# Lowest common denominator

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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 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.

## Middle school instruction

Some K–12 math standards, such as the latest revision of the NCTM math standards and reform mathematics textbooks created since the 1990s, de-emphasize or omit coverage of the LCD entirely in favor of finding any common, but not necessarily the lowest common denominator, or by using less powerful methods such as fraction strips or "benchmark" fractions. The "cross-multiply" method of comparing fractions effectively creates a common denominator by multiplying both denominators together.