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Lowest common denominator

Lowest common multiple of the denominators of a set of fractions

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Inmathematics, the lowest common denominatororleast common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions.

The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator, as in:

${\displaystyle {\frac {1}{2}}+{\frac {2}{3}}\;=\;{\frac {3}{6}}+{\frac {4}{6}}\;=\;{\frac {7}{6}}}$

but it is not always the lowest common denominator, as in:

${\displaystyle {\frac {5}{12}}+{\frac {11}{18}}\;=\;{\frac {15}{36}}+{\frac {22}{36}}\;=\;{\frac {37}{36}}}$

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:

${\displaystyle {\frac {5}{12}}+{\frac {11}{18}}={\frac {90}{216}}+{\frac {132}{216}}={\frac {222}{216}}.}$

With variables rather than numbers, the same principles apply:^[1]

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

Some methods of calculating the LCD are at Least common multiple § Calculation.

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:

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

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

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

It is 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 ${\displaystyle {\frac {5}{12}}+{\frac {6}{12}}={\frac {11}{12}}}$ and that ${\displaystyle {\frac {5}{12}}<{\frac {11}{12}}}$, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what ${\displaystyle {\frac {5}{12}}+{\frac {11}{18}}}$ equals, or whether ${\displaystyle {\frac {5}{12}}}$ is greater than or less than ${\displaystyle {\frac {11}{18}}}$. Any common denominator will do, but usually the lowest common denominator is desirable because it makes the rest of the calculation as simple as possible.^[2]

The LCD has many practical uses, such as determining the number of objects of two different lengths necessary to align them in a row which starts and ends at the same place, such as in brickwork, tiling, and tessellation. It is also useful in planning work schedules with employees with y days off every x days.

In musical rhythm, the LCD is used in cross-rhythms and polymeters to determine the fewest notes necessary to count time given two or more metric divisions. For example, much African music is recorded in Western notation using ¹²
₈ because each measure is divided by 4 and by 3, the LCD of which is 12.

The expression "lowest common denominator" is used to describe (usually in a disapproving manner) a rule, proposal, opinion, or media that is deliberately simplified so as to appeal to the largest possible number of people.^[3]

• Anomalous cancellation
• Greatest common divisor
• Partial fraction decomposition, reverses the process of adding fractions into uncommon denominators