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[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's 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#Computing the least common multiple.

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]

See also[edit]

References[edit]

  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.