How this instrument works
Adding three fractions together, rather than just two, still needs one shared denominator that every original denominator divides into evenly, and the least common denominator is specifically the smallest such value. It is reached by folding the numbers together two at a time: take the least common multiple of the first pair, then take the least common multiple of that result together with the third value. Because any denominator shared by all three must, in particular, be shared by the first two, combining them in this order loses nothing.
This site already carries a least common denominator page built around exactly two fractions. This page extends that identical idea one step further, to three denominators at once, for the genuinely three-way case: combining a third of something with a sixth of it and a tenth of it, say, rather than combining just a pair. The underlying formula is the same lcm identity, applied twice in a row rather than once.
Two edge cases are worth knowing going in. When all three bottom numbers already match, nothing needs converting, and the least common denominator is simply that shared value repeated. When the three share no common factor among them at all, the least common denominator collapses to their straight product — the smallest shared base three mutually unrelated denominators can settle on is nothing less than multiplying all three together.
- Enter the bottom number of your first fraction into the First denominator field.
- Enter the bottom number of your second fraction into the Second denominator field.
- Enter the bottom number of your third fraction into the Third denominator field.
- Read Least common denominator: the smallest value all three fractions can share.
- Rewrite each fraction over that shared value, scaling every top number to match, then add or subtract the three top numbers directly.
Worked example — denominators of 4, 6, and 10
Three fractions carry bottom numbers 4, 6, and 10. Combine the first pair: lcm(4,6) = 12. Combine that result with the third: lcm(12,10) = 60. Sixty is the least common denominator of all three, confirmed since 60 ÷ 4 = 15, 60 ÷ 6 = 10, and 60 ÷ 10 = 6, every one of the three divisions landing on a whole number.
Bottom numbers of 2, 3, and 5 share no common factor among them, so their least common denominator is simply their product: 2 × 3 × 5 = 30. Three matching bottom numbers of 8, 8, and 8 need no conversion at all — the least common denominator of three identical values is just that value, 8, repeated.
Questions
What is the least common denominator of three fractions?
The smallest whole number that all three original denominators divide into evenly, letting every fraction be rewritten with a matching bottom number before their top numbers are added or subtracted directly. For denominators 4, 6, and 10, that shared value is 60.
How do you compute the LCD of three denominators?
Combine two of them first using the least common multiple, then combine that result with the remaining third value using the least common multiple again. lcm(4,6) = 12, then lcm(12,10) = 60 — the order the three are combined in never changes the final answer.
How is this different from this site's two-denominator LCD calculator?
That page stops at exactly two fraction bottoms. This page extends the identical idea to a third value, for the case where three fractions are being added or subtracted at once rather than only a pair.
What if the three denominators share no common factor at all?
Then the least common denominator is simply their straight product. Bottom numbers of 2, 3, and 5 have nothing in common beyond 1, so the smallest shared base they can settle on is 2 × 3 × 5 = 30, with nothing smaller working for all three.
What if all three denominators are already identical?
Then no conversion is needed at all — the least common denominator is just that shared value itself. Three fractions each already carrying a bottom number of 8 have a combined least common denominator of 8, since every one already divides into it evenly.
Does the order the three denominators are entered in change the result?
No. Combining them as lcm(lcm(a,b),c) always returns the same value as any other order, such as lcm(lcm(b,c),a), because the least common denominator of three numbers is a single well-defined value regardless of which pair is folded together first.