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.

Description

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:^[1]

$\frac{a}{bc}+\frac{c}{b^2 d}\;=\;\frac{abd}{b^2 cd}+\frac{c^2}{b^2 cd}\;=\;\frac{abd+c^2}{b^2 cd}$

Some methods of calculating the LCD are at Least common multiple#Computing the least common multiple.

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:

$\frac{2}{3}=\frac{6}{9}=\frac{12}{18}=\frac{144}{216}=\frac{200,000}{300,000}.$

because they are all multiplied by 1 written as a fraction:

$\frac{2}{3}=\frac{2}{3}\times\frac{2}{2}=\frac{2}{3}\times\frac{6}{6}=\frac{2}{3}\times\frac{72}{72}=\frac{2}{3}\times\frac{100,000}{100,000}.$

It's usually easiest to add, subtract, or compare fractions when each is expressed with the same denominator, called a "common denominator". For example, the numerators of fractions with common denominators can simply be added, such 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. Without computing a common denominator, it is not obvious as to what $\frac{5}{12}+\frac{11}{18}$ equals, or whether $\frac{5}{12}$ is greater than or less than $\frac{11}{18}$. 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.^[2]

See also

• Greatest common divisor
• Partial fraction expansion – reverses the process of adding fractions into uncommon denominators.
• Anomalous cancellation

References

1. Brooks, Edward (1901). The Normal Elementary Algebra, Part 1. C. Sower Company. p. 80. Retrieved 7 Jan 2014.
2. "Fractions". The World Book: Organized Knowledge in Story and Picture, Volume 3. Hanson-Roach-Fowler Company. 1918. pp. 2285–2286. Retrieved 7 Jan 2014.