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Lowest common denominator

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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

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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

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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

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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

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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

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References

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