Least Common Denominator Calculator
This least common denominator calculator finds the smallest denominator that two to ten fractions can all share. Enter fractions separated by commas and it returns the LCD, rewrites every fraction over it, shows the multiplier used for each one, works the LCD out by prime factors, and lists the fractions from smallest to largest.
1 1/2 work.How to use the least common denominator calculator
- Type your fractions into the box, separated by commas. It opens with
1/4, 5/6, 3/8. - The least common denominator appears at once: for those three it is 24.
- Below it, each fraction is rewritten over 24, with the multiplier that got it there:
The calculator also shows the prime-factor working for the LCD and puts the fractions in order from smallest to largest: 1/4, 3/8, 5/6. Ordering is easy once the denominators match, because you only compare numerators: 6, 9, 20.
You can enter anything from two fractions to ten. Once you have the fractions over a common denominator, the fraction calculator adds, subtracts, multiplies or divides them, and how to add fractions explains each step of using the LCD by hand.
What is the least common denominator?
The least common denominator of a set of fractions is the smallest number that every one of their denominators divides into exactly. It is the least common multiple of the denominators, given a different name because of the job it does.
You need a common denominator to add, subtract or compare fractions, because those operations only make sense when the pieces are the same size. Any common multiple of the denominators would work as a common denominator. The least one keeps the numbers as small as possible, which means less arithmetic and usually less simplifying at the end. The LCM calculator finds the same number for plain whole numbers.
How to find the least common denominator
Method 1: multiples of the largest denominator
List multiples of the largest denominator and stop at the first one the others divide into. For 4, 6 and 8, the multiples of 8 are 8, 16, 24. Neither 8 nor 16 is divisible by 6; 24 is divisible by 4, 6 and 8, so the LCD is 24. This works well when the denominators are small.
Method 2: prime factors
Write each denominator as a product of primes. The LCD takes every prime that appears, at the highest power it appears anywhere.
This is the method the calculator shows, and it scales to any number of fractions. The prime factorization calculator breaks down a single denominator if you want to check one.
Method 3: product divided by the GCF (two denominators)
For 12 and 18 the GCF is 6, so the LCD is 12 × 18 ÷ 6 = 216 ÷ 6 = 36. The GCF calculator finds that greatest common factor, and how to find the GCF and LCM compares all the methods side by side.
Two shortcuts
- If one denominator divides the other, the larger one is the LCD. For 3 and 9 the LCD is 9.
- If two denominators share no factor other than 1, the LCD is their product. For 5 and 7 it is 35.
Worked examples
Example: 5/12 and 7/18
Prime factors: 12 = 2² × 3 and 18 = 2 × 3². Take 2² and 3².
Since 15/36 is bigger than 14/36, 5/12 is also the larger of the two fractions, by 1/36.
Example: three fractions, 1/6, 3/10 and 4/15
6 = 2 × 3, 10 = 2 × 5, 15 = 3 × 5. Each prime appears at most to the first power.
Multiplying the denominators together would have given 900, thirty times larger than needed.
Example: comparing 7/12 and 5/8
The LCD of 12 and 8 is 24.
The two are close; as decimals they are about 0.5833 and 0.625. A common denominator settles the comparison exactly, without rounding.
Using the LCD to put fractions in order
Ordering fractions by eye is unreliable once the denominators differ. Which is largest out of 2/3, 3/5, 7/10 and 5/8? Rewrite them over a common denominator and the question becomes a comparison of whole numbers.
This is what the calculator does when it sorts your list. Converting to decimals would also work here (0.6, 0.625, about 0.6667 and 0.7), but repeating decimals such as 2/3 need rounding, and two fractions that are very close can round to the same decimal. Over a common denominator the comparison is exact. A fraction to decimal chart is quicker for familiar fractions like these, and the LCD method works for any fractions at all.
Why use the least common denominator and not just any common one?
Multiplying all the denominators together always gives a common denominator. It is just rarely the smallest. For the default fractions the product is 4 × 6 × 8 = 192, eight times the LCD of 24.
Both routes give the same final answer, as they must. The difference is how much work it takes to get there:
With the product you multiply by larger numbers, add larger numbers and then have to simplify. With the LCD the answer usually arrives in lowest terms or close to it. The simplify fractions calculator handles the reduction when you do use a bigger denominator.
LCD reference table for common denominators
| Denominators | GCF | Product | LCD |
|---|---|---|---|
| 2 and 3 | 1 | 6 | 6 |
| 2 and 4 | 2 | 8 | 4 |
| 3 and 4 | 1 | 12 | 12 |
| 3 and 9 | 3 | 27 | 9 |
| 4 and 6 | 2 | 24 | 12 |
| 5 and 7 | 1 | 35 | 35 |
| 6 and 8 | 2 | 48 | 24 |
| 6 and 9 | 3 | 54 | 18 |
| 8 and 12 | 4 | 96 | 24 |
| 9 and 12 | 3 | 108 | 36 |
| 10 and 15 | 5 | 150 | 30 |
| 12 and 16 | 4 | 192 | 48 |
| 12 and 18 | 6 | 216 | 36 |
| 14 and 21 | 7 | 294 | 42 |
| 16 and 24 | 8 | 384 | 48 |
Read across any row and the product divided by the GCF gives the LCD. When the GCF is 1, the product and the LCD are the same number. Fractions over the same LCD are a special case of equivalent fractions: 1/4 and 6/24 are one value written two ways.
Common mistakes
- Multiplying the denominators every time. It gives a common denominator, not the least one, and leaves you simplifying large numbers later.
- Changing the denominator but not the numerator. Rewriting 1/4 as 1/24 changes its value. Whatever multiplies the bottom must multiply the top:
1/4 = 6/24. - Including the numerators. The LCD depends only on the denominators.
- Mixing up the GCF and the LCD. The GCF is the largest number that divides into both; the LCD is the smallest number both divide into. For 12 and 18 they are 6 and 36.
- Not simplifying first when working by hand. 2/4 and 1/6 have an LCD of 12, but 2/4 is 1/2, and 1/2 and 1/6 have an LCD of 6. Both lead to the same answer; the second uses smaller numbers.
- Forgetting the whole parts of mixed numbers. Only the fraction parts need the LCD when adding. The mixed number calculator converts mixed numbers if you would rather work with improper fractions.
Subtraction uses the LCD in exactly the same way as addition; the subtracting fractions calculator shows it step by step.
Frequently asked questions
What is the least common denominator of 1/4, 5/6 and 3/8?
24. The denominators factor as 4 = 2², 6 = 2 × 3 and 8 = 2³, so the LCD is 2³ × 3 = 24. Over 24 the fractions become 6/24, 20/24 and 9/24, using multipliers of 6, 4 and 3. In order from smallest to largest they are 1/4, 3/8 and 5/6.
What is the least common denominator of 1/3 and 1/4?
12. The denominators 3 and 4 share no factor other than 1, so their LCD is simply their product, 3 × 4 = 12. Over 12 the fractions become 4/12 and 3/12, which also shows at a glance that 1/3 is the larger of the two. Their sum is 7/12.
Is the least common denominator the same as the least common multiple?
Yes, applied to the denominators. The LCD of a set of fractions is the LCM of their denominators. The different name reflects the job: an LCM is a property of any set of whole numbers, while the LCD is that number put to work as the shared denominator for adding, subtracting or comparing fractions.
How do you find the least common denominator of three fractions?
Factor every denominator into primes and multiply together each prime at the highest power it appears. For 1/6, 3/10 and 4/15, the denominators are 2 × 3, 2 × 5 and 3 × 5, so the LCD is 2 × 3 × 5 = 30. The same method works for any number of fractions.
What is the LCD of 2/3 and 5/9?
9. Because 3 divides into 9 exactly, the larger denominator already works for both fractions. Only 2/3 needs rewriting: multiply top and bottom by 3 to get 6/9. Then 2/3 − 5/9 = 6/9 − 5/9 = 1/9. Whenever one denominator is a multiple of the other, the larger one is the LCD.
What is the least common denominator of 1/6 and 1/8?
24. The multiples of 8 are 8, 16 and 24, and 24 is the first one that 6 also divides. By prime factors, 6 = 2 × 3 and 8 = 2³, giving 2³ × 3 = 24. The product 6 × 8 is 48, twice as large as needed, because 6 and 8 share a factor of 2.
What is the difference between a common denominator and the least common denominator?
A common denominator is any number every denominator divides into; the least common denominator is the smallest such number. For 1/4 and 1/6, the numbers 12, 24, 36 and 48 are all common denominators, but 12 is the least. Any of them gives the right answer, and the least one involves the smallest numbers.
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