Least Common Denominator Calculator
Find the least common denominator (LCD) of two or more fractions, mixed numbers, whole numbers or decimals, rewrite each one over the LCD, and see the sum and the order from smallest to largest, with steps by prime factors, listing multiples or the GCF formula.
How it works
LCD = least common multiple of all the denominators · new numerator = old numerator × (LCD ÷ old denominator) · LCM(a, b) = a × b ÷ GCF(a, b)
To add, subtract or compare fractions they must share the same denominator, and the smallest one that works is the least common multiple of all the denominators — the least common denominator. You can find it three ways, and the steps show whichever you pick with your numbers. With prime factors, factor each denominator and multiply every prime at the highest power it appears. With listing multiples, write out multiples of each denominator until one number shows up in every list. With the GCF formula, combine the denominators two at a time using LCM(a, b) = a × b ÷ GCF(a, b). Once the LCD is known, each fraction is rewritten by multiplying its top and bottom by LCD ÷ its denominator, which changes how it looks but not its value. You can type mixed numbers (1 1/2), whole numbers and decimals as well as fractions, or switch to “Denominators only” and list just the bottom numbers. The calculator also adds the rewritten fractions and puts them in order from smallest to largest.
Worked example
For 1/4, 1/6 and 2/3, the denominators are 4, 6 and 3. In prime factors, 4 = 2², 6 = 2 × 3 and 3 = 3, so the LCD is 2² × 3 = 12. Rewriting: 1/4 = 3/12, 1/6 = 2/12 and 2/3 = 8/12. Adding the numerators gives 13/12 = 1 1/12, and in order the fractions are 1/6 < 1/4 < 2/3. With a mixed number: 1 1/2, 3/8, 5/6 and 3 become 3/2, 3/8, 5/6 and 3/1, the LCD of 2, 8, 6 and 1 is 24, and they rewrite as 36/24, 9/24, 20/24 and 72/24.
Sources: Lowest common denominator — Wikipedia · Least common multiple — Wikipedia
Frequently asked questions
What’s the difference between the LCD and the LCM?
They’re the same number, viewed differently: the least common denominator of a set of fractions is the least common multiple of their denominators. Switch to “Denominators only” to get the LCM of any list of whole numbers.
Do I have to use the least common denominator?
No — any common denominator works, and multiplying all the denominators together always gives one. The LCD just keeps the numbers as small as possible, so there’s less to simplify at the end.
Why convert to a common denominator at all?
You can only add or compare pieces of the same size. Quarters and sixths are different sizes, but both become twelfths, and twelfths combine directly: 3/12 + 2/12 = 5/12.
Can I enter mixed numbers and decimals?
Yes. Separate the entries with commas and type a mixed number with a space, like 2 3/4. It is changed to the improper fraction 11/4 first. A decimal such as 0.75 becomes 3/4. A whole number like 3 counts as 3/1.
Should I simplify the fractions first?
Usually, yes. For 2/4 and 1/6 the denominators 4 and 6 give an LCD of 12, but 2/4 = 1/2, and 2 and 6 give 6. Switch on “Simplify each fraction first” under More options to do this automatically.
Which method should I use to find the LCD?
Listing multiples is quickest for small denominators like 4 and 6. Prime factors work well for larger numbers like 36 and 48: 2⁴ × 3² = 144. The GCF formula is what most calculators and programs use, and it handles long lists well.
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This calculator is for educational purposes. Double-check important results before acting on them.