Least Common Denominator Calculator: Fractions and Steps
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A least common denominator calculator finds the smallest number that every denominator divides into, which is the least common multiple of the denominators. For 1/4, 5/6 and 3/8 the LCD is 24, because 24 is the first number that 4, 6 and 8 all divide. Rewritten, the fractions are 6/24, 20/24 and 9/24.
The least common denominator of 1/4, 5/6, 3/8 is 24, the least common multiple of 4, 6, 8. Rewritten: 6/24, 20/24, 9/24.
Each fraction with denominator 24
| Number | prime 2 | prime 3 |
|---|---|---|
| 4 = 2² | 2² | − |
| 6 = 2 × 3 | 2 | 3 |
| 8 = 2³ | 2³ | − |
| LCD = 2³ × 3 | 2³ | 3 |
Show steps
- All the numbers are already fractions: 1/4, 5/6, 3/8.
- The LCD is the least common multiple of the denominators: 4, 6, 8. The numerators do not affect it.
- Method 1, prime factorization: 4 = 2², 6 = 2 × 3, 8 = 2³. Take each prime with its highest power: LCD = 2³ × 3 = 24.
- Method 2, listing multiples: count by the largest denominator, 8: 8, 16, 24. The first one that every denominator divides evenly is 24.
- Rewrite each fraction by multiplying top and bottom by LCD ÷ denominator: 1/4 = (1 × 6)/(4 × 6) = 6/24; 5/6 = (5 × 4)/(6 × 4) = 20/24; 3/8 = (3 × 3)/(8 × 3) = 9/24.
- With equal denominators, compare the numerators 6, 20, 9: 1/4 < 3/8 < 5/6.
- To add them, add the numerators: 6 + 20 + 9 = 35, so the sum is 35/24 = 1 11/24.
Any common multiple works as a common denominator (for example the product of the denominators), but the least one keeps the numbers small and often saves a simplifying step.
How to use the least common denominator calculator
- Type 2 to 10 numbers separated by commas, such as 1/4, 5/6, 3/8. Use the Denominators only tab if you just have the bottom numbers.
- Choose whether to use the fractions as entered or simplify them first, and the order you want them listed in.
- Read the LCD, each fraction rewritten over the LCD, the fractions in order and their sum.
- Check the prime factor table, the multiples chart and the number line, and open Show steps for both methods worked out.
What can you enter?
- Fractions: 3/4, 11/6, −5/8. A negative denominator such as 3/−4 is read as −3/4.
- Mixed numbers with a space: 1 1/2 becomes 3/2 and −2 3/4 becomes −11/4.
- Whole numbers: 2 is read as 2/1.
- Decimals: 0.25 is changed to 1/4 and 0.4 to 2/5.
A denominator of 0 is not allowed, and letters or algebraic expressions are not accepted. Numbers can have up to 9 digits each.
How do you find the least common denominator?
The least common denominator is the least common multiple (LCM) of the denominators. Only the denominators matter. Once you have the LCD, multiply the top and bottom of each fraction by the same number, LCD ÷ denominator, so its value does not change.
Method 1: list multiples
Count by the largest denominator until you reach a number the others divide. For 4, 6 and 8: 8, 16, 24. Since 4 and 6 both divide 24, the LCD is 24. This is quick for small denominators but slow for numbers like 18 and 35, whose LCD is 630.
Method 2: prime factorization
Factor each denominator into primes and take every prime with its highest power. For 4 = 2², 6 = 2 × 3 and 8 = 2³, the LCD is 2³ × 3 = 24. This method works for any size of number and any count of fractions.
Shortcut for two numbers: use the GCF
LCM(a, b) = a × b ÷ GCF(a, b). For 6 and 9, the GCF is 3, so the LCD is 6 × 9 ÷ 3 = 18. When the denominators share no factor, as with 7 and 11, the LCD is just their product, 77.
Worked LCD examples
1/4 and 1/6: different denominators
Multiples of 6 are 6, 12, … and 4 divides 12, so the LCD is 12. Then 1/4 = 3/12 and 1/6 = 2/12, and 3/12 + 2/12 = 5/12. Multiplying 4 × 6 = 24 also gives a common denominator, but not the least one.
3/8 and 1/4: one denominator divides the other
4 divides 8, so the LCD is 8, not 32. Then 1/4 = 2/8, and 3/8 − 2/8 = 1/8.
−1/3 and 2: a negative fraction and a whole number
Write 2 as 2/1. The LCD of 3 and 1 is 3, so the fractions become −1/3 and 6/3. Their sum is 5/3, or 1 2/3.
1 1/2, 2 2/3, 3/4 and 5/12: mixed numbers
Convert first: 1 1/2 = 3/2 and 2 2/3 = 8/3. The denominators 2, 3, 4 and 12 have LCD 12, since 12 = 2² × 3. The fractions become 18/12, 32/12, 9/12 and 5/12, so in order 5/12 < 3/4 < 1 1/2 < 2 2/3.
Ordering fractions with the LCD
Once every fraction has the same denominator, the one with the larger numerator is larger. For 7/10, 11/15, 13/18 and 5/12 the LCD is 180, which gives 126/180, 132/180, 130/180 and 75/180. From greatest to least: 11/15 > 13/18 > 7/10 > 5/12.
Common LCDs to know
| Denominators | LCD | Why |
|---|---|---|
| 2 and 3 | 6 | no common factor, so multiply |
| 4 and 6 | 12 | 2² × 3 |
| 6 and 8 | 24 | 2³ × 3 |
| 6 and 9 | 18 | 2 × 3² |
| 8 and 12 | 24 | 2³ × 3 |
| 10 and 15 | 30 | 2 × 3 × 5 |
| 12 and 18 | 36 | 2² × 3² |
| 2, 3, 4, 5, 6 | 60 | 2² × 3 × 5 |
| 2 through 10 | 2,520 | 2³ × 3² × 5 × 7 |
Common mistakes with the least common denominator
- Multiplying the denominators every time. That gives a common denominator, but often not the least one: for 1/4 and 1/6 it gives 24 instead of 12.
- Changing only the denominator. 1/4 is not 1/12. Multiply the numerator by the same factor: 1/4 = 3/12.
- Taking the lowest power in prime factorization. That gives the GCF, not the LCD. Use the highest power of each prime.
- Adding the denominators. 1/4 + 1/6 is not 2/10. Rewrite with the LCD, then add only the numerators.
Least Common Denominator practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Multiples of 8: 8, 16, 24, and 6 divides 24. 48 is a common denominator but not the least; 2 is the GCF.
Only the denominators 3 and 4 matter (1 1/3 = 4/3 and 2 3/4 = 11/4), and LCM(3, 4) = 12.
With LCD 24: 15/24, 14/24 and 16/24. The largest numerator is 16, so 2/3 is greatest.
LCD 30: 3/10 = 9/30 and 4/15 = 8/30, so the sum is 17/30. Adding tops and bottoms (7/25) is a common mistake.
Take each prime with its highest power: 2² × 3² = 36. Taking the lowest powers (2 × 3 = 6) gives the GCF instead.
Quiz complete
Frequently asked questions
What is the difference between the LCD and the LCM?
They are the same number used in different places. The LCM is the least common multiple of any whole numbers; the LCD is the LCM of the denominators of some fractions. The LCD of 1/4 and 1/6 is LCM(4, 6) = 12.
Do the numerators change the LCD?
No. Only the denominators are used. But simplifying a fraction first can change its denominator, and so the LCD: 2/4 and 1/6 have LCD 12 as written, but after simplifying 2/4 to 1/2 the LCD is 6. The calculator lets you choose either way.
What is the LCD of 2/3 and 7/4?
3 and 4 share no factor, so the LCD is 3 × 4 = 12. The fractions become 8/12 and 21/12.
How do you find the LCD of three or more fractions?
Use prime factorization: factor every denominator, then multiply the highest power of each prime. For 4, 6 and 8, that is 2³ × 3 = 24. You can also find the LCM of two denominators, then the LCM of that result with the next one.
Can I use this for algebraic denominators?
No. This calculator accepts numbers only. Algebraic denominators such as x² − 1 and x + 1 need factoring first, with the same rule (take each factor with its highest power) and a note of the values that make a denominator zero.
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