What this calculator does
A lowest common denominator, or LCD, is the smallest number that every denominator in a set of fractions divides into evenly. It is the shared base you need before fractions with different denominators can be added, subtracted or lined up for comparison, since none of those operations work while the parts are sized differently.
The thing people get wrong is multiplying all the denominators together and stopping there. That always gives a common denominator, but usually not the lowest one, which leaves the fractions carrying larger numbers than necessary and the final answer needing extra simplifying. The LCD is the lowest common multiple of the denominators specifically, not just any shared multiple.
The formula
The denominators are combined pairwise using the lowest common multiple: for two numbers, that is their product divided by their greatest common divisor. Each fraction is then rewritten by multiplying its numerator by whatever factor scales its original denominator up to the LCD, which keeps the fraction's value unchanged.
| Term | Meaning |
|---|---|
| LCD | Lowest common denominator: the smallest number every denominator in the set divides into evenly. |
| GCD | Greatest common divisor, used to strip the LCM formula down to its smallest form rather than a plain product. |
| Numerator | The top number of a fraction, rescaled to match once the denominator is changed to the LCD. |
The inputs explained
| Field | What to enter |
|---|---|
| Fractions (e.g. 1/2, 2/3, 5/6) | List two or more fractions separated by commas, each written as numerator/denominator, such as 1/2, 2/3, 5/6. |
When to use it
Adding fractions by hand
Before 1/2 and 2/3 can be added, both need to be rewritten over the same denominator; the LCD gives the smallest version of that shared denominator, keeping the working numbers manageable.
Comparing fractions at a glance
It is hard to tell which of 5/12 or 3/8 is larger just by looking. Rewriting both over their LCD turns the comparison into a straightforward comparison of two numerators.
Checking a student's working
A common mistake is using the product of the denominators instead of the LCD. Running the same fractions through this calculator shows the correct minimum denominator to compare against.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is the lowest common denominator for a set of fractions?
The same tool applied to a few frequently seen sets of fractions.
| Fractions | Lowest common denominator | Fractions rewritten |
|---|---|---|
| 1/2, 2/3, 5/6 | 6 | 3/6, 4/6, 5/6 |
| 1/4, 1/6 | 12 | 3/12, 2/12 |
| 3/8, 5/12, 1/6 | 24 | 9/24, 10/24, 4/24 |
| 1/3, 1/4, 1/5 | 60 | 20/60, 15/60, 12/60 |
Questions
Is the LCD the same as the LCM?
Yes, when it is applied to the denominators specifically. The lowest common denominator of a set of fractions is just the lowest common multiple of their denominators; the "LCD" name simply reflects what the number is being used for.
What if I only have one fraction?
A lowest common denominator only means something when comparing or combining two or more fractions, so at least two are needed. A single fraction already has one denominator and there is nothing to line it up against.
Do the fractions need to be simplified first?
No. The calculator works from whatever denominators are entered, whether or not the fractions are already in lowest terms. Simplifying beforehand can produce a smaller LCD, but is not required for a correct result.
Why is my rewritten fraction not in simplest form?
Rewriting a fraction over the LCD is a different step to simplifying it. The rewritten fractions are deliberately scaled up so every fraction shares the same denominator; simplifying them again afterwards would undo that and defeat the purpose of finding a common base.
Once every fraction shares a denominator, adding or subtracting them is straightforward with the fraction arithmetic calculator. For a plain-number lowest common multiple without the fraction context, see the GCD and LCM calculator.