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Lowest common denominator
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions.
Role in arithmetic and algebra
The same fraction can be expressed in many different forms. As long as the ratio between numerator and denominator is the same, the fractions represent the same number. For example:
\( \frac{2}{3}=\frac{6}{9}=\frac{12}{18}=\frac{144}{216}=\frac{200,000}{300,000}. \)
It's usually easiest to add, subtract, or compare fractions when each is expressed with the same denominator, called a "common denominator". For example, it's obvious that \(\frac{5}{12}+\frac{6}{12}=\frac{11}{12} \) and that \(\frac{5}{12}<\frac{11}{12} \), since each fraction has the common denominator 12. But it's not obvious what \(\frac{5}{12}+\frac{11}{18} \) equals, or whether \( \frac{5}{12} \) is greater than or less than \frac{11}{18}, because the denominators are different. Any common denominator will do, but usually the least common denominator is desirable because it makes the rest of the calculation as simple as possible.[1]
The least common denominator of a set of fractions is the least number that is a multiple of all the denominators: their "least common multiple". The product of the denominators is always a common denominator, as in:
\( \frac{1}{2}+\frac{2}{3}\;=\;\frac{3}{6}+\frac{4}{6}\;=\;\frac{7}{6} \)
but it's not always the least common denominator, as in:
\( \frac{5}{12}+\frac{11}{18}\;=\;\frac{15}{36}+\frac{22}{36}\;=\;\frac{37}{36} \)
Here, 36 is the least common multiple of 12 and 18. Their product, 216, is also a common denominator, but calculating with that denominator involves larger numbers: \(\frac{5}{12}+\frac{11}{18}=\frac{90}{216}+\frac{132}{216}=\frac{222}{216}. \)
With variables rather than numbers, the same principles apply:[2]
\(\frac{a}{bc}+\frac{c}{b^2 d}\;=\;\frac{abd}{b^2 cd}+\frac{c^2}{b^2 cd}\;=\;\frac{abd+c^2}{b^2 cd} \)
Some methods of calculating the LCD are at Least common multiple#Computing the least common multiple.
See also
Greatest common divisor
Partial fraction expansion – reverses the process of adding fractions into uncommon denominators.
Anomalous cancellation
References
"Fractions". The World Book: Organized Knowledge in Story and Picture, Volume 3. Hanson-Roach-Fowler Company. 1918. pp. 2285–2286. Retrieved 7 Jan 2014.
Brooks, Edward (1901). The Normal Elementary Algebra, Part 1. C. Sower Company. p. 80. Retrieved 7 Jan 2014.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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