# Lowest common denominator

In mathematics, the **lowest common denominator** or **least common denominator** (abbreviated **LCD**) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions.

## Description[edit]

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:

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

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:

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

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

## Role in arithmetic and algebra[edit]

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:

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

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 and that , since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what equals, or whether is greater than or less than . 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]}

## Practical uses[edit]

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.

## Colloquial usage[edit]

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]}

## See also[edit]

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

## References[edit]

**^**Brooks, Edward (1901).*The Normal Elementary Algebra, Part 1*. C. Sower Company. p. 80. Retrieved 7 January 2014.**^**"Fractions".*The World Book: Organized Knowledge in Story and Picture, Volume 3*. Hanson-Roach-Fowler Company. 1918. pp. 2285–2286. Retrieved 7 January 2014.**^**"lowest common denominator",*Collins English Dictionary*(accessed February 21, 2018)