thousandths, and it reduces to 333/1000 rather than to 1/3.
How it works
How it is done
- Read each box as the exact decimal that was typed rather than as the binary number a browser stores it in, and count how many decimal places the longer of the two has.
- Multiply both the top and the bottom by ten to the power of that count. Scaling both halves by the same amount leaves the value of the fraction untouched while turning both into whole numbers.
- Run Euclid's algorithm on the two whole numbers with their signs removed — divide the larger by the smaller, then the smaller by the remainder, and keep going until a remainder of zero. The last divisor used is the greatest common divisor.
- Divide the top and the bottom by that divisor. What is left is the fraction in its lowest terms, with any minus sign moved onto the numerator so that -3/4 and 3/-4 come out the same way.
- Divide the numerator by the denominator once more for the mixed number — the quotient is the whole part and the remainder sits over the same denominator.
- Produce the decimal by long division, recording every remainder as it appears. A remainder of zero ends the expansion. A remainder that has already been seen means the digits from that point repeat forever, and they are shown in brackets.
What it assumes
- A minus sign always ends up on the numerator, so -3/4 and 3/-4 both come out as -3/4. They are the same number, and one canonical form is what makes two answers comparable.
- A fraction that divides exactly is written as a plain whole number rather than as something over one, so 10/5 comes out as 2 and never as 2/1.
- Decimal inputs are multiplied by a power of ten before anything else happens, so 2.5/0.75 is treated as 250/75. The answer is exact, but a value with many decimal places produces very large lowest terms.
- The repeating block of a decimal is only bracketed if it both starts and finishes within the number of places set under More options. Beyond that the expansion is cut off with an ellipsis rather than guessed at.
- Whole numbers above 9,007,199,254,740,991 cannot be shown in the numeric boxes without losing digits, so those boxes show a dash. The fraction itself is text and stays exact whatever its size.
- The rounded decimal and the percentage are rounded, to eight and four places. A value smaller than that rounds to zero in those two boxes while the exact form beside them is unchanged.
Common questions
Why does 0.1/0.3 not come out as exactly 1/3 in some tools?
Because 0.1 and 0.3 have no exact binary representation, so a tool working in floating point gets 0.33333333333333337 and cannot see the 1/3. Scaling both parts by the same power of ten first turns it into 1/3, which keeps the arithmetic in whole numbers.
How many decimal places can I use?
As many as you like. Both numbers are scaled by the same power of ten before anything is divided, so 0.125/0.375 becomes 125/375 and then 1/3. Long decimals produce large intermediate numbers but no loss of accuracy, because nothing is rounded on the way.
Sources
Method written and checked by Tessalor on Jul 30, 2026.
The full method, worked example and every assumption behind this figure are on Fraction Simplifier.