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LCD Calculator

Result

60

Least common denominator

Fractions with the common denominator
45/60, 50/60, 42/60

A lowest common denominator is the smallest number that every denominator in a set of fractions divides into evenly. For 3/4, 5/6 and 7/10 it is 60: 60 is 15 fours, 10 sixes and 6 tens, and no smaller number works for all three. Written out, the three fractions become 45/60, 50/60 and 42/60 — same values, one shared denominator, which is the form you need before they can be added, subtracted or compared. The abbreviation LCD is common and is used throughout this page. Two steps get there, and the first one is easy to skip. Every fraction is reduced to simplest form first, because 6/8 and 3/4 are the same number with different denominators, and reading the denominator without reducing gives an answer that is a common denominator but not the lowest one — reduce 6/8 and 10/12 to 3/4 and 5/6, and the answer is 12 rather than 24. Then the denominators are combined: the lowest common denominator of a set of fractions is the least common multiple of their reduced denominators. That is the whole computation, and it is why this page and the least common multiple page arrive at the same number from different directions. With three or more fractions the multiples are folded in two at a time, so 4, 6 and 10 give 12 first and then 60. Reading it off a table is the method to fall back on: list the multiples of each reduced denominator and take the first number they all share. Two special cases come up often enough to know. Whole numbers are accepted and count as fractions with denominator 1, so they never raise the answer — but they are printed back with the common denominator attached, since 2 is 4/2 once everything is over 2. And a denominator that is a multiple of another, 4 among 2 and 4, adds nothing: 2 already goes into 4. The result is exactly the quantity needed for adding fractions with different denominators, and the denominator of the sum is this number.

1/2, 2/3 and 3/4 rewritten over their lowest common denominator

fractiondenominatorwith the common denominator
1/226/12
2/338/12
3/449/12

Read the middle column downwards and the three numbers are 2, 3 and 4. The least common multiple of those is 12, which is the answer, and it is the first number all three divide into: 12 is the sixth multiple of 2, the fourth of 3 and the third of 4. The right-hand column then shows each fraction rewritten over 12, and each entry is worth checking against the one it replaced — 6/12 is a half, 8/12 is two thirds and 9/12 is three quarters, and every one of them still is, because the numerator and the denominator were multiplied by the same number. That multiplication factor is the answer divided by the fraction's own denominator: 12 ÷ 2 = 6 for the first row, 12 ÷ 3 = 4 for the second, 12 ÷ 4 = 3 for the third. Notice that the third denominator, 4, is not the largest contributor even though it is the largest number: what matters is the primes, and 12 needs two 2s from either 2 or 4, plus the 3 that only 3 brings. The table is fixed at these three fractions and does not follow what you typed — the panel above answers your input, this shows the method with the rows short enough to check by eye.

Formula

3/4, 5/6, 7/10 ⇒ lcd = lcm(4, 6, 10) = 60 ⇒ 45/60, 50/60, 42/60

3/4, 5/6, 7/10
The fractions to work with, two to ten of them, each written as numerator/denominator with both parts whole numbers from 1 to 1000000, separated by spaces, commas or semicolons. Whole numbers such as 2 are accepted and read as 2/1. Each fraction is reduced to simplest form before its denominator is used, which is the step that keeps the answer from coming out too large
4, 6, 10
The reduced denominators. Reducing first is what makes these 4, 6 and 10 rather than something larger — 6/8 would contribute 8 if it were not reduced to 3/4. Everything after this point works on these numbers only; the numerators do not affect the answer at all
lcd = lcm(4, 6, 10)
The whole computation in one line: the lowest common denominator is the least common multiple of the reduced denominators. 4 is 2², 6 is 2 × 3 and 10 is 2 × 5, so the primes needed are 2², 3 and 5, giving 4 × 3 × 5 = 60. With more than two fractions this folds two at a time: 4 and 6 give 12, then 12 and 10 give 60
60
The answer. Every reduced denominator divides it evenly, and nothing smaller does — which is what makes it the lowest one and not merely a shared one
n × (60 ÷ d)
How each fraction is rewritten: multiply the numerator by the answer divided by that fraction's own denominator. For 3/4 that is 3 × 15 = 45, so 3/4 becomes 45/60. The value of the fraction does not change — numerator and denominator are multiplied by the same number, which is what equivalent fractions means
45/60, 50/60, 42/60
The rewritten fractions, all over the common denominator. A whole number in the input appears here with the denominator attached rather than reduced back: 2 comes back as 4/2 when the answer is 2, because a row printed as a bare whole number in this column would read as one that was skipped

Adding fractions with different denominators starts here: 3/4 and 5/6 cannot be combined until both are over 12, and the common denominator is the first line of the working. Comparing fractions is the same step — you cannot tell whether 5/6 is larger than 7/10 by looking, but 50/60 against 42/60 settles it immediately, and the denominators agreeing means only the numerators need to be read. Subtracting has the same requirement. Outside the arithmetic classroom, any time quantities are recorded in different fractional units the shared denominator is what lets them be totalled. The table below is the by-hand method: multiples of each reduced denominator, and the first number appearing in every column. Two shortcuts save most of the work, and both come up often: a denominator that is a multiple of another one can be ignored, and denominators that share no prime at all — 6 and 5, say — multiply to give the answer directly. When a fraction is not in simplest form the reduction is not optional, and it is worth doing first rather than last, since it is the one step whose omission produces an answer that looks entirely reasonable.

Worked examples

  1. Three fractions: 3/4, 5/6 and 7/10

    1. All three are already in simplest form, so the denominators to combine are 4, 6 and 10
    2. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
    3. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
    4. Multiples of 10: 10, 20, 30, 40, 50, 60
    5. The first number in all three lists is 60, so the lowest common denominator is 60
    6. 3/4 becomes 45/60 because 60 ÷ 4 = 15 and 3 × 15 = 45
    7. 5/6 becomes 50/60 because 60 ÷ 6 = 10 and 5 × 10 = 50
    8. 7/10 becomes 42/60 because 60 ÷ 10 = 6 and 7 × 6 = 42

    The default. The folded route is shorter than the lists: 4 and 6 give 12, and 12 with 10 gives 60. The rewritten column is the useful check, because each of the three new fractions must equal the one it replaced — 45/60 is 3/4, 50/60 is 5/6 and 42/60 is 7/10, and if any of them were not, the multiplier was applied to only one part of the fraction.

  2. Why the fractions are reduced first: 6/8 and 10/12

    1. 6/8 and 10/12 are not in simplest form
    2. Divide both parts of 6/8 by 2: it becomes 3/4, denominator 4
    3. Divide both parts of 10/12 by 2: it becomes 5/6, denominator 6
    4. The least common multiple of 4 and 6 is 12, so the lowest common denominator is 12
    5. 3/4 becomes 9/12 and 5/6 becomes 10/12
    6. Reading the unreduced denominators instead would have given 8 and 12, and their least common multiple is 24

    The answer that this page exists to get right. 24 is not a wrong common denominator — 6/8 and 10/12 really can both be written over 24 — it is just not the lowest one, and a panel showing 24 would look perfectly normal. Reducing first is what separates them, and it is why the field reduces every fraction before reading its denominator.

  3. A whole number among the fractions: 1/2, 2 and 3

    1. 2 and 3 are read as 2/1 and 3/1, with denominator 1
    2. The denominators to combine are 2, 1 and 1
    3. Every whole number is a multiple of 1, so 1 contributes nothing to the answer
    4. The least common multiple of 2 and 1 and 1 is 2
    5. 1/2 stays 1/2, while 2 becomes 4/2 and 3 becomes 6/2

    The 4/2 and 6/2 are deliberate and are not a failure to reduce. The page is called 'fractions with the common denominator' and every row in it has to carry that denominator, or a row printed as a bare 2 would read as one that had been left alone. The value is unchanged either way — 4/2 is 2 — and this is the only place on the page where an output looks unreduced on purpose.

  4. Near the ceiling: 1/1000 and 2/999983

    1. 1000 is 2³ × 5³ and 999983 is prime, so the two denominators share nothing
    2. With nothing shared the least common multiple is the product: 1000 × 999983 = 999983000
    3. 1/1000 becomes 999983/999983000 because 999983000 ÷ 1000 = 999983
    4. 2/999983 becomes 2000/999983000 because 999983000 ÷ 999983 = 1000 and 2 × 1000 = 2000

    Two denominators that share no prime is the case where the answer is as large as it can get for a given pair, and this example sits just under the ceiling where the page stops. The numerators stay exact here — 999983 is a six-digit value and still lands inside the range a machine holds precisely — but a fraction with a larger numerator would push the product past it, which is why the page refuses beyond a certain denominator rather than printing a rounded rewrite.

Limitations

Each fraction is written numerator/denominator with both parts whole numbers from 1 to 1000000, and there must be between two and ten of them. Decimals are refused rather than converted: 0.75 is not a fraction written as a fraction, and turning it into 3/4 would be guessing at what the writer meant. A negative sign is accepted only in front of the numerator — 3/-4 is refused, because the sign of a fraction belongs in one place and the page will not choose for you. A denominator of zero is refused, since nothing can be divided into zero parts. There is one ceiling and it is not on the input but on the answer: the lowest common denominator must stay at or below 1000000000, and past that the page refuses rather than printing the rewritten fractions. The reason is that each rewritten numerator is the original numerator multiplied by the answer divided by its own denominator, so a large common denominator times a large numerator stops being exact — and an inexact rewrite would look like any other row. Two fractions are the minimum, because a single fraction has nothing to share a denominator with. A fraction repeated in the list changes nothing, and a denominator that is a multiple of another is simply redundant. The reference table below is fixed at 1/2, 2/3 and 3/4 and does not follow your input; the panel above answers your fractions, the table shows the method and the three rewritten rows. The answer is exact, never rounded.

Frequently asked questions

What is the difference between a common denominator and the lowest common denominator?
Any number that every denominator divides into is a common denominator; the lowest one is the smallest such number. For 4, 6 and 10 both 60 and 120 are common denominators, and 60 is the lowest. Only the lowest is worth computing by hand, because a larger one means larger numbers in every rewritten fraction, and larger numbers are where arithmetic mistakes happen. The page always returns the lowest.
Why are the fractions reduced before the denominator is read?
Because two fractions that are the same number can have different denominators. 6/8 and 3/4 are equal, but building a common denominator from 8 gives 24 where the correct answer is 12. The wrong answer is not obviously wrong — 24 is a genuine common denominator of those fractions — so the page reduces every fraction to simplest form first, and the example above pins that case.
How is the lowest common denominator related to the least common multiple?
They are the same computation on different inputs. Take the denominators of the reduced fractions and find their least common multiple; that is the lowest common denominator. It is why the two pages here agree: 4, 6 and 10 have a least common multiple of 60, and a set of fractions with those denominators has 60 as its lowest common denominator. Nothing about the numerators enters the answer.
Why does a whole number come back with a denominator attached?
Because a whole number in the input is read as a fraction over 1, and this page's output column is 'fractions with the common denominator' — every row in it has to carry that denominator. 2 becomes 4/2 when the answer is 2, which is the same value written over the shared denominator rather than a mistake in reducing. It is the only case on the page where an output looks unreduced on purpose.
What is the smallest possible answer?
The largest reduced denominator, when every other denominator divides into it. For 1/2, 1/4 and 3/4 the answer is 4, since 2 divides 4 and 4 is itself. It cannot be smaller than the largest denominator, because the answer has to be a multiple of every one of them, and a multiple of a number is never less than that number.
Can the numerators change the answer?
No. The lowest common denominator is computed from the denominators alone, and 1/4 and 3/4 have the same one. The numerators do appear in the rewritten fractions, where each is multiplied by the answer divided by its own denominator, so they change what the rows say but never how many parts each whole is cut into.

References

Related calculators

LCM CalculatorThe first four common multiples of two to ten numbers.Adding Fractions CalculatorAdd fractions with unlike denominators, simplified.Subtracting Fractions CalculatorSubtract unlike fractions and show the common denominator.Equivalent Fractions CalculatorFractions equal to a given one, up to a limit.